text:
\section{Introduction}
\label{S1}

Percolation theory in a lattice (e.g., the Euclidean lattice $\mathbb{Z}^d$) began with the work of Broadbent and Hammersley in 1957. The principal features of the model are that the space of sites is infinite and  its geometry plays an  essential role. The main problem is to determine if there is an infinite connected component, in which case it is said that percolation occurs. In the first models the connections (bonds) were  only between nearest neighbors.
(See \cite{G, Bo2} for background, and for a physics point of view, \cite{SA}.)
The question of percolation on noneuclidean graphs including nonamenable Cayley graphs given by finitely generated groups was formulated in \cite{BS}.
The study of long range percolation began in the mathematical physics literature (e.g., \cite{AN,  Be, NS, Sc}). In this case connections are allowed between points at any distance from each other with probability depending on the distance. The main problem is the same, and the geometry remains crucial. (See also \cite{BS,Bi,Bis,CGS,T}.)

The theory of random graphs started with the work of  Erd$^{{\rm ''}}\kern -.22cm$os and R\'enyi in 1959. The model consists of a finite number $n$ of vertices  with connection probability $p_n$ between pairs of vertices, depending on $n$ in some way, and there is no
structure on the set of vertices.  The results refer to what happens as $n\rightarrow \infty$, for example with the largest connected component. (See \cite{Bo, JLR, D} for  background.) Results from the theory of random graphs have been useful as technical tools in studies on percolation (e.g. \cite{BB, BLP}).

By introducing a structure on the set of vertices of a random graph or some special form of connection probabilities by means of a kernel which induces a sort of geometry, it is possible to generate a large class of interesting models which share properties of both percolation  and/or (classical) random graphs, and also ``small world'' random graphs (see e.g. the models in \cite{AB,Bo1,TV}).

On the other hand, hierarchical structures arise in the physical, biological and social sciences due to the multiscale organization of many natural objects (see e.g. \cite{BR,RTV}). In particular the hierarchical Ising model which was introduced by Dyson  \cite{Dy} has played an important role in statistical physics (see  \cite{BEI,CE,Si}) and in population genetics (see e.g. \cite{SF}).
Important applications of hierarchical structures have also been made by Kleinberg \cite{K, K2} in the area of search algorithms in computer science. A basic model is the {\it hierarchical group}  $\Omega_N$ of order $N$ (defined in Section 2), which can be represented as the set of leaves at the top of an infinite regular tree, where the distance between two points  is the number of levels from the top to their most recent common node. Such a distance satisfies the strong  triangle inequality
$$
d(x,y) \leq \max \{d(x,z), d(z,y)\}\quad \hbox{\rm for any}\quad x,y,z,
$$
which is the characteristic property of an ultrametric. (See e.g. \cite{Sch} for background on ultrametric spaces.) The main qualitative difference between  Euclidean-type lattices and an ultrametric space such as $\Omega_N$ is that in the former case it is possible to go far by a sequence of small steps, while in the latter case that is not possible, and the only way to go far is to make jumps of ever bigger sizes. This has important consequences for random walks for which there are analogies and differences with the Euclidean case (see e.g. \cite{DGW, DGW1} and references therein) and percolation in ultrametric spaces \cite{DG1}. In particular, percolation in $\Omega_N$ is possible only in the form of long range percolation, that is, with positive probabilities of connections between vertices separated by arbitrarily large distances.

With these precedents it is natural to investigate percolation in ultrametric spaces such as $\Omega_N$ where classical tools do not apply. Our aim is to develop  a mathematical framework that might be useful generally for this kind of model, not thinking about specific motivations from or applications to physics or any other field.

In \cite{DG1} we studied {\it asymptotic percolation} in $\Omega_N$ as $N\rightarrow \infty$
(or {\it mean field percolation}) with connection probabilities of the form $c_k/N^{2k-1}$ between two points separated by distance $ k$, and we obtained a necessary and sufficient condition for percolation. (See Subsection 3.1 for the definition of asymptotic percolation). The
Erd$^{{\rm ''}}\kern -.22cm$os-R\'enyi results on giant components of random graphs were a useful tool, although there are significant differences between classical random graphs and ultrametric ones (e.g., the average length of paths in the giant component of an ultrametric ball is much longer than in the classical case).

In the present paper we study percolation in $\Omega_N$ for fixed $N$ with connection probabilities of the form $c_k/N^{k(1+\delta)}$, $\delta >-1$, between two points separated by distance $k$. In this case percolation means that there is a positive probability that a given point of $\Omega_N$ belongs to an infinite connected component. This is a quite different situation from the asymptotic $N\rightarrow \infty$ model, and new methods must be used. However, properties of  Erd$^{{\rm ''}}\kern -.22cm$os-R\'enyi graphs are again useful, but now it is connectivity results that are of help,  specially the result in the Appendix based on Durrett's approach to connectivity \cite{D}. There are three regimes: $\delta <1$, $ \delta >1$ and $\delta =1$. Roughly speaking, under certain natural assumptions on $c_k$, for $\delta >1$ percolation does not occur, and for $\delta <1$ percolation occurs and the infinite connected component is unique. The most difficult is the critical case, $\delta =1$, where we use certain special forms of $c_k$ and percolation may or may not occur. Our aim is to find forms of $c_k$ such that percolation occurs, or that it does not occur, and in the case of percolation it turns out that the infinite connected component is unique.

We emphasize that taking the limit $N\to\infty$ provides a simplification which allowed us to obtain a sharp result, that is, necessary and sufficient condition for percolation in \cite{DG1}. In the case of finite $N$  that does not seem possible at present, and our work leads to some open problems, in particular regarding the behaviour at the  critical values of the parameters in the form of $c_k$ in the critical case $\delta=1$.  We also consider an intermediate situation  that we call {\it pre-percolation}, which is necessary  for percolation. Pre-percolation occurs in one of our results.

While we were working on this paper we learned about the manuscript of Koval et al \cite{KMT} (for which we thank them), where they also study percolation in $\Omega_N$ for fixed $N$, with connection probabilities of the form $1-\exp \{-\alpha /\beta^k\}$, $\alpha \geq 0, \beta>0$, between two points  separated by distance $k$. Some of their results for $\beta >1$ and  ours may be compared by setting $\beta =N^{1+\delta}$ (see Remark \ref{R3.2}).

For the cases where percolation occurs, specially with $\delta =1$, we use an approach in the spirit of the renormalization  group method of statistical physics  which has been employed, for example, in \cite{CE} for ferromagnetic systems on Dyson's hierarchical lattice $\Omega_2$   and for the study of long range percolation on $\mathbb{Z}^d$ (see  \cite{NS,Sc}).

In Section 2 we describe the model (the hierarchical group $\Omega_N$ and the associated random graph ${\mathcal G}_N$). In Section 3 we recall the result on mean field percolation  \cite{DG1} in order to compare it with a result in the present paper, and we state our   results for $\delta <1$ and $\delta >1$ (Theorem 3.1) and the critical case $\delta =1$ (Theorems 3.3 and 3.5), and we mention some open problems (Subsection 3.4).  Sections 4 and 5 contain the proofs. In an appendix we give a result  on connectivity of random graphs derived from \cite{D}, which is a key ingredient for the proof of percolation  in the critical case.

\vglue .6cm
\noindent
\section{Description of the model}
\setcounter{equation}{0}
\subsection{The hierarchical group $\Omega_N$}
\label{sub:2.1}

 For an integer $N\geq 2$, the {\it  hierarchical group of order $N$}, also called {\it hierarchical lattice of order $N$}, is defined as
\begin{eqnarray*}
\Omega_N &=&\{\mbox{\boldmath $x$} =(x_1, x_2,\ldots): x_i \in \{0,1,\ldots, N-1\}, i=1,2,\ldots, \sum_i x_i<\infty\}
\end{eqnarray*}
with addition componentwise mod $N$; in other words, $\Omega_N$ is a countable Abelian group given by the direct sum of a countable number of copies of the cyclic group of order $N$. The {\it hierarchical distance} on $\Omega_N$ is defined as
$$
d(\mbox{\boldmath $x$}, \mbox{\boldmath $y$}) =\left\{
\begin{array}{lcl}
0 & {\rm if} & \mbox{\boldmath $x$}=\mbox{\boldmath $y$},\\
\max \{i: x_i \neq y_i\} & {\rm if} & \mbox{\boldmath $x$} \neq \mbox{\boldmath $y$}.
\end{array}\right.
$$
It is a translation-invariant metric which satisfies the strong (non-Archimedean) triangle inequality
$$
d(\mbox{\boldmath $x$}, \mbox{\boldmath $y$})\leq \max \{d (\mbox{\boldmath $x$}, \mbox{\boldmath $z$}), d (\mbox{\boldmath $z$}, \mbox{\boldmath $y$})\}\quad \hbox{\rm for any}\quad \mbox{\boldmath $x$}, \mbox{\boldmath $y$}, \mbox{\boldmath $z$}.
$$
Hence $(\Omega_{N}, d)$ is an ultrametric space, and it is well known that it can be represented as the leaves at the top of an infinite regular tree where $N$ branches emerge up from each node, and the distance between two points of $\Omega_N$ is the number of levels from the top to their most recent common node.

For each integer $k \geq 1$, a {\it $k$-ball} in $\Omega_N$, denoted  by $B_k$, is  a set of all points which are at distance at most $k$ from each other.
Any point of a ball can serve as a center. Once a center is chosen, one may speak of the interior and the boundary of the ball.
A $k$-ball contains $N^k$ points, and its boundary  contains $N^{k-1}(N-1)$ points. For $k>1$, a $k$-ball is the union of $N\;(k-1)$-balls, which are at distance $k$ from each other. For $j>k\geq 1$, a $j$-ball is the union of $N^{j-k}$ $k$-balls, which are at distance at least $k+1$ and at most $j$ from each other.
Two balls are either disjoint, or one is contained in the other (this is the reason why connections between nearest neighbours alone cannot produce percolation). For $k\leq j <\ell$, the ($j, \ell]$-{\it annulus} around $B_k$ is the set of all points $\mbox{\boldmath $y$}$ such that $j<d(\mbox{\boldmath $x$}, \mbox{\boldmath $y$})\leq \ell$, where $\mbox{\boldmath $x$}$ is any point in $B_k$. The $(j, \ell]$-annulus is also described as $B_\ell \setminus B_j$ with $B_k \subset B_j \subset B_\ell$, and it contains $N^\ell(1-N^{j-\ell})$ points.  The number of points in a bounded subset $A\subset \Omega_N$ is denoted by $|A|$. The probability that a point chosen at random (uniformly) in $B_k$ belongs to $A\subset B_k$ is $|A|N^{-k}$.

We fix a point of $\Omega_N$ which we denote as $\bold{0}$.  Most of our considerations about percolation will refer to balls containing $\bold{0}$.

\subsection{The random graph ${\mathcal G}_N$}
\label{sub:2.2}
We define an infinite random graph ${\mathcal G}_N$ with the points of $\Omega_N$ as vertices, and for each $k\geq 1$ the probability of connection, $p_{\mbox{\boldmath $x$},\mbox{\boldmath $y$}}$, between $\mbox{\boldmath $x$}$ and $\mbox{\boldmath $y$}$ with  $d(\mbox{\boldmath $x$}, \mbox{\boldmath $y$})=k$ is given by
\begin{equation}
\label{eq:2.1}
p_{(k)}= p_{\mbox{\boldmath $x$},\mbox{\boldmath $y$}}=\min\left(\frac{c_k}{N^{k(1+\delta)}},1\right),
\end{equation}
where $\delta >-1$ and $c_k >0$, all connections being independent. This can be realized in terms of a collection of independent uniform $[0,1]$ random variables
$\{ U_{(\mbox{\boldmath $x$},\mbox{\boldmath $y$})}\}$ by adding an (undirected) edge between $\mbox{\boldmath $x$}$ and $\mbox{\boldmath $y$}$ if and only if $U_{(\mbox{\boldmath $x$},\mbox{\boldmath $y$})}\leq p_{\mbox{\boldmath $x$},\mbox{\boldmath $y$}}$  (see e.g. \cite{JLR}, page 4).

Our aim is to find sufficient conditions on $c_k$ and $\delta$ which imply that percolation occurs, or that it does not occur.

We will study separately  the  cases  $\delta >1$, $\delta <1$,  and   $\delta =1$. As we shall see, the  case $\delta = 1$ requires a more delicate analysis. In this case we take  $c_k$ of the following special forms:
\begin{itemize}
\item[(i)]
\begin{equation}
\label{eq:2.2}
c_k =C_0+C_1\log k+C_2k^\alpha,
\end{equation}
with constants $C_0\geq 0, C_1\geq 0, C_2\geq 0$ and $\alpha >0$.
\item[(ii)] We consider the  hierarchical distances with logarithmic scale
\begin{equation}
\label{eq:2.3}
k_n =k_n(K):=\lfloor K n\log n\rfloor, \quad n=1,2,\ldots,
\end{equation}
$K$ constant $ >0$, so that
\begin{equation}
\label{eq:2.4}
k_{n+1} -k_n \sim K\log n \quad {\rm as}\quad n\rightarrow \infty,
\end{equation}
(where $\sim$ has the usual meaning, see beginning of the proofs),
and we  consider the class of connection rates given by (\ref{eq:2.1}) with $\delta=1$, $c_k$ satisfying
\begin{eqnarray}
\label{eq:2.5}
c_{k_n} &=& C+a \log n \cdot N^{b \log n}
= C+a \log n \cdot n^{b \log N},
\end{eqnarray} and
\begin{equation}
\label{eq:2.6}
c_{k_n} \leq c_j \leq c_{k_{n+1}}\quad {\rm  for} \quad k_n<j <k_{n+1}.
\end{equation}
with constants $C\geq 0$, $a>0$, $b\geq 0$. The constant $K$ is chosen suitably in each case. The value $N=2$ is special because $\log 2  <1$,
and that is why for some results with $N=2$ we set $K> \frac{1}{\log 2}$.
\end{itemize}

The reason for considering $k_n(K)$ and  these forms for $c_{k_n}$ is that they provide  a four parameter family of comparison connection rates (with parameters $K,C,a,b$)  suitable for the renormalization analysis used in the proof
of percolation in Theorem \ref{T3.2}. An intuitive argument for this is given before Theorem \ref{T3.2}. Results of Theorem 3.5 are used to prove Theorem 3.3 which is our main result.

