text:
\section{SAW tree and recursion}
\label{a:saw}

In this section, we first describe how the Ising model on a graph $G$ translates to an Ising model on the SAW tree $T_v$ defined in Section~\ref{s:ssmsawtree}, and then show how to compute the marginal probabilities for $\si_v$ in $T_v$.

\paragraph{Ising model on the SAW tree.}
Given an Ising model on a graph $G=(V,E)$ with inverse temperature $\be$ and external fields $h$, we obtain an
Ising model on $T_{v}$ by taking the same inverse temperature $\be$ and taking the external field at
$(v_{i})_{i=0}^{k}$ to be $h_{v_{k}}$. If we are given boundary conditions $\tau$ on a set of vertices $\pl V$, then we take the boundary condition at
$(v_{i})_{i=0}^{k}$ to be $\tau_{v_k}$ whenever $v_k\in \pl V$.

Fix a lexicographic order on the vertices $V$; this induces an order on the edges incident to any fixed vertex.
For the vertices $\ga$ in $T_v\setminus \hat T_v$ (those representing paths with a cycle), we instead assign them the following boundary condition:
\begin{align*}
\tau_\ga &= \begin{cases}
1, &\text{if the edge closing the cycle is larger than the edge starting the cycle in }\ga\\
-1, &\text{otherwise.}
\end{cases}
\end{align*}

\paragraph{Marginal probabilities on the SAW tree.}
For simplicity of notation, let $p=p_{T_v, \be, h}$ denote the Ising model on the $T_v$.
For convenience, we work with the occupation ratio
\[
R_v := \frac{p(\sigma_v=-1)}{p(\sigma_v=+1)}.
\]
Recalling the definition of Gibbs measure, we have
\[
R_v =e^{-2h_v}\frac{ p'(\sigma_v=-1)}{p'(\sigma_v = +1)} =e^{-2h_v} \prod_{i=1}^{\deg(v)} \frac{p_{u_i}'(\sigma_v =-1)}{p_{u_i}'(\sigma_v =+1)},
\]
where $p'$ is the Gibbs measure after removing $h_v$ (the external field at vertex $v$), and $u_i$ are the child vertices of $v$.
The measure $p'_{u_i}$ is the Gibbs measure defined on the subtree by removing all other subtrees except the one rooted at
vertex $u_i$. Note that this still includes the root vertex $v$.
We define $p''_{u_i}$ as the Gibbs measure on the subtree rooted at $u_i$, excluding $v$. Then
\[
\frac{p_{u_i}'(\sigma_v =-1)}{p'_{u_i}(\sigma=+1)} = \frac{e^{\beta}p''_{u_i}(\sigma_{u_i}=-1)+e^{-\beta}p''_{u_i}(\sigma_{u_i}=+1)}{e^{-\beta}p''_{u_i}(\sigma_{u_i}=-1)+e^{\beta}p''_{u_i}(\sigma_{u_i}=+1)} = \frac{e^{2\beta}R_{u_i}+1}{R_{u_i}+e^{2\beta}}.
\]
Using $R_v = \fc{p_v}{1-p_v}$ and $p_v = \rc{1+\rc{R_v}}$, we can write this in terms of $p_v :=p(\sigma_v = 1) $,
\begin{align}
\label{e:recursion}
p_v  &=
\rc{1+\rc{e^{-2h_v} \prodo id \fc{e^{2\be}R_{u_i}+1}{R_{u_i}+e^{2\be}}}} =
\frac{1}{1+ e^{2h_v}\prod_{i=1}^d \frac{e^{2\beta}
(1-p_{u_i})
+ p_{u_i}
}{
(1-p_{u_i}
+e^{2\beta}p_{u_i}}
}.
\end{align}
This equation provides a recursive method for computing marginal probabilities. Given boundary conditions $\tau$ on $\pl V$, we set $p_v=1$ or 0 according to whether $\tau_v=1$ or $\tau_v=-1$, and then work our way up to the root vertex.

\section{Algorithms}

\label{a:alg}

We explicitly write out the algorithms for approximate sampling (Algorithm~\ref{a:sampling}) and computation of $Z_{G,\be, h}$ (Algorithm~\ref{a:counting}). These algorithms work for both max-degree $\De$ graphs in Theorem~\ref{thm:main} and $\mathcal G(n,\De/n)$ graphs in Theorem~\ref{thm:main2}, in the appropriate regime and with the appropriate constants.
For the sampling algorithm (Algorithm~\ref{a:sampling}), we repeat the following: estimate the marginal probabilities for an unfixed vertex, use it to sample the spin for the vertex, and then add that value to the boundary conditions.
For estimation of $Z_{G,\be, h}$ (Algorithm~\ref{a:counting}), to see that the product $e^{-H_{\be, h}(\si)} \prod_{i=1}^n r_i$ gives the right answer, note that if $p_{v_i}^*$ are the actual probabilities,
\begin{align*}
    r_i^* :&= \begin{cases}
    \rc{p_{v_i}^*}, & \si_i=1\\
    \rc{1-p_{v_i}^*}, & \si_i=-1
    \end{cases}
    \\
    &= p(\si_i' = \si_i | \si_j' = \si_j \text{ for }j<i) =
    \fc{p(\si_j'=\si_j \text{ for } j\le i-1)}{p(\si_j'=\si_j \text{ for } j\le i)}.
\end{align*}
Then we have a telescoping product
\begin{align*}
    e^{-H_{\be, h}(\si)} \prod_{i=1}^n r_i^*
    &=
    Z_{G,\be, h} \cdot p(\si) \prod_{i=1}^n \fc{p(\si_j'=\si_j \text{ for } j\le i-1)}{p(\si_j'=\si_j \text{ for } j\le i)} = Z_{G,\be, h}.
\end{align*}
With appropriate choice of $c$, we can ensure that for each $i$, with probability at least $1-\de/n$, that
$p_{v_i}\in [p_{v_i}^* e^{-\fc{\ep}{2n}}, p_{v_i}^* e^{-\fc{\ep}{2n}}]$ and
$r_i\in [r_i^* e^{-\ep/n}, r_i^* e^{\ep/n}]$. Then with probability at least $1-\de$, the estimate will be contained in $r_i\in [Z_{G,\be, h} e^{-\ep}, Z_{G,\be, h} e^{\ep}]$.

 \renewcommand{\algorithmicrequire}{\textbf{Input:}}
 \renewcommand{\algorithmicensure}{\textbf{Output:}}

 \begin{algorithm}[h!]
 \caption{Approximate sampling from RFIM}
 \begin{algorithmic}[1]
 \Require Random field Ising model $(G, \be, h)$, failure probability $\de$, accuracy $\ep$.
 \Ensure Approximate sample
 \medskip
 \State Order the vertices $v_1,\ldots, v_n$.
 \For{$i = 1 \to n$}
 	\State Construct the SAW tree at $v_i$, $T_{v_i}$.
 	\State Set boundary conditions $\tau_w'=1$ in
 	$T_{v_i}$ for all
 	$w$ such that $d(v_i, w) >
 	c\max\bc{\log\pf n\ep , \log \pf n\de}$ for an appropriately large constant $c$.
 	Set $p_w=1$ for these $w$.
 	\Comment{Note that arbitrary boundary conditions can be chosen.}
 	\State Set boundary conditions corresponding to $\si_j, 1\le j<i$ in $T_{v_i}$.
 	\State Use recursion~\eqref{e:recursion} to compute $p_{v_i}$.
 	\State Set
 	\begin{align*}
 \si_i &= \begin{cases}
 1,&\text{with probability }p_{v_i}\\
 -1,&\text{with probability }1-p_{v_i}.
 \end{cases}
 \end{align*}
 \EndFor
 \State \Return $(\si_1,\ldots, \si_n)$.
 \end{algorithmic}
 \label{a:sampling}
 \end{algorithm}

 \begin{algorithm}[h!]
 \caption{Approximation of partition function for RFIM}
 \begin{algorithmic}[1]
 \Require Random field Ising model on $G$, failure probability $\de$, accuracy $\ep$ (where desired multiplicative accuracy is $e^\ep$).
 \Ensure Approximation of partition function $Z_{G,\be,h}$
 \medskip
 \State Order the vertices $v_1,\ldots, v_n$.
 \For{$i = 1 \to n$}
 	\State Construct SAW tree at $v_i$,  $T_{v_i}$.
 	\State Set boundary conditions $\tau_w'=1$ in $T_{v_i}$ for all
 	$w$ such that $d(v_i, w) >
 	c\max\bc{\log\pf n\ep , \log \pf n\de}$ for an appropriately large constant $c$. Set $p_w=1$ for these $w$.
 	\Comment{Note that arbitrary boundary conditions can be chosen.}
 	\State Set boundary conditions corresponding to $\si_j, 1\le j<i$ in $T_{v_i}$.
 	\State Use recursion~\eqref{e:recursion} to compute $p_{v_i}$.
 	\If{$p_{v_i}\ge \rc 2$}
 		\State Set $\si_i=1$ and $r_i = \fc{1}{p_{v_i}}$.
 	\Else
 		\State Set $\si_i=-1$ and $r_i = \fc{1}{1-p_{v_i}}$.
 	\EndIf
 	\State Include $\si_i$ as boundary condition in $G'$.
 \EndFor
 \State \Return $e^{-H_{\be, h}(\si)} \prod_{i=1}^n r_i$.
 \end{algorithmic}
 \label{a:counting}
 \end{algorithm}