A  set of vertices any two of which are linked by a path of connections is called a {\it cluster}. By transitivity of $(\Omega_{N},d)$ we may focus on clusters containing ${\bf 0}$.
In the proofs  it is implicitly assumed that we consider a sequence of nested balls $(B_k)_{k\geq 1}$ such that
${\bf 0}$ belongs to $B_1$ (or to some $B_k$). We denote by $X_k$ the largest cluster contained in $B_k$, considering only connections within $B_k$ and not through points outside of $B_k$. If there are more than one largest cluster, we choose one of them uniformly from the existing ones. In this way each $k$-ball $B_k$ has a unique attached cluster $X_k$. Note that for $B_k \subset B_{k+j}$, either $X_k \cap X_{k+j}=\phi$ or $X_k \subset X_{k+j}$. Those clusters will be used only in the proofs of sufficient conditions for percolation.  The assumption of connections only within balls makes the renormalization approach quite practical for percolation, and connections through points outside would add to the possibility of percolation.

\begin{definition}
\label{D2.1}
{\rm We say that {\it percolation occurs} in ${\mathcal G}_N$ if there is a positive probability that a fixed point of ${\mathcal G}_N$ (for example
${\bf 0} $) belongs to an infinite cluster.}
\end{definition}

If percolation occurs,  the probability that there is an infinite cluster  is $1$. Indeed,  the event that there is an infinite cluster  is measurable with respect to the tail $\sigma$-algebra generated by the connections involving points outside each $k$-ball (containing $\mbox{\boldmath $0$}$) for every $k$, and the connections involving points  outside a ball are independent of  those inside, so by a $0\!$ - $\!1$ law   the probability that there is an infinite cluster is $0$ or $1$.

In some cases we consider percolation clusters of positive density, that is, $|X_k|N^{-k}$ does not decrease to $0$ as $k\to\infty$.

\begin{remark}\rm{
It follows immediately from the construction in terms of the family $\{U_{(\mbox{\boldmath $x$},\mbox{\boldmath $y$})}\}$ that given two families of connection probabilities $p^1_{\mbox{\boldmath $x$},\mbox{\boldmath $y$}}$,
$p^2_{\mbox{\boldmath $x$},\mbox{\boldmath $y$}}$ with $p^2_{\mbox{\boldmath $x$},\mbox{\boldmath $y$}}\geq p^1_{\mbox{\boldmath $x$},\mbox{\boldmath $y$}}$ for any $\mbox{\boldmath $x$},\mbox{\boldmath $y$}$, percolation for family 1 implies percolation for family 2.
}
\end{remark}

\noindent
\section{Results}
\label{S3}
\subsection{Mean field percolation}
\label{sub:3.1}
\setcounter{equation}{0}
For completeness, we start by recalling the result on  asymptotic percolation as $N\rightarrow \infty$ \cite{DG1}.  This will also be used for comparison with a result below.
The probability of connection between two points separated by distance $k$ is $c_k /N^{2k-1}$. Note that this corresponds to the critical case $\delta =1$ with $c_k$ in (\ref{eq:2.1}) multiplied by $N$, and this may be viewed as a normalization required for obtaining the  result in the limit. {\it Asymptotic percolation} is said to occur if
\begin{eqnarray*}
&& P_{{\rm perc}}: =\inf_{k\to\infty} \liminf_{N\rightarrow \infty} P({\bf 0}\;\hbox{\rm is linked by a path of  connections to a point at distance}\;k)>0.
\end{eqnarray*}
$P_{{\rm perc}}$ is the probability of percolation. For each $k\geq 1$, let $\beta_k\in (0,1)$ satisfy
\begin{equation}
\label{beta}
\beta_k =1-e^{-c_k \beta^2_{k-1} \beta_k}, \quad \beta_0=1,
\end{equation}
where $c_k \beta^2_{k-1}>1$. Note that $\beta_k$ is the well-known survival probability of a Poisson branching process with parameter $c_k\beta^2_{k-1}$. This corresponds to hierarchical level $k$, and the $\beta^2_{k-1}$ comes from the sizes of two connected giant components at the previous level $k-1$. Assume that $c_k \nnearrow \infty$ as $k\rightarrow \infty$, $c
_1>2\log 2$ and $c_2 >8 \log 2$. The results (see \cite{DG1}, Theorem 2.2 and Lemma 2.1) are that asymptotic percolation occurs if and only if
$$
\sum^\infty_{k=1} e^{-c_k}<\infty,
$$
and when it occurs, the probability of percolation is given by
$$
P_{\rm perc} =\prod^\infty_{k=1} \beta_k.
$$
(which is strictly positive if and only if the   exponential series converges), and percolation takes place through a cascade of clusters (in this case giant components) at consecutive hierarchical distances. For example, if $c_k=a \log k$ for large $k$, $a>0$, then asymptotic percolation occurs if and only if $a>1$. See Remark \ref{R3.3}(1) for a partially analogous result with fixed $N$.

\subsection{The  cases $\delta >1$ and $\delta <1$}
\label{sub:3.2}
\begin{theorem}
\label{T3.1}

\noindent
(a) If $\delta >1$ and $\sup_k c_k <\infty$, then  percolation does not occur.

\noindent
(b) If $\delta <1$ and $c=\inf_k c_k$ is large enough, then percolation occurs through a chain of clusters  in $k$-balls, and the percolation cluster is unique.
\end{theorem}

\begin{remark}\label{R3.2}
 {\rm Our results  and those of \cite{KMT} can be compared for $\beta >1$ therein, since in this case their connection probabilities $p_k=1-\exp(-\alpha/\beta^k)\sim\alpha/\beta^k$ as $k\to\infty$. If we set $\beta=N^{1+\delta}$ and let $c_k=c$ not depending on $k$ in (2.1), then  the decay rates agree  with $c$ corresponding  their $\alpha$. The comparison is between Theorem 1 in \cite{KMT} and our Theorem 3.1. We have that for $\delta>1,\beta>N^2$, percolation does not occur with any value of $c$, and for $-1<\delta<1,\; 1<\beta<N^2$, percolation occurs with $c$ sufficiently large, corresponding to $\alpha >\alpha_c(\beta)$ in \cite{KMT}. In addition in \cite{KMT} it is proved that there exists $\alpha_c(\beta)>0$ such that percolation does not occur for $\alpha <\alpha_c(\beta)$.
  Our main objective is to investigate the critical case $\delta=1,\beta=N^2$, for which  percolation does not occur for any $\alpha$ in \cite{KMT}. Our results in this case are stated in the next subsection. }
\end{remark}

\subsection{The case $\delta = 1$}
\label{sub:3.2x}

In the previous subsection we have seen that $\delta=1$ identifies the critical exponential decay rate for percolation. In this subsection we formulate our
main results  that determine the critical polynomial  rate for percolation.

\begin{theorem} \label{MR-1}    Let $\delta =1$ and
\[   c_k= C_0+C_1\log k+C_2k^\alpha \hbox{  with  }\alpha \geq 0,\]
where $C_0\geq 0$, $C_1\geq 0$, $C_2\geq 0$.

\noindent(a) If $\alpha >2$,
then for any $C_1$ there exist $C^*_0>0$ and $C_2^*>0$ such that if  $C_0>C_0^*$ and $C_2> C_2^*$,   percolation occurs and the percolation cluster is unique.

\noindent
(b) If $C_2=0$ and $C_1< N$, then percolation does not occur for any $C_0$.

\noindent
(c) If $\alpha >2$, there exists $C_*>0$ such that if $\max (C_0, C_1,C_2)<C_*$, then percolation does not occur.
\end{theorem}

The proof of this result is based on a renormalization argument that is formulated using the hierarchical distances $k_n(K)$ defined in (\ref{eq:2.3}) and the family of connection rates
$c_{k_n}$ defined in (\ref{eq:2.5}) with parameters $C,a,b$. This is the substance of Theorem 3.5.

In order to express one of the results below we introduce the following notion.

\begin{definition}\label{D3.1x}
{\rm We call {\em pre-percolation} the situation that (with probability $1$) there exists $n_0$ such that
    there is at least one connection from $(k_n,k_{n+1}]$ to $(k_{n+1},k_{n+2}]$ for all $n\geq n_0$.}
\end{definition}

Note that for percolation, in addition to pre-percolation, there would have to be paths connecting points in $(k_{n+1},k_{n+2}]$ which
are connected to $(k_n,k_{n+1}]$ to points in $(k_{n+1},k_{n+2}]$ which are connected to $(k_{n+2},k_{n+3}]$, etc.
\medskip

Before stating the next theorem, let us give an intuitive explanation for the choice of $k_n$ and $c_{k_n}$ in (2.3)-(2.6) with $b>0$, and the assumption $K<b$ (the assumption $\frac{2}{\log N}<K$ is a technical requirement for the method of proof). For this argument only we use the notation $\approx$ for approximate equality for large $n$ without giving it a rigorous meaning. The idea is that $k_n$ is the right scaling and the form of $c_{k_n}$ is the right
one which combines exactly with $k_n$ in order to produce the percolation cluster. We consider the largest clusters $X_{k_n}$ in each one of the $N^{k_{n+1}-k_n}\approx N^{K\log n}$ $k_n$-balls in a $k_{n+1}$-ball, and assume that their sizes are $|X_{k_n}|\approx \beta N^{k_n}$ for some $\beta\in (0,1)$, and that the probability of connection between two points in different clusters is $c_{k_n}/N^{k_{n+1}}$ (which is a lower bound for the actual probabilities). Let $s_n(\beta)$ denote the probability that two such clusters $X_{k_n}$ and $X^\prime_{k_n}$ in disjoint $k_n$-balls are connected. Then
\begin{eqnarray*}
&&s_n(\beta)\approx 1-\left(1-\frac{c_{k_{n}}}{N^{2k_{n+1}}}\right)^{|X_{k_n}||X^\prime_{k_n}|}\\
&&\approx 1-\left(1-\frac{c_{k_{n}}}{N^{2k_{n+1}}}\right)^{\beta^2N^{2k_n}}\\&&
\approx 1-\exp\left(1-\frac{c_{k_{n}\beta^2}}{N^{2(k_{n+1}-k_n)}}\right)\\
&&\approx \frac{C+a\log n\cdot N^{b\log n}}{N^{2K\log n}}\beta^2\\
&& \approx \frac{\beta^2a\log n}{N^{(2K-b)\log n}}=: r_n(\beta),\;{\rm with}\;\; b<2K,
\end{eqnarray*}
(see Lemma 5.8). Consider the E-R random graph
\[ G(N^{K\log n},r_n(\beta)),\]
and write $r_n(\beta)$ as
\[ r_n(\beta)= \frac{\beta^2a\log n}{N^{(K-b)\log n}}\frac{1}{N^{K\log n}}.\]
If $K>b$, then by the E-R theory only order $\log(N^{K\log n})$ of the $X_{k_n}$
are connected, hence the ratio of the size of the largest connected component in the $k_{n+1}$-ball to the size of the ball decreases to $0$
 as $n\to\infty$, so there cannot be a percolation cluster (of positive density).  Therefore we choose $K<b$. Now let $K_1=2K-b$, then $0<K_1<K$. Writing
 $r_n(\beta)$ as
 \[ r_n(\beta)=\frac{\beta^2 a}{K_1\log N}\log N^{K_1\log n}\frac{1}{N^{K_1\log n}}, \]
then by Theorem 2.8.1 of \cite{D} the probability that the graph $G(N^{K_1\log n},r_n(\beta))$ is connected tends to $1$ as $n\to\infty$ if $a>\frac{K_1\log N}{\beta^2}$. Connectivity of that graph whose vertices are the clusters $X_{k_n}$ in the $k_n$-balls in a $k_{n+1}$-ball means that the largest cluster in the $k_{n+1}$-ball contains the largest clusters in all the $k_n$-balls it contains.  If this can be proved for all sufficiently large $n$, then percolation follows.
Note that this argument provides a percolation cluster if $K<b$, but it does not imply that percolation does not occur if $b>K$.

 \medskip
\begin{theorem}
\label{T3.2} Let $\delta =1$.

\noindent

\noindent
(a) Let  $C_2=0$ in (\ref{eq:2.2})  and  $c_{k_n}=C+aN\log n$ in (\ref{eq:2.5}), with $K=1$ and $N\geq 3$  in (\ref{eq:2.3}).
Then

(i)  for $a<1$,  percolation does not occur,

(ii)  for $a >1$, pre-percolation occurs.
\medskip

\noindent
(b) Assume that $\{c_{k_n}\}$ satisfy (\ref{eq:2.5}), (\ref{eq:2.6}) with $b>0$ and  $c_{k_n}=C+a\log n\cdot N^{b \log n}$ where $k_n=k_n(K)$ is given by (\ref{eq:2.3}) and the pair $(K,b)$ satisfy
\begin{equation*}
\frac{2}{\log N}<K<b.
\end{equation*}

(1) Then there exist $C>0$ and $a_*>0$ such that for $a>a_*$ there is a sequence $(\beta_{n})_n$ such that
\begin{equation}
\label{eq:3.1}
\liminf_{n\rightarrow \infty} \beta_{n}>0,
\end{equation}
and for the clusters $X_{k_n}$ in a nested sequence of $k_n$-balls $B_{k_n}$ containing  ${\bf 0}$,

\begin{equation}
\label{eq:3.2}
  P(\hbox{there exists }\; n_{00}\hbox{  such that  }|X_{k_n}|\geq \beta_{n}N^{k_n}\hbox{  for all }n\geq n_{00})=1,
\end{equation}
 percolation occurs, and the percolation cluster is unique.
\medskip

(2)  Assume that $b<  2K-\frac{1}{\log N}$. Then the percolation cluster is given by a ``cascade'' of clusters at distances $k_n$, more precisely,  there exists a (random) number $n_0$ such that for $n\geq n_0$ connections between $X_{k_n}\cap (B_{k_n}\backslash B_{k_{n-1}})$ and $X_{k_{n+\ell}}\cap (B_{k_{n+\ell}}\backslash B_{k_{n+\ell-1}})$ occur only for $\ell=1,2$.

\noindent
(c) In the special case with connection probabilities $c_j=\frac{c_{k_{n}}}{N^{2k_{n+1}}}$ for $k_n+1\leq j\leq k_{n+1}$ and $0<b\leq \frac{2}{\log N} <K$, percolation does not occur.
\end{theorem}

\medskip

\begin{remark}
\label{R3.3}

\noindent
(1) {\rm In Theorem \ref{T3.2} (a) a slightly different result holds for $N=2$ since $\log 2 <1$ so that we need $K>\frac{1}{\log 2}$.}

\noindent(2) {\rm  Note the consistency of Theorem \ref{T3.2} (a) with the example of asymptotic percolation recalled in Subsection \ref{sub:3.1}.  The difference is that  in the finite $N$ case we only have pre-percolation, and  the connections are at hierarchical distances $k \log k$ rather than $k$.}

\noindent
(3) {\rm Theorem \ref{T3.2}(a)(ii) implies that in part (b) pre-percolation occurs with any $a>0$ and $b>0$.}

\noindent

\noindent

\noindent
(4) {\rm The cascade of   clusters in Theorem \ref{T3.2} (b)(2) is analogous to the  cascade of giant components in the  mean field case  \cite{DG1}.}

\noindent
(5) {\rm The formulation in Theorem \ref{T3.2}(b) is used as a technical tool to prove the  result in Theorem \ref{MR-1}, and also provides  a setting to
give a refined result.}

\end{remark}

\subsection{Open problems and related developments.}
\label{R3.4}

\noindent{\em (1)}
In Theorem 3.5(a) we have proved that with $c_{k}=C_{0}+C_{1}\log
k+C_{2}k^{\alpha}$ percolation does not occur if $C_{2}=0$ and $C_{1}%
<{N}$. On the other hand in (b) we proved that for $\alpha>2$ and
$C_{0}$ sufficiently large$,$ percolation occurs. It remains an open question
as to whether percolation can occur for all $\alpha>0$ or even for $C_2=0$
and some $C_{1}$ sufficiently large$.\;$We next explain that to resolve these
questions analogues of well-known results for Erd\H{o}s-R\'enyi graphs would be needed for
a class of  ultrametric random graphs. An  {\em ultrametric random graph} URG(M,d) is a random graph on a finite
set $M$ with ultrametric $d$ and with  connection probabilities $p_{x,y}$ that depend on the ultrametric distance $d(x,y)$.