\section{Introduction}
\label{sec:introduction}

Recent years have seen the development of a rich interplay between statistical physics,
computational complexity, and algorithm design. One central question
is the extent to which phase transitions for discrete statistical
mechanics models are related to the tractability of associated
computational problems. In this paper we are primarily interested in
approximate counting and sampling;
see Section~\ref{sec:defn} for formal definitions.
Results concerning these tasks have
traditionally focused on (i) establishing algorithmic tractability in
so-called `high temperature' (weakly correlated) regimes, and
(ii) establishing the failure of certain algorithmic techniques in
`low temperature' (strongly correlated) regimes. Very recently,
positive algorithmic results have been
obtained at low temperatures~\cite{HelmuthPerkinsRegts,BarvinokRegts,huijben2021sampling},
and occasionally even at all
temperatures~\cite{jerrum1993polynomial,BCHPT,HJP,ALOVII}.

Many of these algorithmic results have
been shown using
a detailed probabilistic and physical understanding of the corresponding
statistical mechanics problems. The intuition gained from this
understanding is typically restricted to specific classes of
graphs, e.g., lattices or specific models of random graphs. One of the
challenges for algorithm design is to go beyond these restricted
classes of graphs, and this can lead to situations in which the
notions of `high temperature', `low temperature', and `phase
transition' are unclear.

This loss of intuition is present
when one asks about the \emph{average-case complexity} of a
well-known \#BIS-hard problem,
the ferromagnetic Ising model
with general vertex-dependent external fields.  Towards understanding this situation, in this paper
we  consider the  Ising model with
vertex-dependent \emph{random} external fields $h_{x}$, $x \in V(G)$,
where the $h_{x}$ are independent and identically distributed (IID)
centered Gaussian.
This model, primarily studied in statistical physics on the integer
lattice $\mathbb Z^d$, is known as the \emph{random field Ising
  model}. A great deal of interest in the random field Ising model has arisen because it behaves differently than the zero-field model with $h_x\equiv0$. To briefly describe this, recall that the zero-field model undergoes a phase transition on $\bb Z^d$ when $d\geq 2$: if $\beta>0$ is small, then correlations decay exponentially and there is a unique infinite volume Gibbs measure. On the other hand, if $\beta$ is large, then correlations do not decay, and multiple Gibbs measures exist. The surprising phenomena is that this picture changes for the random field Ising model: on $\bb Z^d$ for $d\geq 3$ there is still a phase transition if the variance of the random fields $h_x$ is not too large~\cite{BricmontKupiainen}, but on $\bb Z^2$ there is no phase transition if the variance is non-zero~\cite{AizenmanWehr}. In fact, in  recent breakthroughs, it was shown that on $\bb Z^2$ correlations always decay exponentially~\cite{DingXia}, and on $\bb Z^d$, $d \ge 3$, correlations decay exponentially throughout the high-temperature regime~\cite{ding2021new}. It is natural to wonder if there are algorithmic counterparts to these physical phenomena.

While the phase transition phenomena of the previous paragraph concerned small variances, the random field Ising model also exhibits interesting properties (so-called Griffiths singularities) from the point of view of physics when the variance of the external fields is not small.
See Section~\ref{sec:background}. This regime is also \emph{terra incognita} from an algorithmic point of view, and we focus in this paper on the large variances.

It is straightforward, see Section~\ref{sec:largefield}, to design
algorithms for the random field Ising model if one
assumes $|h_{x}|$ is \emph{uniformly} large (depending on the inverse
temperature $\beta$ and on the maximum degree of the graphs being
considered): these large external fields put the system in an
effectively high temperature situation. If, however, $|h_{x}|$ can be
small for some vertices $x$, then
for large inverse temperature $\beta$, highly correlated
subsets of spins may appear --- there can be large `low temperature' islands in a `high
temperature' sea. Our main result shows that if $|h_{x}|$ is
  \emph{typically} large
then for typical realizations of the external fields, the computational
tasks of approximate counting and sampling are tractable. We will
discuss our proof strategy and highlight the barriers faced by other
methods after pausing to give precise formulations of our results.

\begin{definition}
Let $G=(V,E)$ be a finite graph, $h\colon V\to \bb R$, and
$\be\in \bb R$. The \vocab{Ising model} on $G$ with inverse temperature $\beta$ and external fields $h$ is the probability distribution on $\{\pm 1 \}^V$ given by
\begin{equation}
    \label{e:Idef}
    p_{G,\beta,h}(\sigma) = \frac{e^{-H_{\beta,h}(\sigma)}}{Z_{G,\beta,h}}, \quad
    Z_{G,\beta,h} = \sum_{\sigma\in \{\pm 1\}^V} e^{-H_{\beta,h}(\sigma)},
\end{equation}
where the \vocab{Hamiltonian} $H_{\beta,h}$ is the function
\begin{equation*}
    -H_{\beta,h}(\sigma) = \beta\sum_{xy\in E}\sigma_x \sigma_y + \sum_{x\in V} h_x\sigma_x.
\end{equation*}
\end{definition}
A \vocab{random field Ising model} has external fields $h$ that are independent random variables with  prescribed distributions. We use $\P$ to denote the law of the random external fields.  In this paper we will be concerned with random
external fields that are  typically large, which we will model by  centered Gaussians with large variance. Our main theorem in this setting
is the following.
\begin{theorem}
  \label{thm:main}
  For every $\Delta \ge 2$, $\beta \in \bb R$, there exists $H = H(\De,\be)$
  large enough so that for random field Ising models with inverse
  temperature $\beta$ and external
  fields distributed as independent
  $\mathcal{N}(0,H)$ random variables the following holds.  For every graph $G $ of
  maximum degree $\Delta$ on $n$ vertices,
  with probability $1-o(1)$ over the choice of
  random fields, there exists an FPTAS for $Z_{G,\beta, h}$ and
  a polynomial-time sampling scheme.
\end{theorem}

\begin{remark}
  The failure probability (over the randomness of the fields) tends to
  $0$ with the size of the graph $n$.  We can make this failure
  probability arbitrarily small: to achieve failure probability
  $\delta$ requires a factor polynomial in $1/\delta$ in the running
  time of the algorithm.
\end{remark}

\begin{remark}
  The external fields being Gaussian does not play a role in our proof. Theorem~\ref{thm:main} applies more generally to independent
  external fields with
  distributions with the property that
\begin{equation*}
  \label{e:hyp2}
  \P(|h_x|< c)\le p.
\end{equation*}
for
  $c =|\be| \De + \log \De + c_1$ and
  $p = \fc{c_2}{\De^2}$
  for large enough constant $c_1$ and small enough constant $c_2$.
In particular, since $\P_{h\sim N(0,\si^2)} (|h|\le c) \le \sfc{2}{\pi} \fc{c}{\si}$, we see that $H=\Om(\be^2\De^6)$
suffices in the case of Gaussian external fields and $\be$ bounded away from 0.
\end{remark}

\begin{remark}
  \label{rem:check}
  We can efficiently check whether a given instance of the
  external fields satisfy the required conditions in the following
  sense: given $\epsilon>0$ it takes time polynomial in $n$ and
  $1/\epsilon$ to both output an approximation of the partition
  function and to check the conditions that guarantee the
  $\epsilon$-relative accuracy of the approximation.  See
  Proposition~\ref{prop:check}. We emphasize however, that with probability
  $1-o(1)$, a single instance satisfies these conditions for all
  choices of $\epsilon$.
\end{remark}

\begin{remark}
  Theorem~\ref{thm:main} extends in a straightforward manner to the
  setting of edge-specific inverse temperatures $\be_{xy}$ provided
  all $\beta_{xy}$ are bounded in absolute value by a fixed
  $\be>0$. We consider a single inverse temperature for notational
  simplicity.
\end{remark}

\begin{remark}
  Theorem~\ref{thm:main} also applies in the presence of boundary conditions, which arise naturally in our analysis. We define boundary conditions formally in Section~\ref{sec:defn}.
\end{remark}

The key mechanism behind the proof of Theorem~\ref{thm:main} is that
large external fields cause the system to rapidly decorrelate, as
spins tend to align with their external field. We formalize this
decorrelation by generalizing a disagreement percolation argument due
to Camia, Jiang, and Newman~\cite{camia2018note}. The resulting notion
of correlation decay is similar to, but somewhat weaker than, strong
spatial mixing. See Section~\ref{sec:ssmtree}. This correlation decay
property is sufficiently strong to enable a recursive analysis on a
self-avoiding walk (SAW) tree as was pioneered by
Weitz~\cite{weitz2006counting}. Weitz's method for approximately counting weighted independent sets in bounded degree graphs is now known as the `method of correlation decay', and has found numerous applications in approximate counting and sampling and in proving Gibbs uniqueness for spin models on $\mathbb Z^d$, e.g,~\cite{bayati2007simple,sinclair2014approximation,li2013correlation,restrepo2013improved,sinclair2017spatial}.  The analysis of correlation decay algorithms involves proving strong spatial mixing on the SAW tree, usually by means of a contraction argument or a monotonicity argument with respect to boundary conditions.  What is new in our approach is proving a form of strong spatial mixing on the appropriate SAW tree by a probabilistic argument based on percolation theory (more precisely, based on disagreement percolation~\cite{van1993uniqueness,van1994percolation}).

The proof of Theorem~\ref{thm:main} is fairly robust and can be generalized to apply to graphs where the maximum degree is not necessarily bounded. To illustrate this, we use similar ideas to
analyze the random field Ising model on sparse Erd\H{o}s-R\'enyi random graphs. Recall that a graph drawn from $\mathcal G(n,p)$ (an \vocab{Erd\H{o}s-R\'enyi random graph}) is defined as a graph on $n$ vertices $\{v_1,\ldots, v_n\}$ where each potential edge
$\{v_i,v_j\}$, $i\ne j$ independently included
with probability $p$.

\begin{theorem}\label{thm:main2}
For every $\De >1$ and $\beta \in \bb R$, there exists $H$ large enough so that the following holds.  With $G\sim \mathcal G(n,\De/n)$ and independent random external fields distributed as $\mathcal{N}(0,H)$, with probability $1-o(1)$ over the random graph and random fields there exists an FPTAS and polynomial-time sampling scheme for the random field Ising model on $G$ at inverse temperature $\beta$.
\end{theorem}

\begin{remark}
Theorem~\ref{thm:main2} applies more generally to independent external fields with distributions with the property that
\begin{equation*}
  \label{e:hyp}
  \P(|h_x|< c)\le p.
\end{equation*}
for
  $c = c_1(|\be| \De + \log \De + 1)$ and
  $p = \prc{2\De}^{c_2}$
  for constants $c_1,c_2$.
\end{remark}

\subsection{Background}
\label{sec:background}

An important challenge for understanding the relative complexity of approximate
counting was raised by Dyer, Goldberg, Greenhill, and Jerrum in~\cite{dyer2004relative}: how difficult is it to
approximately count independent sets in \emph{bipartite} graphs? This
problem, \vocab{(approximate) \#BIS}, occupies a central place in the analysis
of approximate counting algorithms~\cite{GoldbergJerrum,galanis2016ferromagnetic,liu2014complexity}.
The most relevant fact for us is the \#BIS-hardness of
approximately computing the partition function of the ferromagnetic
Ising model with general vertex-dependent external
fields on bounded-degree bipartite graphs~\cite{cai2016hardness}.

For \#BIS-hardness,
allowing general external fields is
necessary:
it is a classic result
that there are efficient
approximation algorithms for the ferromagnetic Ising model with no
external fields or with consistent external fields (all non-negative or all non-positive) for all values of
$\beta\geq 0$ of the inverse temperature~\cite{jerrum1993polynomial,GoldbergJerrum}. The novelty of
Theorem~\ref{thm:main} is
that it allows for inconsistent
external fields. Standard approaches for analyzing the Glauber
dynamics of the Ising model do not seem capable of proving
Theorem~\ref{thm:main} --- see Remark~\ref{rem:Glauber} below. Note that
if $\be$ is taken sufficiently small, then standard high
temperature methods already apply~\cite{zhang2011approximating}, and
thus the most interesting case of our theorem is when $\be$ is
large.

Another approach to approximation is based on zero-freeness of the
partition function, either via Barvinok's method~\cite{Barvinok} or
cluster expansion methods~\cite{HelmuthPerkinsRegts}.
The main barrier to applying these methods in the case of the random field Ising model is the phenomena of Griffiths
singularities~\cite{vanEnter}.
These singularities arise in spin
systems with random Hamiltonians;
in our case the randomness is contained in the external field.
When the underlying graph is the integer lattice $\mathbb Z^d$, the existence of rare (but
arbitrarily large) regions of atypical behaviour for the random field is
widely believed to lead to thermodynamic functions being infinitely
differentiable but \emph{not}
analytic~\cite{von1995taming,frohlich1984improved,vanEnter}. The
non-analyticity of limiting quantities rules out the existence of
zero-free regions in finite volumes.

\subsection{Future Directions}
\label{sec:disc-future-direct}
Our algorithm is based on Weitz's method of correlation decay on a
computational tree. A natural question is whether Markov chain-based
algorithms
can provide a similar guarantee. See Remark~\ref{rem:Glauber} for indications this may be a subtle question. Our thresholds are certainly improvable, and the
tractability for more moderate values of external field is
unknown, as is tractability in the presence of correlated external fields.

The difference in behaviour for the RFIM in $d=2$ and
$d\geq 3$ suggests that the design of approximate counting algorithms
in the presence of weak disorder is a subtle task, and hence an interesting challenge for future research.
Another interesting direction is to develop algorithms for problems that contain
`high temperature' islands in a `low temperature' sea, i.e., with the
roles of high and low temperature in the present paper exchanged.
Our result does not rely on ferromagnetism, and it is a good
question whether one can obtain a stronger result---such as one that
works for more moderate external fields --- in the ferromagnetic $\beta> 0$ regime.

We end this section by indicating a motivating connection (and potential future direction) between the results of this paper and
\#BIS that does not pass through any formal reductions as in~\cite{cai2016hardness,GoldbergJerrum}. A
difficulty in investigating the complexity of \#BIS is that it is unclear
which instances are hard.  For a single random bipartite graph
(balanced or not), the low temperature behaviour of independent sets
is well understood (see, e.g.,~\cite{mossel2009hardness}):  independent sets typically consist of significantly more
vertices on one side of the bipartition than the other. Thus, in
the search for a hard instance one may be tempted to treat single random
bipartite graphs as gadgets, and to assemble many gadgets together by
adding edges between the gadgets in a bipartite manner. If the density
of added edges is low enough to avoid disrupting the behaviour of
individual gadgets, then in the low temperature regime
the resulting graph heuristically behaves like a ferromagnetic Ising
model. The external field reflects if the constituent graphs are
balanced ($h=0$) or not ($h\neq 0$). Working directly with the Ising
model with an inconsistent magnetic field allows for us to search for
hard instances while bypassing the technicalities that would be
present in making the preceding discussion precise.

\subsection{Organization of the paper}
In Section~\ref{sec:largefield}, we give approximate sampling and counting algorithms in the case that all external fields are large with probability one.
In Section~\ref{sec:ssmtree}, we prove our main theorem (Theorem~\ref{thm:main}). In Section~\ref{sec:extensions}, we prove Theorem~\ref{thm:main2}, the extension of our main result to random graphs. Finally, in Section~\ref{s:non-unif}, we show that our work generalizes one of the main theorems of~\cite{camia2018note} to infinite graphs of max-degree $\Delta$.
In Appendix~\ref{a:saw}, we give details of the SAW tree construction and recursion used by the algorithms, and in Appendix~\ref{a:alg}, we write out the algorithms explicitly.