\medskip

Consider the case $C_{2}=0$. The expected number of in-edges
to the annulus $(k_{n+1}, k _{n+2}]$ from $B_{k_{n+1}}$ is of order $
O(a(1-\frac{1}{N})\log n)$
and the expected number of out-edges to the $(k_{n+2}, k _{n+3}]$ annulus is of order\newline
$O(a(1-\frac{1}{N})\log(n+1)).$ \ In order to determine  the number of in-edges that connect to
an out-edge, it would be necessary to determine the probability that two
randomly chosen vertices in the $(k_{n+1},k_{n+2}]$-annulus are connected by a path in the
associated ultrametric random graph. \ This is
related to the problem of determining the distribution of sizes of the connected components. \ These are
 open problems.

\medskip

Consider the case $C_{2}>0$. In Theorem 3.5(c)  we have proved that the random
graph based on a lower bound for the connection probabilities at distances
$k_{n}+1,\dots,k_{n+1}$ does not exhibit percolation in the case
$0<b\leq\frac{2}{\log N}<K$ \ (corresponding to the case $\alpha<2$). \ In order
to refine the argument and determine the behaviour for the actual connection
probabilities which arise if $0<\alpha<2$ it would be necessary to
determine the size of the largest connected component, that is, the number of
$k_{n}$-balls (more precisely their largest connected components) in the
$k_{n+1}$-ball which are connected and in the largest connected component in
the associated ultrametric random graph  (as $n\rightarrow\infty)$. This is an open problem.

\medskip
\noindent
{\em(2)} It would also be of interest to consider the intermediate case with $\delta=1$ but with connection probabilities of the form
 $\frac{n^{(\log n)^k}}{N^{2k_n}}$.

\medskip

 \noindent
{\em(3)}  One can ask if there can exist a zero-density infinite cluster for parameter values where positive density connected clusters do not exist.

\medskip

\noindent

\noindent
{\em (4)} Berger \cite{Be} has studied the behaviour of  random walk on the infinite cluster of long-range percolation in Euclidean lattices of dimensions $d=1,2$.  It would also be interesting to investigate this behaviour on the infinite clusters obtained in Theorems 3.1(b) and  \ref{T3.2} (b). Long-range random walks on $\Omega_N$ have been studied in \cite{DGW}.

\subsection{The renormalization group approach}

The basic strategy we employ is in the spirit of the renormalization group method of statistical  physics \cite{CE}, which has been used by Newman and Schulman \cite{NS}, Section 2,  in their study of long range percolation in the Euclidean lattice.

Consider the countable ultrametric space $(\Omega_N,d)$.  For each integer $k $ we define an equivalence relation on $\Omega_N$ by
$$ \mbox{\boldmath $x$}\equiv_k \mbox{\boldmath $y$} \hbox{ iff }d(\mbox{\boldmath $x$},\mbox{\boldmath $y$})\leq k,\; \hbox{that is, } \mbox{\boldmath $x$} \hbox{ and }\mbox{\boldmath $y$}\hbox { belong to the same }k\hbox{-ball}.$$
Now consider the set of equivalence classes $\mathcal{E}_k$ furnished with the ultrametric

$$d_k(\widetilde {\mbox{\boldmath $x$}}, \widetilde{\mbox{\boldmath $y$}}):= d(\mbox{\boldmath $x$},\mbox{\boldmath $y$})=0\hbox{ if } d(\mbox{\boldmath $x$},\mbox{\boldmath $y$})\leq k,\\
d_k(\widetilde {\mbox{\boldmath $x$}}, \widetilde {\mbox{\boldmath $y$}}):= d(\mbox{\boldmath $x$},\mbox{\boldmath $y$})-k\hbox{ if } d(\mbox{\boldmath $x$},\mbox{\boldmath $y$})> k,$$
where $\widetilde {\mbox{\boldmath $x$}}$ is an equivalence class containing a point $\mbox{\boldmath $x$}$.
Then the resulting set of equivalence classes with ultrametric $d_k$ can be identified with
$(\Omega_N,d)$.

Given a graph
$$
{\mathcal G}_N= (\Omega_N, {\mathcal E}_N),
$$
with edges $\mathcal E_N$   given by  a symmetric subset of $\Omega _N\times \Omega_N$
 we obtain a new graph  as follows. The set of vertices is the set of all $k$-balls, and the set of edges $\mathcal{E}_{N,1}$
are such that $\widetilde {\mbox{\boldmath $x$}}$ has a connection to $\widetilde{ \mbox{\boldmath $y$}}$ if there is a connection in $\mathcal{G}_N$ between the largest connected subset (cluster) of $\widetilde {\mbox{\boldmath $x$}}$ and the largest connected subset of $\widetilde {\mbox{\boldmath $y$}}$. Using the above identification this defines a new graph $\mathcal{G}^1_N=\Phi\mathcal{G}_N$  on $(\Omega_N,d)$.  Iterating this procedure we obtain a sequence
$${\mathcal G}^k_N= \Phi^k\mathcal{G}_N,\;\; k\geq 1,$$ of graphs all having vertex set $\Omega_N$.  In addition we assign to each vertex $v$ in ${\mathcal G}^k_N$ the $[0,1]$-valued random variable  $$Y_{k}(v):=\frac{|X_{k}(v)|}{N^{k}}$$ where $X_{k}(v)$ denotes the set of vertices in  $\mathcal{G}_N$  contained in the  connected cluster  in the $k$-ball corresponding to $v$ obtained as the union of the clusters in the $(k-1)$-balls it contains.
This construction  defines the renormalization mapping $\Phi: {\mathcal G}_N\rightarrow {\mathcal G}_N$ such that $\Phi^k :{\mathcal G}_N \rightarrow {\mathcal G}^{k}_N$ for each $k$. Note that at each iteration the connection probabilities are the probabilities that the largest connected subsets of equivalence classes are connected and that these probabilities are random and dependent (because the connected components have random sizes), and they change at each iteration.

Rather than working directly with the sequence ${\mathcal G}^{k}_N$ we choose a subsequence $k_n$ and construct a sequence of renormalization maps $\Phi^{k_n}$ such that the number of points in a ball of radius 1 (with respect to the new distances $d_{k_n}$) increases to infinity as $n\to\infty$.
In particular, we will show that there exists an increasing  sequence of integers $(k_n)$ (see (2.3)) and a sequence of graphs $\mathcal{G}^{k_n}$ with $N^{k_n-k_{n-1}}$ vertices constructed recursively as follows, that is,

$$\mathcal{G}^{k_{n+1}}=\Phi_n \mathcal{G}^{k_{n}},$$
where $\Phi_n$ depends on $n$ since it is a mapping from a graph with vertices $\Omega_{N^{k_n-k_{n-1}}}$ to a graph with vertices $\Omega_{N^{k_{n+1}-k_n}}$ and with connection probabilities between vertices that are a function of the distance between them.
 Moreover we can identify $\mathcal{G}^{k_n}$ with a subgraph of $\mathcal{G}_N$ and these subgraphs are a decreasing function of $n$.  We establish percolation by showing that the intersection of these subgraphs starting at a given  point in $\Omega_N$ is non-empty with positive probability.
 The difference now is that $\Phi^{k_{n+1}}$ is not obtained by iteration but by means of $\Phi_n$:
$$ \Phi^{k_{n+1}}=\Phi_n\Phi^{k_n}.$$

Given the sequence $(k_n)$ we can consider the equivalence classes given by

$$ \mbox{\boldmath $x$}\equiv_{k_n} \mbox{\boldmath $y$}\;\; \hbox{iff} \;  d(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) \leq k_n,$$
and define the ultrametric  $d_{k_n}$ by
$$ d_{k_n}(\widetilde {\mbox{\boldmath $x$}},\widetilde {\mbox{\boldmath $y$}})= \ell\hbox{  iff  } k_{n+\ell-1}<d(\mbox{\boldmath $x$},\mbox{\boldmath $y$})\leq k_{n+\ell}, \; \ell\geq 1.$$
We now consider the graph $\mathcal{G}^{k_n}$ whose vertices are the  $d_{k_n}$ equivalence classes.
Two points in $\mathcal{G}^{k_n}$  at $d_{k_n}$-distance $\ell \geq 1$ are connected if there is a $\mathcal{G}_N$-edge joining the largest connected components in these equivalence classes in $\mathcal{G}^{k_{n-1}}$.
Note that there  are $N^{k_{n+1}-k_n}$ points in a ball of $d_{k_n}$-radius 1, $N^{k_{n+2}-k_n}$ in a ball of radius 2, etc. The proof of percolation in the case $\delta <1$ given in Section 4 involves showing that as $n\to\infty$ the graphs $\mathcal{G}^{k_n}$ occupy a certain portion of the $k_n$-balls and an increasing sequence can be linked in a cascade with probability approaching $1$.

In order to apply these ideas to the more delicate critical case $\delta =1$  in Section 5 we define $Y_{k_n}(v),\;v\in \mathcal{G}_{k_n}$ as above.
Then given the random graph $\mathcal{G}_N$, the nontriviality, $\liminf_{n\to\infty}Y_{k_n} >0$,  has probability 0 or 1.  Our goal is to find a sufficient condition for this to be 1.
In order to achieve this our strategy  is to look for a pair of sequences
$k_n\to\infty$,  $\liminf_{n} \beta_n >0$, such that the probability that $Y_{k_n}\geq \beta_n$ converges to $1$ as $n\to\infty$.  This program will be carried out in Subsection 5.2 using as basic tools large deviation estimates for binomial distributions and probability bounds for the connectivity of an Erd\H{o}s-R\'enyi graph.

\section{Proofs for the  cases $\delta>1$ and $\delta <1$}
\label{S4}
\setcounter{equation}{0}

We first mention a few notational points.  In some  places in the proofs in this and the following section where a number appears which should be a non-negative integer and it is not necessarily so, it should be interpreted as its integer part.

 $a_n\sim b_n$ means that $\frac{a_n}{b_n}\to 1$ as $n\to\infty$, $a_n>>b_n$ means that $b_n=o(a_n)$,  and \[a_n \lesssim b_n {\rm \;means\; that\;}
0\leq \liminf_n \frac{a_n}{b_n}\leq \limsup_n \frac{a_n}{b_n} \leq 1.\]

\begin{definition}
\label{D4.1}  {\rm
For $0<\gamma <1$, we say that a $k$-ball} $B_k$ is $\gamma$-good
{\rm if its attached cluster} $X_k$  {\rm satisfies} $|X_k|\geq N^{\gamma k}$.
\end{definition}

If ${\bf 0} \in B_k$ and  $B_k$ is $\gamma$-good,  then the probability that ${\bf 0} \in X_k$ is greater than or equal to  $ N^{(\gamma -1)k}$.

\bigskip

\noindent
{\bf Proof of Theorem 3.1.}

\noindent
(a) If  suffices to show that
\begin{equation}
\label{eq:4.1'}
P(B_k\;\hbox{\rm is connected to its complement for infinitely many}\;k)=0.
\end{equation}

For $j\geq k$,
\begin{eqnarray*}
\lefteqn{P(B_k\;\hbox{\rm is connected to the}\;(j,j+1]
\hbox{\rm -annulus around it})}\\
&=& 1-\biggl(1-\frac{c_{j+1}}{N^{(j+1)(1+\delta )}}\biggr)^{N^k N^j (N-1)}\\
&&(\hbox{\rm using}\; 1-(1-x/y)^z<xz/y, \quad 0<x<y, \quad z\geq 2)\\
&<& c_{j+1}\frac{N^kN^j (N-1)}{N^{(j+1)(1+\delta )}}\\
&\leq & M\frac{N^k}{N^{j\delta}},\quad M\;\hbox{\rm is a constant }(\hbox{since }\sup_k c_k<\infty),
\end{eqnarray*}
hence
$$
P(B_k \;\hbox{\rm is connected to its complement)}\;\leq M
\sum_{j\geq k}\frac{N^k}{N^{j\delta}},
$$
and since
$$
\sum_{k\geq 1} \sum_{j\geq k} \frac{N^k}{N^{j\delta}}<\infty,
$$
(\ref{eq:4.1'})  follows by Borel-Cantelli.

\noindent
(b) For each $n$, let $k_n$ be given by:
\begin{equation}
\label{eq:4.1}
k_n =\lfloor n\log n\rfloor ,
\end{equation}
hence
\begin{equation}
\label{eq:4.2'}
k_{n+1} -k_n \sim \log n \quad{\rm as}\; n\rightarrow \infty.
\end{equation}
Choose $\gamma$ so that
\begin{equation}
\label{eq:4.2}
\frac{1+\delta}{2}<\gamma <1.
\end{equation}
Then the probability of connection between two  clusters $X_{k_n}$ and $X'_{k_n}$ in (disjoint) $\gamma$-good $k_n$-balls in  a $k_{n+1}$-ball  is bounded below by
\begin{equation}
\label{eq:4.3'}
 1-\biggl(1-\frac{c}{N^{k_{n+1} (1+\delta)}}\biggr)^{|X_{k_n}||X'_{k_n}|} \geq 1-\biggl(1-\frac{c}{N^{k_{n+1}(1+\delta)}}\biggr)^{N^{2\gamma k_n}}.
\end{equation}

There are $N^{k_{n+1}-k_n}\;\; k_n$-balls in a $k_{n+1}$-ball. If at least $N^{\gamma (k_{n+1}-k_n)}$ of  them are $\gamma$-good and their  clusters are connected within the $k_{n+1}$-ball, then the size of the  cluster   $X_{k_{n+1}}$ in the $k_{n+1}$-ball is
 greater than or equal to $ N^{\gamma k_{n+1}}
$, so the $k_{n+1}$-ball is $\gamma$-good.