\subsection{Preliminaries and notation}
\label{sec:defn}

\paragraph{Approximate counting and sampling.}

A \vocab{fully polynomial-time approximation scheme (FPTAS)} for a function $Z(G)$ is a deterministic algorithm that given a graph $G$ and a tolerance $\epsilon>0$ outputs a number $\hat Z$ such that $e^{-\epsilon}\hat Z \leq Z\leq e^{\epsilon}Z$, with running time polynomial in $1/\epsilon$ and $|V(G)|$. A \vocab{polynomial time sampling scheme} for a distribution $\mu_G$ is a randomized algorithm that, given $G$ and a tolerance $\epsilon>0$ outputs a sample from a distribution $\hat\mu$ such that $\|\hat\mu - \mu_G\|_{\textrm{TV}}\leq \epsilon$, with running time polynomial in $1/\epsilon$ and $|V(G)|$.

\paragraph{Notation.} Throughout we implicitly restrict our attention to
  connected graphs, as all of the algorithmic tasks we consider factor
  over connected components. We let $\mathcal G_\De$ denote the set of graphs with maximum degree at most $\De$, and write $\deg(v)$ for the degree of a vertex $v$. Given a graph $G=(V,E)$, we denote by $d(v,w)$ the distance between vertices $v,w$ on the graph, i.e., the length $\ell$ of the shortest  path $v_0,\ldots, v_\ell$ with $v_0=v$ and $v_\ell=w$, and $(v_i,v_{i+1}) \in E$.
For a set $S\sub V$, let $d(v,S):= \min_{w\in S} d(v,w)$. For a vertex $v$ let $N(v)$ denote the set of neighbors of $v$. Let $N(v,\ell)$ denote the set of vertices at distance exactly $\ell$ in  a graph $G$.

The \vocab{Ising model with boundary condition $\tau\in \{\pm 1\}^V$ on $B\subset V$} is defined by the formulas in~\eqref{e:Idef} but with the restriction that $\sigma$ is a spin configuration that agrees with $\tau$ on $B$. We write $p^{\tau}_{G,\be,h}$ for the law of this model.
Given an Ising model with fixed $\tau, \beta, h$, and letting $\si'\in \{\pm 1\}^\La$ for a subset $\La\sub V$, let $p_v^{\si'}$ be the marginal probability of spin 1 at vertex $v$ conditioned on $\si_\La = \si'$, i.e.,
\begin{equation}
\label{e:isingmarginal}
p_v^{\si'}:= p^{\tau}_{G, \be, h}(\si_v=1|\si_{\La} = \si').
\end{equation}
In~\eqref{e:isingmarginal} and above we have written $\sigma_A = (\sigma_x)_{x\in A}$ to denote the spins at the vertices $A\subset\Lambda$.

For functions $f,g \colon \bb R\to\bb R$ we write $f=O(g)$ if there exists $C>0$ such that $|f(x)|\leq C |g(x)|$ for all $x$ large enough, and $f=\Omega(g)$ if $g=O(f)$.

\section{Deterministic large external fields}
\label{sec:largefield}
In this section we give approximate sampling and counting algorithms
in the case that $|h_x| \ge c(\beta,d)$ for all $x \in V$.
 This case in which \emph{all} external fields are large with probability $1$ provides some intuition for the main result by indicating how the
presence of large fields facilitates correlation decay. However, the
simple proof we provide here
does not work without a uniform bound on the external fields,
see Remark~\ref{rem:Glauber} below.  Define
\begin{equation}
  \label{eq:M}
M(\De, h, \be) =\abb{ \rc{1+e^{-2\be \De - 2h}} - \rc{1+e^{2\be \De - 2h}}}.
\end{equation}
The quantity $M$, and particularly upper bounds on $M$, will be important for our
analysis in this and subsequent sections.
\begin{lemma}\label{l:M}
For any $\be \in \bb R$,  $\De\ge 0$, and $\ep>0$, if $|h|\ge |\be| \De + \rc 2 \log\prc \ep$, then $M(\De, h, \be)< \ep$.
\end{lemma}
\begin{proof}
Consider the terms $\rc{1+e^{-2\be \De - 2h}}$ and $\rc{1+e^{2\be \De - 2h}}$.
If $h\ge |\be| \De + \rc 2 \log\prc \ep$, then both terms are $\ge
\rc{1+\ep}$, and if $h\le -(|\be| \De + \rc 2 \log\prc \ep)$, then both
terms are $\le \rc{1+\rc{\ep}}= \fc{\ep}{1+\ep}$. Since both terms are
bounded between 0 and 1, $M(\De, h, \be)\le \fc{\ep}{1+\ep}<\ep$ follows.
\end{proof}

The following monotonicity property follows by differentiating in $\Delta$.
\begin{lemma}
\label{lem:Mmon}
 For fixed $h,\beta$, $M(\Delta,h,\beta)$ is non-decreasing in $\Delta$.
\end{lemma}

The following bounds the influence of boundary conditions on the marginal probability at $v$.
\begin{lemma}\label{l:marg-M}
Let $v\in V$, $\La\sub V$ be a set not containing $v$, and
let $\si_\La, \tau_\La\in \{\pm 1\}^{\La}$.  Then
\begin{equation*}
|p^{\si_{\La}}_v - p^{\tau_{\La}}_v| \le M(\deg(v), h_v, \be).
\end{equation*}
\end{lemma}
\begin{proof}
We first prove the result when $\La$ contains all the neighbors of $v$.
In this case, considering only the relevant part of the Hamiltonian and temporarily abbreviating $\sigma=\sigma_{\Lambda}$ on the right-hand side,
\begin{equation*}
p_v^{\si_{\Lambda}} = \fc{e^{\sum_{y\in N(v)} \be \si_y+h_v}}{e^{\sum_{y\in N(v)} \be \si_y +h_v} + e^{\sum_{y\in N(v)} -\be \si_y-h_v}}.
\end{equation*}
The maximum and minimum possible values of this are $\rc{1+e^{-2\be\deg(v) - 2h_v}}$ and $\rc{1+e^{2\be\deg(v) - 2h_v}}$. Hence $|p_v^{\si_\La} - p_v^{\tau_\La}|$ is bounded by $M(\deg(v),h_v,\be)$.
For general $\La$, by conditioning on the value of $\si$ on $N(v)$, we can write $p_v^{\si_\La}$ as a weighted average of $p_v^{\si_{N(v)}}$, so the lemma follows in this case as well.
\end{proof}

This bound allows us to prove rapid mixing of Glauber dynamics via the
method of path coupling~\cite{bubley1997path}.  This in turn gives us
randomized polynomial-time approximate counting and sampling
algorithms when the external field is \emph{uniformly}
large.
We recall that the Glauber dynamics are a
time-homogeneous Markov chain $(\sigma(n))_{n\in \bb N}$ that evolves by uniformly selecting a vertex $x\in V$, and then updating
the spin at $x$ according to the marginal distribution at $x$ conditioned on the spins of its neighbors, i.e., $\P(\sigma_{x}(n+1)=1) =
p_{\beta,h}^{\tau}(\sigma_{x}=1|\sigma_{V\setminus
  \{x\}}=\sigma(n)_{V\setminus
  \{x\}})$. The other spins are unchanged in this step.

\begin{theorem}
  \label{thm:PC}
 Fix $\De\in\{2,3,\dots\}$.
  If $|h_{x}|\geq h_{0}(\Delta,\beta)=\Delta|\be|+
  \frac{1}{2}\log\Delta$, then the mixing time of the Glauber dynamics
  for the Ising model on $G\in\mathcal{G}_{\Delta}$ with $|V(G)|=n$ is $O( n \log n)$.
\end{theorem}
\begin{proof}
  Fix $G\in \mathcal{G}_{\Delta}$. As described above, in a single
  step of the Glauber dynamics we pick $x \in V(G)$ uniformly at
  random and then update the spin $\sigma_x$ conditioned on the spins
  of $N(x) = \{y\in V \mid \{x,y\}\in E(G)\}$.

  Consider two configurations $\sigma, \sigma'$ that disagree only at
  vertex $y$.  We couple two copies of the Glauber dynamics starting
  from $\sigma(0) = \sigma$ and $\sigma'(0)=\sigma'$ respectively by
  picking the same vertex $x$ to update and updating to the same spin
  with as high probability as possible. We analyze how the Hamming
  distance between the two configurations changes in a single step of
  the chain.
  \begin{enumerate}
  \item If $x= y$, the vertex at which $\si$ and $\si'$ disagree, then
    with probability one $\sigma(1)= \sigma'(1)$, and the Hamming
    distance decreases by $1$. This occurs with probability $1/n$.
  \item If $d(x,y) >1$, then both chains see the same boundary conditions
  and so make the same update. The Hamming distance does not change.
  \item If $x \in N(y)$, then the two chains see different boundary
    conditions.
    By Lemmas~\ref{lem:Mmon} and~\ref{l:marg-M} the difference in the
    probability of updating to $+1$ is at most $M(\De, h_v,\be)$,
    and this is an upper
    bound on the expected change in Hamming distance.  This occurs with
    probability $\De/n$.
  \end{enumerate}
  Let $\de(\sigma,\sigma')$ be the expected change in Hamming distance
  between $\sigma$ and $\sigma'$ after one step
  of the coupled chains. Since $|h_{v}|\geq h_{0}(\Delta,\beta)$,
  Lemma~\ref{l:M} and the considerations above give
  \begin{equation*}
    \de(\sigma,\sigma') \le - \frac{1}{n} \left[ 1 - \Delta
      M(\De,h_{0},\beta)  \right] <0.
  \end{equation*}
  The theorem follows by path coupling, see,
  e.g.,~\cite[Corollary~14.7]{levin2017markov}.
\end{proof}

\begin{remark}
  \label{rem:Glauber}
  Analyzing the Glauber dynamics without a uniform lower bound on
  $|h_{x}|$ would require addressing the fact that the dynamics are
  not contractive at each step. To see this, suppose that
  $h_{x}\in \{-h,0,h\}$, and note $M(\Delta,0,\be)\approx 1$ if $\be$ is
  large. In this case when the dynamics act on vertices with $h_{x}=0$ there
  is no contraction. This reflects the fact that $\be$
  is above the uniqueness threshold for the Ising model on the $\Delta$-regular tree with no
  external field.

  For some types of
  disordered systems, rigorous results that rule out exponential
  relaxation have been obtained in infinite
  volume, see, e.g.,~\cite{cesi1997relaxation}.
\end{remark}

\section{Random field Ising model}
\label{sec:ssmtree}

In this section we prove Theorem~\ref{thm:main}. We will that show strong spatial mixing, a strong form of
correlation decay, holds with high probability on a tree when
the \emph{typical} value of $|h_x|$ is large enough. Recall that $p_v^\si$ denotes the probability that $\sigma_{v}=1$ under boundary conditions $\si$.
\begin{definition}
  Let $G=(V,E)$ be a graph, and let $v\in V$ be a vertex.  We say that
  \vocab{strong spatial mixing (SSM) with rate $\alpha(\cdot)$ and
    min-distance $\ell_{0}$} holds for $v$
  if for any $\La\sub V$ and any two configurations $\si_{\La}, \tau_{\La}\in\{\pm 1 \}^{\Lambda}$,
\begin{equation*}
|p_v^{\si_{\La}} - p_v^{\tau_\La}| \le \alpha (d(v,\La'))
\end{equation*}
whenever $d(v,\La')\ge \ell_0$,
where $\La'\subeq \La$ is the subset on which $\si_{\La}$ and $\tau_\La$ differ.
\end{definition}

The standard definition of \vocab{strong spatial mixing with
    rate $\alpha$} corresponds to taking $\ell_{0}=0$. Taking $\ell_{0}$
  non-zero is a weaker condition.
The preceding definition is partly inspired by Camia, Jiang, and Newman~\cite{camia2018note}, who obtained a certain \emph{non-uniform} spatial mixing result on $\bb Z^d$.
For algorithmic purposes a uniform spatial mixing result is necessary, and we establish such a result in  Lemma~\ref{l:tree-mix} using ideas similar to those of~\cite{camia2018note}. The cost of uniformity is that we obtain SSM with min-distance $\ell_0$ of order $\log n$. Section~\ref{s:ssmsawtree} extends this SSM result to SAW trees, and we prove Theorem~\ref{thm:main} in Section~\ref{s:proofmain}.

For completeness, we show how our generalization of the technical result in~\cite{camia2018note} leads to a generalization of~\cite[Theorem~6]{camia2018note} in Section~\ref{sec:NUSM}. This section does not play a role in our algorithmic results.

\subsection{Disagreement percolation and spatial mixing}

Lemma~\ref{l:bound-by-perc} below relates the distance between marginal distributions to
the probability of disagreement percolation from the boundary to the region of
interest. First, we define the relevant notion of a percolation process.
\begin{definition}
  Let $G=(V\cup \partial V,E)$ be a finite graph.

  Define (inhomogeneous, independent)
  \vocab{site percolation} with probabilities $p_x\in [0,1]$, for each $x\in V$, as the following
  process. Let $T\in \{0,1\}^{V\cup \pl V}$, where
  $(T_x)_{x\in \pl V}$ are given boundary conditions on $\partial V$ and
  $(T_x)_{x\in V}$ are independent Bernoulli random variables with
  $\P(T_x=1)=p_x$.  We denote the
  law of $(T_x)_{x\in V}$ by $P_p$.
\end{definition}
In the preceding definition the boundary condition is implicit in the notation $P_p$; we will explicitly highlight the boundary condition in what follows. As is standard in percolation, for disjoint $A,B\sub V$, we write $A\lra B$ if
  there exists a path $v_0,v_1,\ldots, v_d$ with $v_0\in A$ and
  $v_d\in B$, such that $T_{v_i}=1$ for each $0\le i\le d$.

\begin{lemma}[{cf. \cite[Lemma 5]{camia2018note}}]\label{l:bound-by-perc}
  Given an Ising model on a connected
  graph $G=(V\cup\partial V,E)$, let $A\sub V$, and let
  $\eta, \xi$ be two boundary conditions on $\partial V$.  Let $P_p$ be the law of a site percolation $T$ with boundary condition
  $T_x=\one[\eta_x \ne \xi_x]$ for $x\in \pl V$, and
  $p_x = M(\deg(x), h_x, \be)$ for all other vertices.  Then
\begin{equation*}
d_{TV}(p^{\eta}_{\be, h}(\si_A\in \cdot ),p^{\xi}_{\be, h}(\si_A\in \cdot))
\le  P_{p}(\pl V \lra A).
\end{equation*}
\end{lemma}
\begin{proof}
Order the vertices of $V=\{x_1,x_2,\ldots\}$ in such a way that $x$ precedes $y$ in the ordering if $d(x,\pl V)<d(y,\pl V)$. We couple draws $\si^{(1)}$, $\si^{(2)}$, $S_x$ by drawing $\si^{(1)}_x,\si^{(2)}_x, S_x$ sequentially according to an exploration process. $S$ will be a site percolation process with boundary condition given by
\begin{align*}
S_x &= \one [\eta_x\ne \xi_x]\text{ when }x\in \pl V.
\end{align*}
For $t\in \bb N_0$, let $W_t$ denote the set of sites explored up to and including time $t$, and let $V_t:=\{x\in W_t:S_x=1\}$. We now inductively define the explored set.
\begin{itemize}
\item
Let $W_0:=\pl V$.
\item
For each $t\ge 0$, reveal the first unexplored site $x$ (according to
the chosen ordering) that is adjacent to $V_t$. Note that this eventually
exhausts the graph since $G$ is connected. We set the values
of $\si_x^{(i)}$ to have the correct marginal distributions, and to be
a maximal coupling. More precisely, let $U_x$ be a independent uniform
random variable in $[0,1]$, let $\nu^{(1)}=\eta$, $\nu^{(2)}=\xi$, and let
\begin{align*}
\si_x^{(i)} &= \begin{cases}
1, & U_x\le p^{\nu^{(i)}}_{\be, h}(\si_x=1|\si_{W_t} = \si_{W_t}^{(i)})\\
-1, &\text{otherwise,}
\end{cases}\\
S_x &= \one[\si_x^{(1)}\ne \si_x^{(2)}].
\end{align*}
Then let $W_{t+1}:=W_t\cup \{x\}$.
\end{itemize}
Note that conditioned on $\si_{W_t}^{(i)},i=1,2$,
we have $\si_x^{(1)}\ne \si_x^{(2)}$ with probability at most
$M(\deg(x), h_x, \be)$, and as a result the site percolation process $T$ with the same boundary condition as $S$ stochastically
  dominates $S$.
With this coupling, $\si_A^{(1)}\ne \si_A^{(2)}$ only if $\pl V \lra A$ in $S$; by stochastic domination this is at most the probability that $\pl V \lra A$ in $T$.
\end{proof}