For each $n$, let $p_n$ denote the probability that $B_{k_n}$ is $\gamma$-good:
\begin{equation}
\label{eq:4.3}
p_n =P(|X_{k_n}|\geq N^{\gamma k_n}).
\end{equation}
Then, writing ${\rm Bin} (n, p, \geq k)=\sum^n_{j=k}({n\atop j})p^j (1-p)^{n-j}$ and ${\rm Bin}(n, p, <k)=1-{\rm Bin}(n,p, \geq k)$, we have, by (\ref{eq:4.3'}), (\ref{eq:4.3}), and independence of connections at different hierarchical distances,
\begin{eqnarray}
\label{eq:4.4}
&& p_{n+1} \geq  {\rm Bin} \biggl(N^{k_{n+1}-k_n},p_n, \geq N^{\gamma (k_{n+1}-k_n)}\biggr)
 \biggl[1-\biggl(1-\frac{c}{N^{k_{n+1}(1+\delta)}}\biggr)^{N^{2\gamma k_n}}\biggr]^{N^{\gamma (k_{n+1}-k_n)}}.
\end{eqnarray}
Now,
\begin{eqnarray}
\label{eq:4.4''}
\lefteqn{1-\biggl(1-\frac{c}{N^{k_{n+1}(1+\delta)}}\biggr)^{N^{2\gamma k_n}}}\nonumber\\
&\geq & 1-\biggl(1-\frac{cN^{2\gamma k_n -(1+\delta)k_{n+1}}}{N^{2\gamma k_n}}\biggr)^{N^{2\gamma k_n}}\nonumber\\
&&(\hbox{\rm using}\; (1-x/y)^y <e^{-x},\quad x>0,\, y>0),\nonumber\\
&>& 1-\exp \{-c N^{2\gamma k_n-(1+\delta)k_{n+1}}\},
\end{eqnarray}
\noindent
hence
\begin{eqnarray}
\label{eq:4.4'}
&&\biggl[1-\biggl(1-\frac{c}{N^{k_{n+1}(1+\delta)}}\biggr)^{N^{2\gamma k_n}}\biggr]^{N^{\gamma (k_{n+1}-k_n)}}\nonumber\\
&&>(1-\exp \{-c N^{2\gamma k_n-(1+\delta)k_{n+1}}\})^{N^{\gamma (k_{n+1}-k_n)}}.
\end{eqnarray}
(\ref{eq:4.1}) and (\ref{eq:4.2}) imply that
\begin{equation}
\label{eq:4.5}
2\gamma k_n-(1+\delta) k_{n+1} > \varepsilon n\log n
\end{equation}
for some $0<\varepsilon <1$ and  sufficiently large $n$. Then, from
(\ref{eq:4.4'}),
\begin{equation}
\label{eq:4.6}
\biggl[1-\biggl(1-\frac{c}{N^{k_{n+1}(1+\delta)}}\biggr)^{N^{2 \gamma k_n}}\biggr]^{N^{\gamma (k_{n+1}-k_n)}} > 1-\varepsilon_n,
\end{equation}
where
\begin{equation}
\label{eq:4.7}
\varepsilon_n :=1-\biggl(1-\exp \{ -c N^{2\gamma k_n -(1+\delta )k_{n+1}}\}\biggr)
^{N^{ \gamma (k_{n+1}-k_n)}},
\end{equation}
and we have that
\begin{equation}
\label{eq:4.8}
\sum_n \varepsilon_n <\infty ,
\end{equation}
because, from (\ref{eq:4.7}), (\ref{eq:4.2'}),   and (\ref{eq:4.5}),
\begin{eqnarray*}
\lefteqn{
\varepsilon_n < N^{\gamma (k_{n+1}-k_n)}\exp \{-c N^{2\gamma k_n -(1+\delta ) k_{n+1}}\}}\\
&\lesssim & N^{\gamma \log n}\exp \{-cN^{\varepsilon n\log n}\}\\
&=&n^{\gamma \log N}\exp \{-cn^{\varepsilon n\log N}\}
\end{eqnarray*}
for all sufficiently large $n$, which is summable.

From (\ref{eq:4.4}) and (\ref{eq:4.6}),
\begin{equation}
\label{eq:4.9}
p_{n+1}> {\rm Bin} (N^{k_{n+1}-k_n}, p_n, \geq N^{\gamma (k_{n+1}-k_n)}) (1-\varepsilon_n).
\end{equation}

Now we assume that there exists $n_0$ such that
\begin{equation}
\label{eq:4.10}
 N^{k_{n_0+1}-k_{n_0}} p_{n_0}>>N^{\gamma (k_{n_0+1}-k_{n_0})},
\end{equation}
(this assumption will be verified below), and we suppose that $p_{n_0}>\eta >0$. Then,  by a large deviation inequality \cite{JLR} (p. 26, (2.6)),
\begin{eqnarray}
\label{eq:4.11}
&& {\rm Bin} \left(N^{k_{n_0+1}-k_{n_0}}, p_{n_0}, \geq N^{\gamma (k_{n_0+1}-k_{n_0})}\right)\\
&&= 1-{\rm Bin} \left(N^{k_{n_0+1}-k_{n_0}}, p_{n_0}, <N^{k_{n_0+1}-k_{n_0}}p_{n_0}
-(N^{k_{n_0+1}-k_{n_0}}p_{n_0} -N^{\gamma (k_{n_0+1} -k_{n_0})})\right)\nonumber\\
&&\geq  1-\exp
\biggl\{-\frac{(N^{k_{n_0+1}-k_{n_0}} p_{n_0} -N^{\gamma (k_{n_0+1}-k_{n_0})})^2}{2N^{k_{n_0+1}-k_{n_0}}p_{n_0}}\biggr\},\nonumber
\end{eqnarray}
hence, by (4.15),
\begin{equation}
\label{eq:4.12}
{\rm Bin} \left(N^{k_{n_0+1}-k_{n_0}}, p_{n_0}, \geq N^{\gamma (k_{n_0+1}-k_{n_{0})}}\right)\geq 1-\delta_{n_0},
\end{equation}
where we may
 take
\begin{equation}
\label{eq:4.13}
0<\delta_{n_0} <(1-\varepsilon_{n_0}-\eta) /(1-\varepsilon_{n_0})
\end{equation}
(with $\varepsilon_{n_0}$ and $\eta$ small enough so that $\varepsilon _{n_0}+\eta <1)$. Then, from (\ref{eq:4.9}), (\ref{eq:4.12}) and (\ref{eq:4.13}),
\begin{equation}
\label{eq:4.14}
p_{n_0+1}> (1-\delta_{n_0} )(1-\varepsilon_{n_0})>\eta,
\end{equation}
so, iterating the argument,
\begin{equation}
\label{eq:4.14'}
p_n >\eta \quad \hbox{\rm for all}\quad n\geq n_0,
\end{equation}
and therefore (\ref{eq:4.10}) will in fact hold for all sufficiently large $n_0$.

From (\ref{eq:4.14}),
\begin{equation}
\label{eq:4.15}
1-p_{n_0+1}\leq \delta _{n_0} +\varepsilon_{n_0}.
\end{equation}
Choosing $\delta_{n}, n\geq n_0$, so that $\sum\limits_{n\geq n_0} \delta_{n}<\infty$, we then have from (\ref{eq:4.15}), together with (\ref{eq:4.8}),
$$
\sum_{n\geq n_0} (1-p_n)<\infty.
$$
Therefore, by Borel-Cantelli,
\begin{equation}
\label{eq:4.22}
P(\hbox{\rm all but finitely many}\;B_{k_n}\;{\rm are}\;\gamma\hbox{\rm -good})=1.
\end{equation}

On the other hand, the probability that the clusters $X_{k_n}$ and $X_{k_{n+1}}$ of  the $\gamma$-good  balls $B_{k_n}$ and
$B_{k_{n+1}}, B_{k_n} \subset B_{k_{n+1}}$, are not  connected is, for large $n$, by (\ref{eq:4.1}) and  (\ref{eq:4.5}), bounded above by
\begin{eqnarray*}
\biggl(1-\frac{c_{k_{n}}}{N^{k_{n+1} (1+\delta)}}\biggr)^{|X_{k_n}||X_{k_{n+1}}|} &\leq & \biggl( 1-\frac{c}{N^{k_{n+1}(1+\delta)}}\biggr)^{N^{\gamma (k_n +k_{n+1})}}\\
&<& \exp \{-c N^{\gamma (k_n+k_{n+1})-k_{n+1}(1+\delta)}\}\\
&\lesssim & \exp \{-c N^{(2\gamma k_n-(1+\delta )k_{n+1})}\} \\
&<&\exp \{-cN^{\varepsilon n \log n}\}\\
&=& \exp \{-cn^{\varepsilon n\log N}\},
\end{eqnarray*}
which is summable. Therefore, by Borel-Cantelli,
\begin{equation}
\label{eq:4.23}
P \hbox{\rm (all but finitely many pairs}\; X_{k_n}\;{\rm and}
\; X_{k_{n+1}}\;\hbox{\rm  are connected}) =1.
\end{equation}

Since there is a positive probability that $\mbox{\boldmath $0$}$ belongs to some $X_{k_n}$ (the probability is greater than or equal to $ N^{(\gamma-1)k_n})$ in the  chain of $\gamma$-good  $B_{k_n}$'s with connected clusters $X_{k_n}$, then by (\ref{eq:4.22}) and (\ref{eq:4.23})  there is an infinite cluster  that contains $\mbox{\boldmath $0$}$ with positive probability. So, percolation occurs.

It remains to verify assumption (\ref{eq:4.10}). We do this by means of a connectivity result for E-R random graphs. Writing $c_k/N^{k(1+\delta)}$ in (\ref{eq:2.1}) as $\widetilde{c}_k/N^k, \widetilde{c}_k = c_k /N^{\delta k}$, we have that for all $k\leq n$,
$$
\widetilde{c} _k > \overline{c}_n \log N^n\quad {\rm with}\quad \overline{c}_n =\frac{c}
{N^{ \delta n} \log N^{n}} .
$$
We consider the E-R random graph $G(N^n, \overline{c}_n/N^n)$ whose vertices are the points of an $n$-ball. If
\begin{equation}
\label{eq:4.21}
c>N^{\delta n} \log N^{ n}
\end{equation}
for some very large $n$, then the probability that  $G(N^{n}, \overline{c}/N^{n})$ is connected is close to $1$, by Theorem 2.8.1 in \cite{D}. It is possible to choose $c$ large enough so that (\ref{eq:4.21}) holds because the only restriction on $c$ is $c<N^{(1+\delta)n}$. This implies that the probability that all the points in  the $n$-ball are  connected is close to $1$. Then, taking $n=k_{n_0}$, (\ref{eq:4.10}) is true.

Finally, the uniqueness follows from   Theorem 2 in  \cite{KMT} (see Remark \ref{KMTr}).

 \hfill $\Box$

\begin{remark}\label{KMTr} \rm{
Uniqueness of the infinite cluster is proved in \cite{KMT}  (Theorem 2). They prove that $(\Omega_N,d)$ can be embedded into $\mathbb{Z}$ such that any ball of radius $r$ will be represented by $N^r$ consecutive integers, and the collection of balls of radius $r$ partitions $\mathbb{Z}$ and  that the embedding is  stationary and ergodic.  The uniqueness then follows from Gandolfi et al. \cite{GKN}, Theorem 0, on the uniqueness of the infinite cluster for long range percolation on $\mathbb{Z}$
satisfying the positive finite energy condition. The proof uses only the properties that the connection probabilities between vertices $\mbox{\boldmath $x$},\mbox{\boldmath $y$}$  are strictly positive
and depend only on $d(\mbox{\boldmath $x$},\mbox{\boldmath $y$})$, and therefore is also applicable to our case.
An intuitive argument for uniqueness of the percolation cluster (of positive density), using the argument of the theorem, is that two chains of nested balls will eventually intersect, and by ultrametricity from then on they coincide, so, if their largest clusters occupy a sizeable part of the balls, they will eventually be the same.
}
\end{remark}

\section{Proofs for the critical case $\delta =1$}
\label{S5}
\setcounter{equation}{0}
\subsection{Lemmas for the proof of Theorem 3.5.}
\label{sub:5.1}

\begin{lemma}
\label{L5.1}
Let $b=0$ and with $aN$ instead of $a$ in (\ref{eq:2.5}), and $K=1$ in (\ref{eq:2.3}).

\noindent
(a) Let $A_{n,j}$ denote the event that a $k_n$-ball $B_{k_n}$ (containing ${\bf 0}$) is connected to the $(k_{n+1} +j-1, k_{n+1}+j]$-annulus around $B_{k_n}$, $j=1,\ldots ,k_{n+2}-k_{n+1}$. Then
\begin{equation}
\label{eq:5.5x}
 P(A_{n,j})\sim
{\frac{aN(1-N^{-1})\log n}{N^{j} N^{\log n}}} \quad \hbox{  as  } n\to\infty.
\end{equation}

\noindent
(b) Let $\widehat{A}_{n,j}$ denote the event that there in no connection between the  $k_{n+1}$-ball $B_{k_{n+1}}$ and the $(k_{n+1} +j-1, k_{n+1}+j]$-annulus around   $B_{k_{n+1}}$, $j=1, \ldots, k_{n+2} -k_{n+1}$. Then
\begin{equation}
\label{eq:5.6}
P(\widehat{A}_{n,j})\sim n^{-aN(1-N^{-1})/N^{j}}\quad {\it as}\quad n\rightarrow \infty .
\end{equation}

\noindent
(c) Let $A_n$ denote the event that there is no connection between the interior of a $k_{n+1}$-ball $B_{k_{n+1}}$ and the
$(k_{n+1}, k_{n+2}]$-annulus around $B_{k_{n+1}}$. Then
\begin{equation}
\label{eq:5.7}
P(A_n) \sim n^{-a}\quad {\it as}\quad n\rightarrow \infty,
\end{equation}
and this implies that

(i) if $a<1$, then with probability $1$ there are infinitely many pairs  $((k_n, k_{n+1}],\, (k_{n+1},k_{n+2}])$ of successive annuli that are not connected,

(ii) if $a>1$, then with probability $1$ there are at most finitely many pairs  $((k_n, k_{n+1}],\, (k_{n+1},k_{n+2}])$ of successive annuli that are not connected.

\noindent
(d) Let $A_{n,j,\ell}$ denote the event that  a $k_n$-ball $B_{k_n}$ is connected to the $(k_{n+\ell}+j-1, k_{n+\ell}+j]$-annulus around $B_{k_n}$, $j=1, \ldots, k_{n+\ell +1}-k_{n+\ell}$. Then for $\ell \geq 1$,
\begin{equation}
\label{eq:5.8}
P(A_{n,j,\ell})\sim \frac{aN(1-N^{-1}) \log (n+\ell)}{N^{j} (n+\ell)^{\ell \log N}}\quad {\it as}\quad n\rightarrow \infty.
\end{equation}
(Note that (\ref{eq:5.6}) is a special case with $\ell=1$).

\noindent
(e) Let $\widetilde{A}_n$ denote the event that there are  connections from a $k_n$-ball $B_{k_n}$ to the complement of a $k_{n+1}$-ball $B_{k_{n+1}}$, with $B_{k_n}\subset B_{k_{n+1}}$. Then
\begin{equation}
\label{eq:5.9}
P(\widetilde{A}_n) \lesssim\; a  \sum^\infty_{\ell =1}
\frac{\log (n+\ell)}{(n+\ell)^{\ell \log N}} \quad {\it as}\quad n\rightarrow \infty.
\end{equation}
(f) Let $A$ denote the event that there are no connections from a $k_n$-ball $B_{k_n}$ to the complement of a $k_{n+1}$-ball $B_{k_{n+1}}$, with $B_{k_n} \subset B_{k_{n+1}}$, for all but finitely many $n$. Assume in addition that in the case $N=2$, $K$ in (2.3) is taken so that $K>\frac{1}{\log 2}$. Then for any $a>0$,
\begin{equation}
\label{eq:5.10}
P(A)=1.
\end{equation}
\end{lemma}
\noindent
{\bf Proof.}

\noindent
(a) By (\ref{eq:2.5}) and (\ref{eq:2.6}), $c_{k_{n+1}+j}\sim c_{k_n}$ as $n\to\infty$, hence
\begin{eqnarray*}
P(A_{n,j}) &= & 1-\biggl( 1-
\frac{c_{k_{n+1}+j} }{N^{2(k_{n+1}+j)}}\biggr)
^{N^{k_n}N^{k_{n+1}+j} (1-N^{-1})}\\
&\sim & 1-\exp \biggl\{ -
\frac{aN(1-N^{-1})\log (n+1) }
{N^{j} N^{k_{n+1}-k_n}}\biggr\}\\
&\sim & 1-\exp \biggl\{-
\frac{aN(1-N^{-1})\log (n+1) }
{N^{j}N^{\log n}}\biggr\},
\end{eqnarray*}
and then (\ref{eq:5.5x}) follows.

\noindent
(b) Since there are $N^{k_{n+1}-k_n -1}\sim N^{\log n}\;\; k_n$-balls in the interior of the $k_{n+1}$-ball, then from (\ref{eq:5.5x}),
\begin{eqnarray*}
P(\widehat{A}_{n,j}) & \sim & \biggl(1-\frac{aN(1-N^{-1}) \log n}{N^{j}N^{\log n}}\biggr)^{N^{\log n}}\\
&\sim & \exp \biggl\{ -\frac{aN(1-N^{-1})\log n}{N^{j}}\biggr\}\\
&=& n^{-aN(1-N^{-1}) /N^{j}}.
\end{eqnarray*}
(c) By (\ref{eq:5.6}) and independence,
\begin{eqnarray*}
P(A_n) &\sim & \prod^{k_{n+2}-k_{n+1}}_{j=1} n^{-aN(1-N^{-1})/N^{j}}\\
&= & n^{-aN (1-N^{-1}) \sum^{\log (n+1)}_{j=1} N^{-j}}\\
&=& n^{-a (1-N^{-\log n})}\\
&\sim & n^{-a}.
\end{eqnarray*}
Since $\sum_n n^{-a} <\infty$ if and only if $a>1$, then by independence and the (second) Borel-Cantelli lemma, for $a<1$, with probability $1$ there are infinitely many successive $(k_n, k_{n+1}]$-annuli that are not connected, and  by the (first) Borel-Cantelli lemma, for $a>1$, with probability $1$ there are at most finitely many successive $(k_n, k_{n+1}]$-annuli that are not connected.

\noindent
(d) By (\ref{eq:2.5}) and (\ref{eq:2.6}),
\begin{eqnarray*}
P(A_{n,j, \ell}) &\sim & 1-\biggr(1-
\frac{C+aN\log (n+\ell)}{N^{2(k_{n+\ell}+j)}}\biggr)^{N^{k_n}N^{k_{n+\ell}+j}(1-N^{-1})}\\
&\sim & 1-\exp \biggl\{ -\frac{aN(1-N^{-1})\log (n+\ell)}
{N^{j}N^{k_{n+\ell}-k_n}}\biggr\}\\
&\sim& \frac{aN(1-N^{-1})\log (n+\ell)}{N^{j} N^{\ell \log (n+\ell)}}\\
& =&
\frac{aN(1-N^{-1})\log (n+\ell)}{N^{j} (n+\ell )^{\ell \log N}}.
\end{eqnarray*}
(e) By (\ref{eq:5.8}),
\begin{eqnarray*}
P(\widetilde{A}_n) &\sim &\sum^{\infty}_{\ell =1} \sum^{k_{n+\ell +1}-k_{n+\ell}}_{j=1}
\frac{aN(1-N^{-1})\log (n+\ell)}{N^{j} (n+\ell )^{\ell \log N}}\\
&\sim & aN \sum^\infty_{\ell =1}(N^{-1}-N^{-\log (n+\ell)}) \frac{\log (n+\ell)}{(n+\ell )^{\ell \log N}}\\
& \lesssim & a \sum^{\infty}_{\ell =1}\frac{\log (n+\ell)}{(n+\ell )^{\ell \log N}}.
\end{eqnarray*}
(f) By (\ref{eq:5.9}),
\begin{eqnarray*}
\sum^\infty_{n=1} P (\widetilde{A}_n) &\lesssim & a \sum^\infty_{n=1} \sum^\infty_{\ell =1} \frac{\log (n+\ell)}{(n+\ell )^{\ell \log N}}\\
&\lesssim & aN \sum^\infty_{j=2} \frac{\log j}{j^{\log N}}<\infty ,
\end{eqnarray*}
and the result follows by Borel-Cantelli.

This proof has been done for $K=1$ and $\log N>1$, hence $N\geq 3$.  For $N=2$ we have $k_{n+\ell}-k_n\sim K\log(n+\ell)$ (see (\ref{eq:2.4})), which yields $K\log N$ instead of $\log N$ in the last step, so we take $K>\frac{1}{\log 2}$ for summability.
 \hfill $\Box$

\begin{lemma}
\label{L5.7} Let $K$ and $b$ be as in part (b)(2) of Theorem \ref{T3.2}, that is, $0<b<2K-\frac{1}{\log N}$.
     Let $A_{n,j}$ denote the event that the cluster $X_{k_n}$ in a $k_n$-ball $B_{k_n}$ is connected to the $(k_{n+j}, k_{n+j+1}]$-annulus around $B_{k_n}$, $j\geq 2$. Then there is a positive constant $M$ such that
$$
P(A_{n,j}) \lesssim \frac{M\log n}{n^{(Kj-b)\log N}},
$$
as $n\to \infty$, and
$$\sum_{n=1}^\infty \sum_{j=2}^\infty P(A_{n,j})<\infty.$$
Hence with probability $1$ there exists a (random) number $n_0$ such that for all $n\geq n_0$ the connections between the clusters $X_{k_n}$ restricted to the $(k_n,k_{n+1}]$-annuli do not skip over two successive  annuli, that is, there are no connections between the annulus $(k_{n-1},k_n]$ and the annuli $(k_{n+2},k_{n+3}]$, $(k_{n+3},k_{n+4}]$, etc.
\end{lemma}
\noindent
{\bf Proof.}
By (\ref{eq:2.5}) and (\ref{eq:2.6}),

\begin{eqnarray*}
P(A_{n,j})
&=& \sum_{\ell =k_{n+j}+1}^{k_{n+j+1}}\left[1-\left(1-\frac{c_\ell}{N^{2\ell}}\right)\right]^{|X_{k_n}|N^\ell(1-N^{-1})}\\
&\leq& (C+a\log (n+1+j)\cdot N^{b\log (n+1+j)})N^{k_n}\sum_{\ell =k_{n+j}+1}^{k_{n+j+1}}\frac{1}{N^\ell}\\
&\lesssim & M\frac{\log n\cdot N^{b\log n}}{N^{k_{n+j}-k_n}}\lesssim M\frac{\log n}{N^{(Kj-b)\log n}}.
\end{eqnarray*}

It is easy to show that the assumptions on $K$ and $b$ imply that
$$
\sum_{n=1}^\infty\sum^\infty_{j=2} \frac{\log n}{n^{(Kj-b) \log N}}<\infty,
$$
and then the result  follows by Borel-Cantelli.

\hfill $\Box$
\vglue .25cm
  The main tools for proving part (b) of the theorem  are a large deviation inequality for the binomial distribution and a connectivity result for an E-R random graph. Recall that a graph is said to be connected if it has only one connected component and no isolated vertices.
\medskip

We first recall the large deviation bound for the binomial distribution  \cite{JLR} (Corollary 2.4).
\begin{lemma}
\label{L5.2} Let $Y_n$ be Bin$(n, 1-p)$ and $c>1$. Then for $x\geq cn(1-p)$,
$$
P(Y_n \geq x)\leq e^{-h(c)x},
$$
where $h(c)=\log c-1+1/c>0$.
\end{lemma}
\begin{corollary}
\label{C5.3}
There exist $\kappa >0$ and $\varepsilon >0$ such that for $0<\sigma < \min(\frac{p}{1-p},\varepsilon)$,
\begin{equation}
\label{eq:5.2}
P\biggl(1-\frac{Y_n}{n} \leq p-\sigma (1-p)\biggr)\leq e^{-\kappa \sigma^2 (1-p) n}.
\end{equation}
\end{corollary}
\noindent
{\bf Proof.} It is easy to see that there exist $\kappa >0$ and $\varepsilon>0$ such that $h(c)\geq \kappa (c-1)^2$ for $1<c<1+\varepsilon$. Then the result follows from the lemma putting $c=1+\sigma$. \hfill $\Box$

\begin{definition}{\rm
\label{D5.4} For $0<\beta <1$, we say that a $k$-ball $B_k$ is} $\beta$-good {\rm if its
   cluster $X_k$  satisfies $|X_k|\geq \beta N^k$.}
\end{definition}

Note that if ${\bf 0} \in B_k$ and $B_k$ is $\beta$-{\it good}, then  the probability that
${\bf 0} \in X_k$  is $\geq \beta$.

\begin{lemma}
\label{L5.5'}
 Assume that for some $\beta>0$ and some $n_0$,
\begin{equation}
\label{assumption}
P(|X_{k_n}|\geq \beta N^{k_n}\quad \hbox{\it for all}\quad n\geq n_0)=p_*>0.
\end{equation}
Then percolation occurs.
\end{lemma}
\noindent
{\bf Proof.} By transitivity we may assume that $\mathbf{0}$ belongs to the $k_n$-ball whose largest cluster is $X_{k_n}$. Then
 by the assumption (\ref{assumption}) we have that
$$\liminf_{k\rightarrow \infty} P({\bf 0}\in X_{k_n})\geq \beta p_*>0,
$$
which implies percolation. \hfill $\Box$

\begin{lemma}
\label{L5.5} Let  $0<b  <2K$ in (\ref{eq:2.5}) with  $k_n$ as in (\ref{eq:2.3}), and let $0<\beta <1$. Let $X_{k_n}$ and $X'_{k_n}$ be the largest clusters in  two (disjoint) $\beta$-{\it good} $k_n$-balls in a $k_{n+1}$-ball. Then
\begin{equation}
\label{eq:5.3'}
P(X_{k_n}\;{\rm and}\;X'_{k_n}\;\hbox{\it are connected within the}\; k_{n+1}\hbox{\it -ball})   \gtrsim r_n(\beta) \hbox{ as }n\to\infty,
\end{equation}
 where
\begin{equation}
\label{eq:5.3}
r_n (\beta) = \frac{\beta^2 a\log n}{N^{(2K-b)\log n}}.
\end{equation}
\end{lemma}
\noindent
{\bf Proof.}  By (\ref{eq:2.6}) and (\ref{eq:2.5}),  $k_n<d(X_{k_n},X'_{k_n})\leq k_{n+1}$, so
\begin{eqnarray*}
&& P( X_{k_n}\;{\rm and}\;{X'}_{k_n}\;\hbox{\it are connected within the}\; k_{n+1}\hbox{\it -ball})\\
 & & \geq  1 -\biggl(1-\frac{c_{k_{n}}}{N^{2k_{n+1}}} \biggr) ^{|X_{k_n}||{X'}_{k_n}|}\\
& & \geq 1-\biggl(1-\frac{c_{k_{n}}}{N^{2k_{n+1}}}\biggr)^{\beta^2 N^{2k_n}}\\
&&> 1 -\exp \biggl\{-\frac{c_{k_{n}}\beta^2}{N^{2(k_{n+1}-k_n)}}\biggr\}\\
&&\sim  1-\exp \biggl\{ -
\frac{\beta^2 (C+a \log n\cdot N^{b\log n})}{N^{2K\log n}}\biggr\}\\
&&\sim r_n(\beta).
\end{eqnarray*}
\hfill$\Box$

\noindent
\subsection{Proof of Theorem \ref{T3.2}}

\noindent

\bigskip

\noindent
(a)  (i)  Lemma \ref{L5.1}(f)  guarantees that with probability 1 there exists $n_0$ such that if  $n>n_0$ there are no connections between the $(k_n,k_{n+1}]$-annulus and the complement of $B_{k_{n+2}}$.  Moreover, if $a<1$, by Lemma \ref{L5.1}(c)(i) there are infinitely many $n$ such that the $(k_n,k_{n+1}]$-annulus and the $(k_{n+1},k_{n+2}]$-annulus are not connected.   This    implies that with probability 1 there there exists some $n\geq n_0$ such that there are no connections from $B_{k_{n+1}}$ to the exterior.
\medskip

\noindent
(ii) The pre-percolation statement  follows from Lemma \ref{L5.1}(c)(ii).

\bigskip

\bigskip

\noindent
(b)
(1)  We begin by indicating the main ideas of the proof. We consider a sequence of nested balls $B_{k_n}$ (containing ${\bf 0}$) and  their largest  clusters $X_{k_n}$. Recall that each $k_{n+1}$-ball is comprised of $N^{k_{n+1}-k_n}$ (disjoint) $k_n$-balls.

At each stage we will focus on the subset of the $k_n$-balls in a $k_{n+1}$-ball that are $\beta_n$-{\it good} (i.e., $|X_{k_n}|\geq \beta_n N^{k_n}$, see Definition 5.5), where $(\beta_n)_n$ is a sequence of numbers in $(0,1)$ to be determined below. By construction the events that different $k_n$-balls are $\beta_n$-good are independent, and by transitivity they all have the same probability
 \begin{equation}
\label{eq:6.1}
p^G_n (\beta_n) = P(|X_{k_n}|\geq \beta_n N^{k_n}).
\end{equation}
Let ${\mathcal N}_n$ denote the number of $\beta_n$-good $k_n$-balls and recall (\ref{eq:5.3'}) and (\ref{eq:5.3}).
Now we consider the E-R random graph $G({\mathcal N}_n, r_n (\beta_n))$ whose vertices are the ${\mathcal N}_n\;\beta_n$-good $k_n$-balls in the $k_{n+1}$-ball with connection probability $r_n(\beta_n)$.