We now use Lemma~\ref{l:bound-by-perc} to show
how an assumption that the external field is
typically large results in a strong spatial mixing property on trees. We quantify typically large by requiring the following condition on the external field distribution $h$ (for a parameter $h_0$ to be specified):
\begin{equation}
    \label{e:lf}
    \P\pa{|h|< h_0}\le
    \rc{16 \De^2}.
\end{equation}
\begin{lemma}
  \label{l:tree-mix}
  Let $G$ be a tree with max degree $\De$ and root vertex $v$.
  Let $h_0$ be such that
  $M(\Delta, h_0,\be) < \De^{-2}$.
  Suppose $h_x$ are such that
  along each path from any $w$ to $v$, the $h_x$ are independent and satisfy~\eqref{e:lf}.
Then with probability at least $1-\delta$ over the $h$, there is a $c_1>0$ such that strong spatial mixing holds with rate
\begin{equation*}
\alpha (\ell) = e^{-c_1 \ell}
\end{equation*}
for $v$ and for min-distance $\ell_0:=\log_2 \pf{1}{2\delta}$.
\end{lemma}

\begin{remark}
  We can take $c_1 = -\rc2 \log(M(\De, h_0,\be)\De^2)$.
  Examining the proof shows that the right-hand side of~\eqref{e:lf} can be improved to $O\prc{\De^{1+\ep}}$ for any $\ep>0$ at the cost of a tighter bound on $M(\De, h_0,\be)$, by replacing $\fc{\ell}2$ in~\eqref{e:CH} by $(1-O(\ep)) \ell$.
\end{remark}
\begin{proof}
First, we fix $\ell\ge \ell_0$ and consider the case when $\si_\La$ and $\tau_\La$ disagree at exactly 1 vertex $w$ at distance $\ell$ from $v$. There is a unique path $\ga_{wv}$ joining $w$ and $v$. If $\ga_{wv}$ contains another vertex $w'\in\La$ besides $w$, then the probability of the spin at $u$ is conditionally independent of the spin at $w$ given the spin at $w'$, so $p^{\si_\La}_u=p^{\tau_\La}_u$. Otherwise, we apply Lemma~\ref{l:bound-by-perc}: letting $p_x=M(\deg(x), h_x,\be)$ for $x\in V\bs \La$, we have
\begin{align*}
|p_v^{\si_\La} - p_v^{\tau_\La}| &\le
    P_{p}(w\lra v)
    \le \prod_{u\in \ga_{wv}} M(\Delta, h_u,\be),
\end{align*}
since percolation occurs only if all sites on the path $\ga_{wv}$ have value 1; the second inequality is by Lemma~\ref{lem:Mmon}.
In the general case, by changing the vertices at distance $\ell$ one at a time, we get that
\begin{align*}
|p_v^{\si_\La} - p_v^{\tau_\La}| &\le \sum_{w:d(v,w)=\ell} \prod_{u\in \ga_{wv}} M(\De,h_u,\be).
\end{align*}
To estimate this we next establish that most $h_u$ are large.  Let $p = \mathbb{P}(|h|<h_0)$. Using the upper bound $p\le \rc{16\De^2}$ from \eqref{e:lf} and
the Chernoff-Hoeffding bound we obtain that, with high probability, most of the $h_u$'s along a path are large.
\begin{align}
\label{e:CH}
\P\pa{\abb{\{u\in \ga_{wv}:|h_u|\ge h_0\}}\ge \fc \ell2} &\ge
1-e^{-\pa{\rc 2 \log \pf{1/2}p + \rc 2 \log\pf{1/2}{1-p}}\ell}\\
\nonumber
&\ge 1-e^{\pa{\log 2 - \rc2\log \prc{p}}\ell} \\
\nonumber
&\ge 1-2^\ell p^{\rc 2\ell }
\ge 1-\prc{\De}^\ell 2^{-\ell}.
\end{align}

Under this event and by the hypothesis that $M(\Delta,h_0,\beta)<\Delta^{-2}$, we have that there exists $c_1>0$ such that
\begin{equation*}
\prod_{u\in \ga_{wv}} M(\Delta,h_u,\be) \le \pf{e^{-2c_1}}{\De^2}^{\ell/2}
= \fc{e^{-\ell c_1}}{\De^\ell}.
\end{equation*}
In particular, $c_1=-\rc2 \log(M(\De, h_0,\be)\De^2)$ works.
Hence, doing a union bound over all paths, we get that with probability at least
$1- 2^{-\ell}$,
\begin{align*}
|p_v^{\si_\La} - p_v^{\tau_\La}| &\le e^{-\ell c_1}.
\end{align*}
for any $\si_{\La},\tau_{\La}$ disagreeing in $\La'$, where $d(\La,\La')=\ell$. Now taking a union bound over $\ell\ge \ell_0$, this holds for all $\ell\ge \ell_0$ with probability
\begin{align*}
1-\sum_{\ell=\ell_0}^{\iy} 2^{-\ell}
\ge 1-2 \cdot 2^{-\ell_0} \ge
1-\delta.
\end{align*}
where the last inequality uses the definition of $\ell_0$.
\end{proof}

\subsection{Strong spatial mixing on the SAW tree}
\label{s:ssmsawtree}

Our next corollary will have nearly the same conclusion as Lemma~\ref{l:tree-mix},
but it concerns a specific tree of self-avoiding walks
(SAW tree).
To prepare for this, we recall the construction of the SAW tree for Ising models,
see~\cite[Appendix~A]{liu2019fisher} or~\cite{zhang2011approximating},
which follows Weitz's original construction for the
hard-core model~\cite{weitz2006counting}. As this construction is
well-known and clear expositions exist in the literature, we will be
somewhat brief.

A \vocab{self-avoiding walk of length $k$}, $(v_{i})_{i=0}^{k}$, is a
sequence of adjacent vertices, each $v_{i}$ distinct. The set of
self-avoiding walks started at a fixed vertex $v$ has a natural rooted
tree structure: the root is the length $0$ walk consisting of
$v_{0}=v$ alone, and the children of length $k$ self-avoiding walk
$(v_{i})_{i=0}^{k}$ are the length $k+1$ self-avoiding extensions
$(v_{i})_{i=0}^{k+1}$ with $v_{k+1}$ adjacent to $v_{k}$. Call this
tree $\hat T_v$. The \vocab{self-avoiding walk tree $T_{v}$ rooted at
  $v$} is obtained from $\hat T_v$ as follows. To each vertex
$(v_{i})_{i=0}^{k}$ in $\hat T_v$ append additional leaf vertices
$w_{1},\dots, w_{j}$, one for each
$v_{k+1}\in V, v_{k+1}\neq v_{k-1}$, such that $(v_{i})_{i=0}^{k+1}$
is \emph{not} self-avoiding, i.e., $v_{k+1}$ completes a cycle of
length at least three. The tree $T_{v}$ is finite, and one obtains an
Ising model on $T_{v}$ by taking the external field at
$(v_{i})_{i=0}^{k}$ to be $h_{v_{k}}$.

The key result is then that if one defines a boundary condition $\tau$
on the leaves of $T_{v}$ correctly (i.e., $\tau_{w}\in \{-1,+1\}$ for
each leaf $w$, see Appendix~\ref{a:saw}
or \cite{liu2019fisher,zhang2011approximating} for details of the construction),
then the distribution of
$\sigma_{v}$ on $T_{v}$ with boundary condition $\tau$ is identical to
the distribution of $\sigma_{v}$ on $G$:
\begin{lemma}
  \label{lem:Wlemma}
  There is a choice of spins $\tau_{w}$ for the leaves $w$ of $T_{v}$
  such that the marginal distribution of the spin at the root is
  precisely the same as the marginal distribution of $\sigma_{v}$ on
  the graph $G$.
\end{lemma}
The exact way in which the boundary condition $\tau$ is determined
will not play a role in what follows, so we will not discuss
this in the main text.
The preceding construction generalizes to the
situation in which there is a boundary condition $\xi$ for the Ising
model on $G$: in this case the corresponding spins for the Ising model
on $T_{v}$ are fixed to agree with $\xi$. Lemma~\ref{lem:Wlemma} holds
in this more general setting as well.