The key idea of the proof is to establish that with probability $1$ there is a (random) number $n_0$ such that for $n\geq n_0$,
\begin{equation}
\label{eq:6.3}
G({\mathcal N}_n, r_n (\beta_n))\quad\hbox{\rm is connected},
\end{equation}
which implies that
\begin{equation}
\label{eq:6.4}
|X_{k_{n+1}}|\geq {\mathcal N}_n \beta_n N^{k_n}.
\end{equation}
We denote by $E_n$ the event
\begin{equation}
\label{eq:6.5}
E_n =\{G({\mathcal N}_n, r_n (\beta_n ))\quad \hbox{\rm is connected}\}
\end{equation}
and
\begin{equation}
\label{eq:6.6}
p^A_n (\beta_n)=P(E_n).
\end{equation}
In order to prove (\ref{eq:6.3}) for all large $n$, by Borel-Cantelli it suffices to show that
\begin{equation}
\label{eq:6.7}
\sum_n (1-p^A_n (\beta _n))<\infty.
\end{equation}
We denote by $F_n$ the event
\begin{equation}
\label{eq:6.10}
F_n =\{{\mathcal N}_n \geq (1-\varepsilon_n) p^G_n (\beta_n )N^{k_{n+1}-k_n}\},
\end{equation}
where $\varepsilon_n \in (0,1)$ and $p^G_n (\beta_n)$ is given by (\ref{eq:6.1}), and
\begin{equation}
\label{eq:6.9}
p^B_n (\beta_n, \varepsilon_n)=P(F_n),
\end{equation}

The next key idea is to choose a sequence of numbers $\varepsilon_n \in (0,1)$ of the form $\varepsilon_n=n^{-(1+\theta)}$, $\theta$ to be chosen below, with $\sum_n \varepsilon_n <\infty$ such that
\begin{equation}
\label{eq:6.8}
\sum_n (1-p^B_n (\beta_n, \varepsilon_n))<\infty.
\end{equation}

If both events $E_n$ and $F_n$ occur, then by (\ref{eq:6.4})
$$
|X_{k_{n+1}}|\geq (1-\varepsilon_n) p^G_n (\beta_n) \beta_n N^{k_{n+1}},
$$
so the $k_{n+1}$-ball is $\beta_{n+1}$-good with
\begin{equation}
\label{eq:6.11}
\beta_{n+1} = (1-\varepsilon_n)p^G_n (\beta_n) \beta_n.
\end{equation}
Therefore, since  $E_n$ and $F_n$ are independent (because $E_n$ is defined in terms of distance $k_{n+1}$, and $F_n$ in terms of distance $k_n$), then

\begin{equation}
\label{eq:6.12}
p^G_{n+1} (\beta_{n+1}) \geq p^B_n (\beta_n, \varepsilon_n) p^A_n (\beta_n).
\end{equation}
We will show that for sufficiently large values of  $C$ and $a$ there exists a sequence
$\beta_n$ such that
\begin{equation}
\liminf_n \beta_n>0.
\end{equation}
and
\begin{equation}
\label{eq:6.13}
\sum_n (1-p^G_n (\beta_n))<\infty,
\end{equation}
in order to obtain the results (\ref{eq:3.1}) and (\ref{eq:3.2}).

Since the quantities involved in the scheme described above are interdependent, we need  to overcome the interaction among them. We proceed as follows:
\begin{itemize}
\item[$\bullet$] We  set $\varepsilon_n=n^{-(1+\theta)}$ for some
$0<\theta <\frac{K\log N}{2}-1$ (recall that $K>\frac{2}{\log N}$),  hence
\begin{equation}
\label{eq:5.26}
K\log N >2(1+\theta).
\end{equation}
\item[$\bullet$] In Steps 1 and 2 below we will obtain  estimates
\begin{equation}
\label{eq:6.14}
 \sum_{n\geq n_0} \biggl(1-p^B _n \biggl(\beta_n, \frac{1}{n^{1+\theta}}\biggr)\biggr)<
\infty\quad \hbox{ if  }p^G_n(\beta_n)\geq\frac{1}{2}\hbox{  for all  }n\geq n_0,
\end{equation}
and
\begin{equation}
\label{eq:6.15}
\sup_{\beta \geq \frac{1}{5}} \sum_{n\geq n_0} (1-p^A_n (\beta))<\infty.
\end{equation}

We will then show that we can choose  $n_0$, $C$ and $a$ such that $\beta_n\geq \frac{1}{5}$ and $p^G_n(\beta_n)\geq \frac{1}{2} $ for all $n\geq n_0$.

\end{itemize}

 To complete the proof we proceed step by step. We first verify (\ref{eq:6.14}) and (\ref{eq:6.15}).

\noindent
Step 1. Assume that  $p^G_n(\beta_n)\geq \frac{1}{2}$ for all $n\geq n_0$. We first focus on one $k_{n+1}$-ball. It contains $N^{k_{n+1}-k_n}\;k_n$-balls. The probability that a $k_n$-ball is $\beta_n$-good is given by $p^G_n (\beta_n)$ (see (\ref{eq:6.1})). The number ${\mathcal N}_n$ of  $\beta_n$-good $k_n$-balls in the $k_{n+1}$-ball is ${\rm Bin} (N^{k_{n+1}-k_n}, p^G_n (\beta_n))$. From (\ref{eq:6.10}), (\ref{eq:6.9}) and Corollary \ref{C5.3},

\begin{eqnarray*}
1-p^B_n (\beta_n, \varepsilon_n) &=& P\biggl(\frac{{\mathcal N}_n}{N^{k_{n+1}-k_n}} <(1-\varepsilon_n) p^G_n (\beta_n)\biggr)\\
&=& P\biggl(1-\frac{N^{k_{n+1}-k_n}-{\mathcal N}_n}{N^{k_{n+1}-k_n}} <
p^G_n (\beta_n) -\sigma_n (1-p^G_n (\beta_n))\biggr)\\
&\leq & \exp \{-\kappa \sigma^2_n (1-p^G_n (\beta_n))N^{k_{n+1}-k_n}\},
\end{eqnarray*}
where
$$
\sigma_n =\varepsilon_n \frac{p^G_n(\beta_n)}{1-p^G_n (\beta_n)},
$$
hence
$$
1-p^B_n (\beta_n, \varepsilon_n) \leq  \exp\biggl\{-\kappa \frac{(p^G_n (\beta_n))^2}{1-p^G_n (\beta_n)} \varepsilon^2_n N^{k_{n+1}-k_n}\biggr\}.
$$
Since we assumed that $p:=p^G_n (\beta_n) \geq \frac{1}{2}$, then
$p^2 \geq \frac{1}{2} (1-p)$, so
\begin{equation}
\label{eq:6.16}
1-p^B_n (\beta_n, \varepsilon_n)\leq \exp\biggl\{- \frac{\kappa}{2}\varepsilon^2_n N^{k_{n+1}-k_n}\biggr\}.
\end{equation}
Using $\varepsilon_n=n^{-(1+\theta)}$ from (\ref{eq:6.16}) and (\ref{eq:2.4}),
\begin{equation}
\label{eq:6.17}
1-p^B_n \biggl(\beta_n, \frac{1}{n^{1+\theta}}\biggr)\leq \exp \biggl\{-
\frac{\kappa}{2}\frac{1}{n^{2(1+\theta)}} N^{K\log n}\biggr\} =
\exp \biggl\{-\frac{\kappa}{2} n^{(K\log N-2(1+\theta))}\biggr\}.
\end{equation}

We can then conclude from (\ref{eq:6.17}) and (\ref{eq:5.26}) that
$$
\sum_n \biggl(1-p^B_n \biggl(\beta_n, \frac{1}{n^{1+\theta}}\biggr)\biggr)<\infty,
$$ if $ p^G_n(\beta_n)\geq \frac{1}{2}$ for all $n\geq n_0$,
so (\ref{eq:6.14}) will be verified.

\bigskip
\noindent
Step 2. We  prove the result for a fixed $b$ satisfying
\begin{equation}\label{eq:3.2x} \frac{2}{\log N}<K<b<2K-\frac{1}{\log N},
\end{equation}
and then recalling Remark 2.2 observe that a simple coupling argument shows that the result remains true for all larger values of $b$.

Assume that $\beta_n\geq \frac{1}{5}$ for $n\geq n_0$. This assumption will be verified in Step 3.
 Define another constant
 \begin{equation}
 K_1=2K-b,
 \end{equation}
so that
\begin{equation}\label{old5.33}
\frac{1}{\log N}<K_1<K.
\end{equation}

Recalling (\ref{eq:5.3}), (\ref{eq:6.5}) and conditioning on the event
$\{\mathcal{N}_n\geq N^{K_1\log n}\}$,  we have
$$ p^A_n(\beta_n)\geq P\left(G\left(\mathcal{N}_n,r_n(\beta_n)\right)\hbox{ is connected}|\;\mathcal{N}_n\geq N^{K_1\log n}\right)P(\mathcal{N}_n\geq N^{K_1\log n}),$$
and therefore
\begin{eqnarray}\label{eq:5.32}
&&1-p^A_n(\beta_n)\leq P\left(G\left(\mathcal{N}_n,r_n(\beta_n)\right)\hbox{ is not connected}|\;\mathcal{N}_n\geq N^{K_1\log n}\right)\\
&&\qquad\qquad\qquad\qquad  +1- P(\mathcal{N}_n\geq N^{K_1\log n}).\nonumber
\end{eqnarray}
Since $p^G_n(\beta_n)\geq\frac {1}{2}$ for large $n$, $\varepsilon_n\to 0$ and $K_1<K$ by (\ref{old5.33}), then, for large $n$,
$$
N^{K_1\log n}<(1-\varepsilon_n)p^G_n(\beta_n)N^{K\log n},$$
so, by Step 1,
\begin{equation}\label{eq:5.33x}
\sum_n(1-(P(\mathcal{N}_n\geq N^{K_1\log n}))<\infty.
\end{equation}

Using $N^{K_1\log n}\leq\mathcal{N}_n\leq N^{K\log N}$, assuming $\beta_n\geq\frac{1}{5}$ and taking $a>25K\log N$, and applying the inequality in the Appendix we obtain
\begin{eqnarray*}
&& P\left(G\left(\mathcal{N}_n,r_n(\beta_n)\right)\quad\hbox{\rm is not connected}|\;\mathcal{N}_n\geq N^{K_1\log n}\right) \\
&&\leq M[(\log n)^{13}n^{K_1\log N\cdot(1-a/25K\log N)}+n^{-K_1\log N}+e^{-L(\log n)^3 n^{2 K_1\log N}}],
\end{eqnarray*}
where $M$ and $L$ are positive constants.
Since $K_1\log N>1$, the sum of the second terms converges, and the sum of the third term also converges.  The sum of the first terms converges if
$$ a>25\left(K\log N+\frac{K}{K_1}\right)=: a_*.$$

Hence for $a>a_*$, together with (\ref{eq:5.32}), (\ref{eq:5.33x}) we have
$$ \sum_n(1-p^A_n(\beta_n))<\infty$$
uniformly for $\beta_n\geq \frac{1}{5}$.
which implies (\ref{eq:6.15}).

\noindent
Step 3. We must show that the assumptions  on $\beta_n$ and $p^G_n(\beta_n)$  used in steps 1 and 2 are self-consistent, that is, we can choose $n_0$, $\beta_{n_0}$, and $p^G_{n_0}(\beta_{n_0})$  such that
\begin{eqnarray}
&&\label{eq:5.33} p^G_{n}(\beta_{n})\geq \frac{1}{2} \hbox{   for all  }n\geq n_0,\\
&&\label{eq:5.34} \beta_n\geq\frac{1}{5} \hbox{   for all  }n\geq n_0.
\end{eqnarray}
\medskip

We proceed as follows.  Given $\theta$ satisfying (\ref{eq:5.26}) and recalling  (\ref{eq:6.14}), (\ref{eq:6.15}) we can choose $n_0$ such that the following  products satisfy:
\begin{eqnarray}
  &&\label{eq:5.35}\prod_{n\geq n_0}\biggl(1-\frac{1}{n^{1+\theta}}\biggr)\geq \biggl(\frac{4}{5}\biggr)^{1/3},\\
  &&\text{and for any  }k\in\mathbb{N}\nonumber\\
&&\label{eq:5.37}\prod_{n=n_0}^{n_0+k}p^B_n\biggl(\beta_n,\frac{1}{n^{1+\theta}}\biggr) \geq \biggl(\frac{4}{5}\biggr)^{1/3}\\&&\qquad\qquad \hbox{ provided that }p^G_n(\beta_n)\geq \frac{1}{2}\hbox{   for }n= n_0,\dots,n_0+k\nonumber\\
 &&\label{eq:5.36}\prod_{n= n_0}^{n_0+k}p^A_n(\beta_n) \geq \biggl(\frac{4}{5}\biggr)^{1/3}\\&&\qquad\qquad \hbox{ provided that }\beta_n\geq \frac{1}{5}\hbox{   for  }n= n_0\dots,n_0+k.\nonumber
\end{eqnarray}

Now choose
\begin{equation}\label{eq:3.n0} \beta_{n_0}\geq \frac{1}{2}\end{equation}
and
choose $C$ in (\ref{eq:2.5}) sufficiently large so that (see (\ref{eq:6.1}))
\begin{equation}
\label{eq:5.30}
p^G_{n_0} (\beta_{n_0})\geq \frac{2}{3}.
\end{equation}

Then by (\ref{eq:6.11}) and (\ref{eq:5.35})
\begin{eqnarray} &\beta_{n_0+1} =\left(1-\frac{1}{n_0^{1+\theta}}\right)p_{n_0}^G(\beta_{n_0})\beta_{n_0}
 \\ &\quad\;\geq  \biggl( \frac{4}{5}\biggr)^{1/3}\cdot \frac{2}{3}\cdot \frac{1}{2} >\frac{1}{5},\nonumber
\end{eqnarray}
and by (\ref{eq:6.12}), (\ref{eq:5.37}) and (\ref{eq:5.36}),  we  have
\begin{eqnarray}
&&\label{eq:5.41}p^G_{n_0+1}(\beta_{n_0+1})\geq p^B_{n_0}\biggl(\beta_{n_0},\frac{1}{n_0^{1+\theta}}\biggr)p^A_{n_0}(\beta_{n_0})\\
&&\geq \biggl(\frac{4}{5}\biggr)^{2/3}\geq \frac{1}{2}.\nonumber
\end{eqnarray}
Now assume that
\begin{equation}
\label{eq:5.42}
\prod_{\ell=1}^{k-1}p^G_{n_0+\ell}(\beta_{n_0+\ell})\geq \biggl(\frac{4}{5}\biggr)^{2/3},
\end{equation}
and $\beta_{n_0+\ell}\geq \frac{1}{5}$ for $\ell=1,\dots,k-1$.
\medskip

Then by (\ref{eq:6.12}),  (\ref{eq:5.37}) and (\ref{eq:5.36}) we  have
\begin{eqnarray}
&&\label{eq:5.41x}\prod_{\ell =1}^{k}p^G_{n_0+\ell}(\beta_{n_0+\ell})\\&&\geq \prod_{\ell =0}^{k-1} p^B_{n_0+\ell}\biggl(\beta_{n_0+\ell},\frac{1}{(n_0+\ell)^{1+\delta}}\biggr)p^A_{n_0+\ell}(\beta_{n_0+\ell})\nonumber\\
&&\geq \biggl(\frac{4}{5}\biggr)^{2/3}\geq \frac{1}{2}.\nonumber
\end{eqnarray}
Moreover by (\ref{eq:6.11}), (\ref{eq:5.35}), (\ref{eq:5.30}) and (\ref{eq:5.42}),
 \begin{eqnarray*}
&&\beta_{n_0+k}=\beta_{n_0}\prod_{j=1}^k\frac{\beta_{n_0+j}}{\beta_{n_0+j-1}}\\
&&=\beta_{n_0}\prod_{j=0}^{k-1}\biggl(1-\frac{1}{(n_0+j)^{1+\theta}}\biggr)\prod_{j=0}^{k-1}p^G_{n_0+j}(\beta_{n_0+j})\\
&&\geq \frac{1}{2}\cdot \biggl(\frac{4}{5}\biggr)^{1/3}\cdot  \frac{2}{3}\cdot   \biggl(\frac{4}{5}\biggr)^{2/3} >\frac{1}{5}.
\end{eqnarray*}

Therefore by induction we have
\begin{equation}\label{eq:5.43} \prod_{n\geq n_0+1}p^G_n(\beta_n)\geq\biggl(\frac{4}{5}\biggr)^{2/3}
\end{equation}
and
\begin{equation}\label{eq:5.44}\beta_n\geq \frac{1}{5} \text{ for all  }n\geq n_0.
\end{equation}

Step 4. For  $a>a_*$, $n_0$ and $C$ chosen above we then have
$$
\beta_n \geq \frac{1}{5}\quad{\rm and}\quad p^G_n (\beta_n)\geq \frac{1}{2}\quad\hbox{\rm for all}\quad n\geq n_0.
$$
This together with (\ref{eq:5.35}), (\ref{eq:5.36})  yields the estimate (\ref{eq:6.13}) and
\begin{equation}
\prod_{n\geq n_0} p^G_n(\beta_n)>0,
\end{equation}
which implies the assumption of Lemma \ref{L5.5'}.
Then
(\ref{eq:6.13}) implies (\ref{eq:3.2}) (for some $n_{00}\geq n_0$) and (\ref{eq:5.44}) implies (\ref{eq:3.1}). Percolation then follows from (\ref{eq:3.1}), (\ref{eq:3.2}) and Lemma \ref{L5.5'}.
The uniqueness of the infinite cluster again follows from \cite{KMT}, Theorem 2.

(2) The proof follows immediately from Lemma  \ref{L5.7}.

\medskip

\noindent
(c)   Consider the  case $0<b\leq \frac{2}{\log N}<K$ but modifying the model by replacing the actual connection probabilities at distances $k_n+1,\dots,k_{n+1}$ with the lower
bound $\frac{c_{k_{n}}}{N^{2k_{n+1}}}$. In this case the lower bound on the connection probabilities in Lemma \ref{L5.5} can be replaced by the upper bound

\[ P( X_{k_n}\;{\rm and}\;{X'}_{k_n}\;\hbox{\it are connected within the}\; k_{n+1}\hbox{\it -ball})\leq \widetilde r_n\quad \rm{as}\;\;n\to\infty,\]
where
\[ \widetilde r_n(\beta)=  \frac{a\log n}{N^{(K-2/\log N)\log n}}\frac{1}{N^{K\log n}}.\]
We can then consider the E-R graph $G(\mathcal{N}_n,\widetilde r_n(\beta))$.
Assuming that $\mathcal{N}_n$ is of order $ N^{K\log n}$, in this case by Erd\H{o}s-R\'enyi theory the resulting random graph has only of order $\log(N^{K\log n})$ good $k_n$-balls in in the largest connected component in the $k_{n+1}$-ball . This  would imply that the limiting density of the largest connected component  in the $k_n$-balls decreases to 0 as $n\to\infty$ so that percolation does not occur.

To make this precise consider $G(\mathcal{N}_n,\lambda/\mathcal{N}_n)$ with $0<\lambda <1$ and  $|\mathcal{C}_n|$  the size of the largest connected component. We have
\[ \alpha(\lambda) =\lambda-1-\log \lambda >0,\]
and there exists $n_0$ such that for $n\geq n_0$,   $\frac{a\log n}{N^{(K-2/\log N)\log n}}<\lambda$. Then, given $\varepsilon >0$, for $n\geq n_0$,
\[ P(|\mathcal{C}_n|\geq (1+\varepsilon)(\log \mathcal{N}_n)/\alpha(\lambda))\leq \mathcal{N}_n^{-(1+\varepsilon)}/\lambda,\]
(see \cite{D}, page 39).

We then have that the probability that there are more than $\frac{(1+\varepsilon)}{\alpha(\lambda)}\log \mathcal{N}_n$ good $k_n$-balls in the largest connected component in the $k_{n+1}$-ball for infinitely many $n$ is $0$.  Therefore there cannot be an infinite connected component (with positive density).

\hfill $\Box$

\subsection{Proof of Theorem \ref{MR-1}}
\noindent
(a)
Note that if $c_k=C_0+C_1\log k+ C_2k^\alpha$, with $\alpha >2$, then in Theorem 3.5 we can choose $b,K$ such that
\[ \frac{2}{\log N} <K<b< \frac{\alpha}{\log N}.\]
Then
\begin{eqnarray*}  &&c_{k_n}=C_0+ C_1\log\lfloor Kn\log n\rfloor+ C_2\lfloor K^\alpha n^\alpha(\log n)^\alpha\rfloor
\geq C+a\log n\cdot n^{b\log N}\end{eqnarray*}
for sufficiently large $C_0$ and $C_2$, where $C$ and $a$ are as in Theorem 3.5(b).
The proof follows then from (2.3), (2.5), (2.6) and the assumptions on $b$ in Theorem 3.5(b) and Remark 2.2.

\medskip

\noindent
(b) If $C_2=0$ and $C_1<N$, then $c_{k_n}= C_0+C_1 (\log K+ \log n+\log\log n)\leq \widetilde C_0+aN\log n$  for some $0<a<1$ and $\widetilde C_0> C_0$ (with $K=1$ if $N\geq 3$). The result then
follows from Theorem 3.5(a)(i).

\medskip

\noindent
(c) The existence of $C_*$ follows by the argument in \cite{KMT} (Theorem 1(b)) as follows. The expected number of edges from a given
vertex is (see (2.1))
\[ \sum_{k=1}^\infty(N-1)N^{k-1}p_{(k)} \leq \sum_{k=1}^\infty (N-1)N^{k-1}\frac{(C_0+ C_1\log k+C_2k^\alpha)}{N^{2k}}\]
which is less than $1$ for sufficiently small $C_0,C_1,C_2$. The result follows by coupling the largest connected cluster containing a given point with a subcritical branching process  (see e.g. \cite{JLR}, page 109).
\hfill $\Box$

\bigskip

\noindent
{\bf Appendix. Connectivity of a random graph}
\vglue .25cm
Consider the E-R random graph $G(n, \frac{a \log n}{n})$, $a>0$. Using a random walk approximation for cluster growth in a susceptible-infected-removed epidemic model, Durrett \cite{D} proves the known result that $P(G(n, \frac{a \log n}{n})$ is connected) $\rightarrow 1$ as $n\rightarrow \infty$ if $a>1$. Putting together the parts of the proof  one obtains (see p. 64) the lower bound for $a>1$,
\begin{eqnarray*}
\lefteqn{P\biggl(G\biggl(n, \frac{a\log n}{n}\biggr)\;\quad \hbox{\rm is connected}\biggr)}\\
&\geq& \biggl[\biggl(1-\frac{14 (a \log n)^{13}e^{(13a\log n)/n}}{n^a}\biggr)\biggl(1-\frac{1}{n^{2.1}}\biggr)\biggl(1-\frac{1}{n^2}\biggr)\biggr]^n \\ && \cdot (1-e^{-(\log n)^3/100})^{n(n-1)}.
\end{eqnarray*}
Then using the inequalities $1-x>e^{-2x},\; 0<x<0.7968$, and $1-e^{-x}<x,\; x>0$ it follows that for $a>1$,
\begin{eqnarray*}
\lefteqn{P\biggl(G\biggl(n, \frac{a\log n}{n}\biggr)\;\quad \hbox{\rm is not connected}\biggr)}\\
&\leq & M[(\log n)^{13}n^{1-a}+n^{-1}+\exp(-L(\log n)^{13}n^2)],
\end{eqnarray*}
where $M$ and $L$ are positive constants.

\bigskip
\noindent
{\bf Acknowledgment.} We thank an anonymous referee for comments that helped us to improve the paper.

redpajama_set_name: RedPajamaArXiv


ok.. this is one of my most favorite projects, mostly because it was like 30 minutes from end to end for the reward of a completed project.
I have a full tutorial at my blog.. but I'm already working on a crazy floor lamp with 5 record shades and a glitter base... teheheheh..
Sewing skills now available in Mii form!
that's a really neat idea!
Oh, this is fabulous! Thanks for sharing.
God doesn't send people to hell. He simply honors their choice.


This restaurant, located at the Avenue des Champs Elysées in Paris, is one of this company's flagship restaurants, with global relevance due to the architecture and size of the restaurant, and to the exclusive location near famous sights in the city center of Paris. It is therefore essential that the restaurant takes care of its public image.
Secath Concept, an engineering company located in Paris, contacted AAF to assist in solving air filtration issues that the restaurant was having, especially problems with odour, as offices in the neighborhoods nearby were complaining about pungent odours from rancid cooking oil.
Instead of simply offering odour control gas-phase filters to Secath Concept, AAF filtration experts first did a comprehensive technical audit.
It is not a revolutionary finding that the engineering consultants, together with AAF experts, agreed to recommend the installation of a multi-stage filtration system.
Besides the standard function of a filter system, namely to filter particles, this system needed the ability to filter gaseous contaminants responsible for the restaurant's offending odours. To eliminate these odours, analysis determined that the chemical filtration stage of the system needed to filter nitrogen oxide (NO2), Volatile Organic Compounds (VOC's) and ozone (O3).
After taking all filtration and engineering issues into consideration, AAF recommended a three-stage filter system with four filters per stage.
Crucial to the design of the filter housing was the total weight of the unit, as it had to be installed on the rooftop of the building.
The selected VariSorb® XL filter effectively removes VOCs, sulphur dioxide, nitrogen dioxide, and ozone. Moreover, the corrosion-free, non-metal, and lightweight construction of the filter, compared to filter canisters, helped to achieve the goal of the total weight of the filtration unit.
The VariSorb XL filter is designed to improve indoor air quality through the effective removal of indoor and outdoor gaseous contaminants, including VOCs, sulphur dioxide, nitrogen dioxide, and ozone.


To create a platform for academia, industry and R&D to share knowledge.
To create a forum for interaction among academicians, industrialists, scientists and engineers in the broad areas of advanced computing and communications.
To collaborate with universities, societies, institutions and other bodies in India and aboard.
To organize conferences, workshop, seminars, courses, tutorials and exhibitions in the area of advanced computing and communications technologies.
To undertake specific actions that can guide national policies and programmes.
To publish books, journals and newsletters related to advanced computing and communications.
To organize lectures by scientists and experts and to disseminate their ideas and concepts among the Indian advanced computing and communications community.
To undertake or promote studies and study tours to examine specific areas of advanced computing and communications.
To institute stipends, scholarships, fellowships, prizes, etc. to financially assist outstanding students and individuals in accordance with their merits.
To institute awards to honor individuals for their outstanding contributions in the field of advanced computing and communications.
To publish books, journals and newsletters relating to advanced computing and communications.
To do such other things as may be conducive or incidental to the attainment of any or all of the above objectives.


Participants will take a quiz and see how they score in their philanthropic actions. Each set of questions moves them from individual involvement to group philanthropic actions.
The facilitator explains that this activity will help the participants understand more about their current and past involvement in philanthropic activities. The participants will learn more about the concepts that make up philanthropic action.
Each set of actions includes one for individual action and one for moving beyond "self" to involving others. The actions involve the giving of one's time, talent, and treasure for the common good. This activity reinforces the definition of philanthropy used throughout the lessons: thoughtful, intentional, and ongoing giving of your time, talent, and treasure for the common good.
The facilitator explains that points will be given for TIME, TALENT, and TREASURE actions. The facilitator will call out a series of actions and if the participants have performed the action, they are to place the appropriate number of points (as stated by the facilitator) in the blank next to the question. At the end of the questions, the participants will total the numbers for a "philanthropic action score".
Facilitator note: Once the participants have totaled their scores, ask the participants to raise their hands if they have scored at least 25, 50, or 100 to see the ranges. Take a few minutes to discuss the questions.
Celebrate the philanthropic actions that the youth have accomplished. It is important to connect the dots of how these questions represent the sharing of someone's time, talent and treasure; therefore, by taking these actions, they are philanthropists.


b. you have lent money to the company.
When you buy a bond, you are lending money to the company. The company promises to pay you interest and to return your money on a date in the future. This promise generally makes bonds safer than stocks, but bonds can be risky. To assess how risky a bond is you can check the bond's credit rating. Unlike stockholders, bond holders know how much money they will make, unless the company goes out of business. If the company goes out of business, bondholders may lose money, but if there is any money left in the company, they will get it before stockholders.


Erik Haula contract will be expiring at the end of this season. The Nashville Predators have 3 more attempts available to try to sign my client to an extension before he goes off to free agency.
The market value for his services is currently a 2-year contract worth a total of $1,538,000, which equals $769,000 per season.
[2017-11-23 8:50:58 PM] - Erik Haula of Predators was sent to pro.
[2017-09-25 12:21:03 AM] - Erik Haula of Senateurs was sent down to farm.
[2017-09-27 9:09:44 PM] - Erik Haula of Senateurs was sent to pro.
[2017-10-14 3:56:42 AM] - Erik Haula of Senateurs was sent down to farm.
[2017-10-28 10:39:34 PM] - Erik Haula of Senateurs was sent to pro.
[2017-11-10 8:47:49 PM] - Erik Haula of Senateurs was sent down to farm.
[2017-11-22 9:59:37 PM] - Erik Haula was added to Predators.
[2017-11-22 9:59:39 PM] - TRADE : From Senateurs to Predators : Erik Haula (67).
[2017-09-25 00:21:03] - Erik Haula of Senateurs was sent down to farm.
[2017-09-27 21:09:44] - Erik Haula of Senateurs was sent to pro.
[2017-10-14 03:56:42] - Erik Haula of Senateurs was sent down to farm.
[2017-10-28 22:39:34] - Erik Haula of Senateurs was sent to pro.
[2017-11-10 20:47:49] - Erik Haula of Senateurs was sent down to farm.
[2017-11-22 21:59:37] - Erik Haula was added to Predators.
[2017-11-22 21:59:39] - TRADE : From Senateurs to Predators : Erik Haula (67).
[2017-11-23 20:50:58] - Erik Haula of Predators was sent to pro.
[2017-06-26 11:58:34] - Erik Haula signed with Senateurs for 710Â 000 $ for 2 year(s) with 140Â 000 $ in bonus.
[2017-06-26 11:58:34] - Erik Haula average salary was modified from 450Â 000 $ to 710Â 000 $.
[2017-06-26 11:58:34] - Erik Haula was added to Senateurs.


text:
{% extends "templates/layout-question.html" %}

{% block body_classes %}version-1{% endblock %}

{% block page_title %}
  Do you have any children under the age of 18 living in your household?
{% endblock %}

{% block question_number %}11{% endblock %}
{% block previous_url %}housing{% endblock %}
{% block next_url %}children-under-18{% endblock %}

{% block question_content %}

	    	<input type="radio" id="children-under-18-yes" name="children-under-18" value="children-under-18-yes">
	      	Yes

	      	<input type="radio" id="children-under-18-no" name="children-under-18" value="children-under-18-no">
	      	No

{% endblock %}

redpajama_set_name: RedPajamaGithub


Have you experienced a past life regression? Are past lives real? Does it matter? As a therapy, visiting a past life can be very effective. Not all hypnotherapists conduct past life regression therapy sessions. However, do you know what to do when your client spontaneously goes to an apparent past life? Will you lose credibility if you don't know how to facilitate their session? How do you help a client if you haven't been trained in past life regression?
Do you really want to send them off to another hypnotherapist? Learn these valuable techniques and help more clients.
You may begin by moving yourself to a time when you were in your teens. You will find that it's very easy to do. Choose a memory of that time. Choose a scene or an event that you would like to relate from your teen age years. With a private client: Bring the saliva to your mouth and let your lips become moist so that you may speak easily…. and now, you're in your teens. Describe what you're experiencing — what you're doing, or seeing, or hearing, or feeling, or sensing?
That's good. Now, continue moving to a time when you were about three or four or five years old. Again, choose an experience and focus on the memory……. You're now three or four or five years old. What's happening?
You're doing very well. Now continue to move beyond that time. Move to the time of your birth. If you want to do so, you may pause to note the time of your birth. (Ask any questions which seem necessary.) ……. and now, move beyond the womb experience. Move to the place where you were before you entered the body. Some people refer to it as a Blue Mist. It could be a little different for you. It is abounding space; a room without walls or ceiling; a timeless place. It is a place of inner peace and gentle rhythms.
Prefer the hard copy version?