\begin{corollary}
  \label{cor:ssmdSAW}
  Let $G$ be a graph with max degree $\De$ on $n$ vertices.   Suppose that the distribution of $h_x$ satisfies \eqref{e:lf} for $h_0$ such that $M(\Delta,h_0,\beta)<\De^{-2}$.
  Then there exists a constant $c_1>0$ so that with probability $ 1-o(1)$ over the realization of the external fields, for every vertex $v\in V$ the SAW tree $T_v$ at
  $v$ satisfies strong spatial mixing with rate
  \begin{equation*}
    \alpha (\ell) = e^{-c_1\ell}
  \end{equation*}
  for $v$ and for min-distance $\ell_0:=\fc{\log n}{c_1}$.
\end{corollary}

\begin{proof}
  Note that in the SAW tree rooted at $v$ the value of
  $h_x$ is repeated at some vertices, but there are no repetitions
  along any path to   the root $v$, because this path corresponds to a self-avoiding
  walk. Therefore
  Lemma~\ref{l:tree-mix} applies to the SAW tree rooted at
  $v$. To obtain the corollary, apply the result of
  Lemma~\ref{l:tree-mix} with $\delta$ taken to be
  $\fc{\delta}n$ for each vertex $v\in
  V(G)$. The result follows by a union bound over all vertices of $G$, choosing $\delta=o(1)$, and decreasing $c_1>0$ if necessary.
\end{proof}

\subsection{Proof of Theorem~\ref{thm:main}}
\label{s:proofmain}

We use Weitz's approach to approximate counting~\cite{weitz2006counting}.    In brief, to approximate the partition function of a graph in $ \mathcal G_\De$ with arbitrary boundary conditions it suffices to approximate the marginal of any vertex of any graph in $\mathcal G_\De$ with arbitrary boundary conditions by writing the partition function as a telescoping product; see Appendix~\ref{a:saw} for the calculation.  Similarly, given the ability to approximate marginals, one can sample by setting one spin at a time according to its marginal and updating the boundary conditions. Both the algorithms for counting and for sampling are written out in Appendix~\ref{a:alg}.

Recall,
see~\cite{weitz2006counting}
or~\cite[Theorem~2.8]{sinclair2014approximation} that Weitz proved
that for any two-state spin system, strong spatial mixing (SSM) on the
$\Delta$-regular tree implies the existence of an FPTAS on all graphs
of degree at most $\Delta$. In the following, we briefly recall this
algorithm and its analysis. We will see that weaker notion of SSM at
distance $\ell_{0}$ is sufficient to carry out the analysis.

By the SAW tree construction discussed in
  Section~\ref{s:ssmsawtree}, to compute the marginal distribution of
  the spin at a fixed vertex $v$ it would suffice to compute the
  marginal distribution of the spin at the root of $T_{v}$. Given the
  tree structure, this is a recursive computation. The running time,
  however, is exponential in the depth of the recursion, which could
  be as large as $n$. Weitz observed that one can truncate the tree
  $T_{v}$ at logarithmic depth if  correlations decay exponentially fast in the depth of the tree, as the
  analysis of the recursive computation will not be sensitive to the
  value of the spins at large distances. The running time of the
  recursion on this truncated tree is linear in the size of the tree, which is polynomial in $n$.

To make this precise for the hard-core model, Weitz proved that when $\lambda < \lambda_c(\Delta)$, SSM with rate $\alpha(t) = e^{- \Omega(t)}$ holds on the SAW
tree of a graph of maximum degree $\Delta$. Consequently, to
obtain an $\epsilon/n$-approximate evaluation of the marginal of the root of the SAW tree, one can truncate the SAW tree at
depth $\ell = O(\log(n/\epsilon))$
(see~\cite[Section~5]{weitz2006counting}). The running time
of this algorithm is polynomial in $n$.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Given the discussion above and Lemma~\ref{lem:Wlemma}, it is clear that Corollary~\ref{cor:ssmdSAW} suffices for verifying
that validity of the algorithm in the setting of the RFIM: for a fixed $0<\delta<1$, we have
SSM at distance $\ell_{0}=\frac{\log n}{c_1}$.
Hence by taking $\ell'=\max\{\ell,\ell_{0}\}$ and truncating at depth $\ell'$ we obtain the desired polynomial-time
algorithm.
\end{proof}

Lastly, we check that the event on which
the algorithm
correctly outputs the desired approximation can be
identified in polynomial time as was described in Remark~\ref{rem:check}.
\begin{proposition}
  \label{prop:check}
  Fix $\epsilon>0$. Given $h$, there is a polynomial time algorithm in
  $1/\epsilon$ and $n$ that determines if the
  output of the algorithm from Theorem~\ref{thm:main} is an
  $\epsilon$-approximation to the partition function of the Ising
  model with external fields $h$.
\end{proposition}
\begin{proof}
  For each $v\in V$, it takes polynomial time to construct the SAW
  tree $T_{v}$ to depth $\ell'=\max\{\ell,\ell_{0}\}$, where $\ell$ is
  the constant from the proof of Theorem~\ref{thm:main} above. Constructing the
  SAW tree for each $v$ to this depth thus takes polynomial time as
  well. Each tree has polynomially many leaves, and for each leaf to
  check that the path from root to leaf has at least half of its
  vertices with external field at least $h_{0}$ in magnitude takes
  linear time.
\end{proof}

\section{Random field Ising model on random graphs}
\label{sec:extensions}

This section establishes Theorem~\ref{thm:main2}.
The arguments are similar to those that established
Theorem~\ref{thm:main}, and we focus our exposition on the new
aspects.

A graph drawn from $\mathcal G(n,p)$ (an Erd\H{o}s-R\'enyi random graph)
is defined as a graph on $n$ vertices $\{v_1,\ldots, v_n\}$ where each pair of vertices $\{v_i,v_j\}$, $i\ne j$ forms an edge independently with probability $p$. We will take $p=\De/n$, so that the average degree of a vertex is $p(n-1)\approx \De$. However, the maximum degree of $\mathcal G(n,\De/n)$ is $\Te\pf{\log n}{\log\log n}$ with high probability.

The key observation that allows us to apply our results in this setting is that an Erd\H{o}s-R\'enyi random graph has bounded connective constant with high probability~\cite{sinclair2013spatial,sinclair2017spatial}.
\begin{lemma}[{\cite[in Proof of Theorem~1.2]{sinclair2013spatial}}]
\label{l:cc}
Let $\De >1$.
Suppose $G$ is distributed as $\mathcal{G}(n,\Delta/n)$, and let $\gamma,\nu>0$.
With probability at least $1-n^{-\nu}$, for all $\ell \ge \fc{\nu+2}{\log(1+\gamma/2)}\log n$ and all vertices $v$, the SAW tree $T_v$ satisfies
\begin{equation*}
|N(v,\ell)| \le [\De(1+\gamma/2)]^\ell, \qquad
\sum_{d=1}^\ell |N(v,d)| \le \fc{\De}{\De-1}[\De(1+\gamma/2)]^\ell.
\end{equation*}
\end{lemma}

The following lemma is the analogue of Corollary~\ref{cor:ssmdSAW}. The key additional idea is to classify a vertex as bad if its degree is large, and that large degrees occur with low probability in $\mathcal{G}(n,p)$. Note that degrees of vertices are not independent, so we instead use the Chernoff–Hoeffding inequality on the edge count.
\begin{lemma}\label{l:ER}
 Let $\De>1$ and $G\sim \mathcal G(n,\Delta/n)$ be a random graph on $V$. There are constants $c_3, c_4, c_5$ such that the following hold. Fix a vertex $v\in V$. Let $c_1,c_2$ be any large enough constants. Let $h_0$ be such that $M(e^{c_2c_3}\Delta, h_0, \beta) \le \fc{e^{-c_1}}{c_4\Delta^2}$ and $\P(h_u<h_0)\le \pf{1}{2\De}^{c_2c_5}$.
 Then with probability $\ge 1-\de$ over $G$ and the realization of the external field $h$, the following hold.
 \begin{enumerate}
     \item The SAW tree $T_v$ at $v$ satisfies strong spatial mixing with rate $\al(\ell) = e^{-c_1\ell/2}$ for $v$ and for min-distance $\ell_0=\fc{\log\pf{2}{\de}}{c_2}$.
     \item For all $\ell \ge \ell_0$, on the SAW tree $T_v$, $\sum_{d=1}^{\ell} |N(v,d)|\le (c_4\De)^{\ell/2}$.
 \end{enumerate}
This holds for the SAW trees at all vertices with the same rate and min-distance $\ell_0=\fc{\log\pf{2n}{\de}}{c_2}$.
\end{lemma}
\begin{proof}
Call a path $v_0,\ldots, v_\ell$ on the SAW tree starting at $v_0=v$ \emph{bad} if one of the following holds:
\begin{enumerate}
\item
At least $\fc 14\ell$ of the vertices $v_0,\ldots, v_{\ell-1}$ have degree  greater than $c_3\Delta$.
\item
At least $\fc 14\ell$ of the values $|h_{v_i}|$ satisfy $|h_{v_i}|> h_0$.
\end{enumerate}
The first step in the proof is to rule out the existence of bad paths with high probability by a union bound argument. To this end, we first bound the probability (over the randomness in $G$ and $h$) that a fixed sequence $v_0,\ldots, v_\ell$ of vertices is a bad path. Since each edge is included in $G$ with probability $\fc{\De}n$, the probability of this fixed path
being a path in the SAW tree is $\pf{\De}{n}^\ell$.