Mecklenburg County, York County, Stanly County, Gaston County, Lancaster County and Union County.
I've been a full time Real Estate Broker for over 24 years and I'm originally from Roanoke, Virginia where I was an award winning Agent with Long & Foster Real Estate. I began my career in Charlotte in 2007 and work with Wilkinson & Associates ERA.


text:
"""TraceMiner Node2Vec and RNN on Pytorch."""

import json
import os
import time

from absl import app
from absl import flags
from absl import logging
import numpy as np
import pandas as pd
from sklearn import metrics
from tensorflow_enterprise_addons.cloudtuner import optimizer_client
import torch
from torch import nn

if not __package__:
  import models  # pylint: disable=g-bad-import-order,g-import-not-at-top
  import utils  # pylint: disable=g-bad-import-order,g-import-not-at-top
  import utils_gcs  # pylint: disable=g-bad-import-order,g-import-not-at-top
else:
  from gnns_for_news import models  # pylint: disable=g-bad-import-order,g-import-not-at-top
  from gnns_for_news import utils  # pylint: disable=g-bad-import-order,g-import-not-at-top
  from gnns_for_news import utils_gcs  # pylint: disable=g-bad-import-order,g-import-not-at-top

FLAGS = flags.FLAGS
flags.DEFINE_string('task', 'cat', ('task: sr(subreddit classification), '
                                    'cat(url categorization), '
                                    'fake(fake news detection)'))
# TODO(huangdiana): convert all default args for strings to ''.
flags.DEFINE_string('gcs_path_in', None, 'gcs bucket input directory')
flags.DEFINE_string('gcs_path_out', None, 'gcs bucket output directory')
flags.DEFINE_string('local_path_in', './fake_input/', 'local path for input')
flags.DEFINE_string('local_path_out', './fake_output/', 'local path for output')
flags.DEFINE_string('g_emb', None, 'graph embedding file')
flags.DEFINE_string('seq_file', None, 'post sequence file')
flags.DEFINE_string('balance_df', '', 'the balanced dataset with url ids')
# RNN, LSTM parameters
flags.DEFINE_string('model', 'rnn', 'rnn, lstm')
flags.DEFINE_float('train_ratio', 0.8, 'training data ratio')
flags.DEFINE_float('val_ratio', 0.1, 'validation data ratio')
flags.DEFINE_integer('batch_size', 64, 'bacth size for rnn')
flags.DEFINE_integer('dim', 128, 'graph embedding dimension')
flags.DEFINE_integer('hid_dim', 32, 'hidden dimension in RNN, LSTM')
flags.DEFINE_integer('num_layers', 2, 'number of layers in RNN, LSTM')
flags.DEFINE_boolean('bi', False, 'birectional')
flags.DEFINE_float('dropout', 0.8, ('dropout rate, will be auto-set to 0.0 if'
                                    'num_layers=1'))
flags.DEFINE_integer('epochs', 40, 'epochs')
flags.DEFINE_float('lr', 0.002, 'learning rate')
flags.DEFINE_integer('print_step', 10, 'print step during training')
flags.DEFINE_boolean('save_model', False, 'save model')
flags.DEFINE_string('name', '', 'specify mode name')

# XCloud parameters
flags.DEFINE_string('trial_name', None,
                    'Identifying the current job trial for addMeasurement')
flags.DEFINE_string('study_config', None,
                    'Study configuration for CAIP Optimizer service')
flags.DEFINE_boolean('use_optimizer', False,
                     'Use dynamic job generation for hparam optimization')

# Flag specifications
flags.mark_flag_as_required('gcs_path_in')
flags.mark_flag_as_required('gcs_path_out')
flags.mark_flag_as_required('g_emb')
flags.mark_flag_as_required('seq_file')

def print_gpu_info():
  """print gpu info."""
  logging.info('torch.__version__: %s', torch.__version__)
  logging.info('torch.cuda.device_count(): %s', torch.cuda.device_count())
  logging.info('torch.cuda.is_available(0): %s', torch.cuda.is_available())
  if torch.cuda.is_available():
    logging.info('torch.cuda.current_device(): %s', torch.cuda.current_device())
    logging.info('torch.cuda.get_device_name(0): %s',
                 torch.cuda.get_device_name(0))

def train(model_tm, x, y, train_batches, criterion, optimizer,
          device, print_step):
  """Trains Traceminer model.

  Args:
    model_tm: An instance of RNN or LSTM class.
    x: A list of url sequences containing post author embeddings. Each url
      sequence is a numpy.array of size (number of users, graph embedding size).
    y: A one-dimensional numpy.array of labels.
    train_batches: A list of one-dimension numpy.arrays. Each numpy.array has
      indexes of input x, label y for each batch during training.
    criterion: Model loss function.
    optimizer: Model optimizer.
    device: A string of device type, cpu or gpu.
    print_step: Number of steps for logging information.
  """
  model_tm.train()
  for i, batch in enumerate(train_batches):
    batch_x = [torch.from_numpy(x[idx]) for idx in batch]
    batch_x = utils.data_padding(batch_x).to(device)
    batch_y = torch.from_numpy(y[batch]).to(device)
    optimizer.zero_grad()
    output = model_tm(batch_x)
    loss = criterion(output, batch_y)
    loss.backward()
    optimizer.step()
    pred = nn.functional.log_softmax(output, 1)
    acc = metrics.accuracy_score(batch_y.data.cpu().numpy(),
                                 torch.argmax(pred, 1).data.cpu().numpy())
    if i % print_step == 0:
      logging.info('Train Batch [%s/%s] | Loss: %.4f | Train Acc : %.4f',
                   i, len(train_batches), loss.item(), acc)

def val(model_tm, x, y, val_batches, device):
  """Evaluates Traceminer model on validation dataset.

  Args:
    model_tm: An instance of RNN/LSTM class.
    x: A list of url sequences containing input features. Each url
      sequence is a numpy.array of size (number of users, user embedding).
    y: A one-dimensional numpy.array of labels.
    val_batches: A list of one-dimension numpy.arrays. Each numpy.array has
      indexes of input x, label y for each batch during validation.
    device: A string of device type, cpu or gpu.

  Returns:
    val_f1: F1 score for the validation dataset, used as metrics for
    hyperparamter searching.
  """
  model_tm.eval()
  pred_y, true_y = [], []
  with torch.no_grad():
    for batch in val_batches:
      batch_x = [torch.from_numpy(x[idx]) for idx in batch]
      batch_x = utils.data_padding(batch_x).to(device)
      batch_y = torch.from_numpy(y[batch]).to(device)
      output = model_tm(batch_x)
      pred_y += torch.argmax(output, 1).data.cpu().tolist()
      true_y += batch_y.data.cpu().tolist()
  if len(set(y)) == 2:
    logging.info(('Validation Acc : %.4f | F1 : %.4f'),
                 metrics.accuracy_score(true_y, pred_y),
                 metrics.f1_score(true_y, pred_y))
    val_f1 = metrics.f1_score(true_y, pred_y)
  else:
    logging.info(('Validation Acc : %.4f | Validation Micro-F1 : %.4f | '
                  'Macro-F1 : %.4f | Weighted Macro-F1 : %.4f'),
                 metrics.accuracy_score(true_y, pred_y),
                 metrics.f1_score(true_y, pred_y, average='micro'),
                 metrics.f1_score(true_y, pred_y, average='macro'),
                 metrics.f1_score(true_y, pred_y, average='weighted'))
    val_f1 = metrics.f1_score(true_y, pred_y, average='macro')
  return val_f1

def test(model_tm, x, y, test_batches, device):
  """Evaluates Traceminer model on test dataset.

  Args:
    model_tm: An instance of RNN/LSTM class.
    x: A list of url sequences containing input features. Each url
      sequence is a numpy.array of size (number of users, user embedding).
    y: A one-dimensional numpy.array of labels.
    test_batches: A list of one-dimension numpy.arrays. Each numpy.array has
      indexes of input x, label y for each batch during testing.
    device: A string of device type, cpu or gpu.
  """
  model_tm.eval()
  pred_y, true_y = [], []
  with torch.no_grad():
    for batch in test_batches:
      batch_x = [torch.from_numpy(x[idx]) for idx in batch]
      batch_x = utils.data_padding(batch_x).to(device)
      batch_y = torch.from_numpy(y[batch]).to(device)
      output = model_tm(batch_x)
      pred_y += torch.argmax(output, 1).data.cpu().tolist()
      true_y += batch_y.data.cpu().tolist()
  if len(set(y)) == 2:
    logging.info(('Test Acc : %.4f | F1 : %.4f'),
                 metrics.accuracy_score(true_y, pred_y),
                 metrics.f1_score(true_y, pred_y))
  else:
    logging.info(('Test Acc : %.4f | Test Micro-F1 : %.4f | '
                  'Test Macro-F1 : %.4f | Test Weighted Macro-F1 : %.4f'),
                 metrics.accuracy_score(true_y, pred_y),
                 metrics.f1_score(true_y, pred_y, average='micro'),
                 metrics.f1_score(true_y, pred_y, average='macro'),
                 metrics.f1_score(true_y, pred_y, average='weighted'))

def main(_):
  if not os.path.exists(FLAGS.local_path_in) or FLAGS.use_optimizer:
    utils_gcs.download_files_from_gcs(FLAGS.local_path_in, FLAGS.gcs_path_in)
  logging.info('Data downloaded successfully!')

  sequence_df = pd.read_hdf(
      os.path.join(FLAGS.local_path_in, FLAGS.seq_file), 'df')
  if FLAGS.balance_df:
    balance_df = pd.read_hdf(
        os.path.join(FLAGS.local_path_in, FLAGS.balance_df), 'df')
    sequence_df = sequence_df[sequence_df['url'].isin(balance_df['url'])]
  embeddings_dict = utils.get_n2v_graph_embedding(
      os.path.join(FLAGS.local_path_in, FLAGS.g_emb), graph_gen=False,
      normalize_type='minmax')
  x_sequence, y_label, label_list = utils.load_input_with_label(
      sequence_df, embeddings_dict, FLAGS.task)

  train_idx, val_idx, test_idx = utils.split_data_idx(
      len(x_sequence), FLAGS.train_ratio, FLAGS.val_ratio)
  train_batches = np.array_split(train_idx, len(train_idx) / FLAGS.batch_size)
  val_batches = np.array_split(val_idx, len(val_idx) / FLAGS.batch_size)
  test_batches = np.array_split(test_idx, len(test_idx) / FLAGS.batch_size)

  # model training/testing
  logging.info('FLAGS.epochs: %s', FLAGS.epochs)
  logging.info('FLAGS.batch_size: %s', FLAGS.batch_size)
  logging.info('FLAGS.learning_rate: %s', FLAGS.lr)

  dropout = 0.0 if FLAGS.num_layers == 1 else FLAGS.dropout

  print_gpu_info()
  device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')
  logging.info('Current device is %s', device.type)
  if FLAGS.model == 'rnn':
    tm_model = models.RNN(
        in_dim=FLAGS.dim, hid_dim=FLAGS.hid_dim, num_label=len(label_list),
        num_layers=FLAGS.num_layers, dropout=dropout).to(device)
  elif FLAGS.model == 'lstm':
    tm_model = models.LSTM(
        in_dim=FLAGS.dim, hid_dim=FLAGS.hid_dim, num_label=len(label_list),
        num_layers=FLAGS.num_layers, dropout=dropout,
        bi_direct=FLAGS.bi).to(device)
  criterion = nn.CrossEntropyLoss()
  optimizer = torch.optim.Adam(tm_model.parameters(),
                               lr=FLAGS.lr, weight_decay=1e-6)
  if FLAGS.use_optimizer:
    # example trial_name:
    # 'projects/{project_id}/locations/{region}/studies/{study_id}/trials/{trial_id}'
    trial_name_split = FLAGS.trial_name.split('/')
    project_id = trial_name_split[1]
    region = trial_name_split[3]
    study_id = trial_name_split[-3]
    trial_id = trial_name_split[-1]
    logging.info('project_id: %s, region: %s, study_id: %s, trial_id: %s',
                 project_id, region, study_id, trial_id)
    ml_client = optimizer_client.create_or_load_study(
        project_id, region, study_id, json.loads(FLAGS.study_config))
  for epoch in range(FLAGS.epochs):
    logging.info('Epoch %s', epoch)
    start_time = time.time()
    train(tm_model, x_sequence, y_label, train_batches, criterion,
          optimizer, device, FLAGS.print_step)
    val_f1 = val(tm_model, x_sequence, y_label, val_batches, device)
    test(tm_model, x_sequence, y_label, test_batches, device)
    if FLAGS.use_optimizer:
      elapsed_secs = int(time.time() - start_time)
      metric_list = [{'metric': 'valf1', 'value': float(val_f1)}]
      ml_client.report_intermediate_objective_value(epoch, elapsed_secs,
                                                    metric_list, trial_id)

  logging.info('Experiment finished.')

  if FLAGS.save_model:
    filename = '%s_%s_%s' % (FLAGS.task, FLAGS.model, FLAGS.name)
    utils.save_model(tm_model, optimizer, FLAGS.local_path_out, filename)
    utils_gcs.upload_files_to_gcs(local_folder=FLAGS.local_path_out,
                                  gcs_path=FLAGS.gcs_path_out)

if __name__ == '__main__':
  app.run(main)