Next we bound the probability of event (i). Given a fixed subset $S$ of $\{v_0,\ldots, v_{\ell-1}\}$ with $\ce{\ell/4}$ vertices, we bound the probability that all its vertices have degree greater than $e^{c_2c_3}\Delta$. For this to happen, there must be at least $\fc{e^{c_2c_3}\Delta}2\cdot \fc{\ell}{4}$ edges in $S\times (V\bs S)$, which has cardinality $O(\ell n)$. By the Chernoff-Hoeffding bound, the probability of this is
$e^{-\Omega(\Delta \ell c_2c_3)}$
for large enough $c_3$. By a union bound over appropriate subsets of $S$ (less than $2^\ell$ in number), the probability of (i) is still $e^{-\Omega(\Delta \ell c_2c_3)}$.

The probability of event (ii) is bounded exactly as in Lemma~\ref{l:tree-mix}: letting $p=\pf{1}{2\De}^{c_2c_5}$, it is bounded by
$p^{O(\ell)} \le (2\De)^{-\Om(c_2c_5\ell)}$. Thus, the probability of a fixed sequence $v_0,\ldots, v_\ell$  being a bad path is
\begin{align*}
\pf{\De}{n}^\ell \cdot \pa{e^{-\Omega(\Delta \ell c_2c_3)} +  (2\De)^{-\Om(c_2c_5\ell)}}\le \rc2
\prc{n}^{\ell}\cdot e^{-c_2\ell }.
\end{align*}
for large enough $c_3,c_5$.

By a union bound over possible paths of length $\ell$, of which there are most $n^\ell$, the probability that a bad path of length $\ell\ge \ell_0$ exists is at most $e^{-c_2\ell_0 }$. This is at
most $\fc{\de}2$ when $\ell_0=\log\pf2{\de}/c_2$.

Next, observe that we can choose $c_4$ sufficiently large so that by Lemma~\ref{l:cc}, the probability that $\sum_{d=1}^\ell |N(v,d)|\le (c_4 \De)^{\ell/2}$ for all $\ell\ge \ell_0$
is at least $1-\fc{\de}2$. (We apply Lemma~\ref{l:cc} directly when $\De\ge 2$; otherwise to avoid the $\rc{\De-1}$ factor, we note that $\sum_{d=1}^\ell |N(v,\ell)|$ is stochastically dominated by its value when $\De=2$.)
Let $\mathcal{E}$ be the event that there are no bad paths of length $\ell\ge \ell_0$, and
$\sum_{d=1}^\ell |N(v,d)| \le (c_4 \De)^{\ell/2}$
for each $\ell\ge \ell_0$. By a union bound, $\mathcal{E}$ has probability at least $1-\de$.

If a path from $v$ to $w$ is not bad, then letting $\ell = d(v,w)$, at least $\fc 12 \ell$ of the vertices $u$ on the path satisfy $\deg(u)\le e^{c_2c_3}\Delta$ and $|h_u|\le h_0$. For any such $u$,  $M(\deg(u), h_0, \beta)\le \fc{e^{-c_1}}{c_4\Delta^2}$ by Lemmas~\ref{l:M} and~\ref{lem:Mmon}.
As in the proof of of Lemma~\ref{l:tree-mix},
let $\ga_{wv}$ denote the unique path from $w$ to $v$.
Then, we have that if $\si_\La$, $\tau_\La$ are boundary conditions differing only at $w$, then
\begin{align*}
\P(\si_v\ne \tau_v|\si_\La, \tau_\La) &=
\prod_{u\in \ga_{wv}}M(\deg(u), h_{v_k},\be)\le \pf{e^{-c_1}}{c_4\Delta^2}^{\ell/2} = \fc{e^{-\ell c_1/2}}{c_4^{\ell/2} \De^\ell}.
\end{align*}
Fix $\ell \ge \ell_0$. On the event $\mathcal{E}$,
by changing the vertices at distance $\ell$ one at a time, we have
\begin{equation*}
|p_v^{\si_\La} - p_v^{\tau_\La}| \le \sum_{w:d(v,w)=\ell} \sum_{u\in \ga_{wv}} M(\deg(u), h_u,\be)
\le (c_4 \De)^{\ell/2} \cdot \fc{e^{-\ell c_1/2}}{c_4^{\ell/2} \De^\ell} \le e^{-\ell c_1/2},
\end{equation*}
and we obtain the same conclusion for all $v$ on the event that no bad paths exist starting from any $v$.
Finally, by replacing $\de$ by $\fc{\de}n$ and union-bounding over all vertices, there are no bad paths in the SAW tree at $v$ for each vertex $v$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main2}]
Theorem~\ref{thm:main2} follows from Lemma~\ref{l:ER} after noting the high-probability bound on the neighborhood of a vertex $v$ given by Lemma~\ref{l:cc}.  The recursive computation of the marginals on the SAW tree still runs in polynomial time  because the size of the $\ell'$-neighborhood of $v$ in the SAW tree $T_v$ is  $\sum_{d=1}^{\ell'}|N(v,d)| \le (c_4\De)^{\ell'/2}$, which is polynomial in all parameters.
\end{proof}

\section{Non-uniform spatial mixing on infinite graphs}
\label{sec:NUSM}\label{s:non-unif}
The following non-uniform spatial mixing result generalizes~\cite[Theorem~6]{camia2018note}. In this section we work in the context of infinite graphs; we always fix boundary conditions on sets $B$ such that $V\setminus B$ is finite.

\begin{theorem}[{cf.\ \cite[Theorem~6]{camia2018note}}]\label{t:cjn}
Consider the RFIM with IID Gaussian external fields.
Let $c_2>0$. There exists $c_1(\De, \be, c_2)$ such that for $\Var(h_x)\ge c_1(\De, \be,c_2)$, for any $A$ and $B$ such that $V\bs B$ is finite, for almost all realizations $h$,
\begin{align*}
\sup_{\eta, \xi \in \{\pm 1\}^{B}}
d_{TV}(p^{\tau\wedge \eta}_{\be, h}(\si_A\in \cdot ),p^{\tau \wedge \xi}_{\be, h}(\si_A\in \cdot))
&\le
\sum_{x\in \pl A, y\in \pl B} c_3(x,h) e^{-c_2d(x,y)}.
\end{align*}
where
\begin{align*}
(\tau\wedge\eta)(x)
&= \begin{cases}
\tau(x), & x\in V\setminus B\\
\eta(x), & \text{otherwise,}
\end{cases}
\end{align*}
and similarly for $\xi$.
Here, for a set $A\sub V$, $\pl A \subset V\setminus A$ denotes the subset of $A$ whose neighbors are not all contained in $A$.
\end{theorem}

Recall the inhomogenous site percolation processes $P_p$ with $p_x = M(\deg(x),h_x,\beta)$ introduced in Lemma~\ref{l:bound-by-perc}.
\begin{lemma}[{cf. \cite[Lemma 6]{camia2018note}}]
\label{l:6}
Consider the measure $P_p$ averaged over the randomness in $h$,  $\ol P_p(\cdot) = \int_{\bb R^V}P_p(\cdot) \P(dh)$ where $\P$ is the law of $h$.
Let $c_2>\log 2$. There exists $h_0=h_0(\De, \be,c_2)$ so that when $\P(|h_u|<h_0)\le p:=\fc{e^{-2c_2}}{4\De^2}$,
\begin{align*}
\ol P_p(x\lra y)
&\l
 4e^{-c_2d(x,y)}.
\end{align*}
\end{lemma}

\begin{proof}
Choose $h_0 = |\be| \De + \log(\De) + c_2$, so by Lemma~\ref{l:M}, $M(\De, h_0, \be) < \fc{e^{-2c_2}}{\De^2}$.
Consider a path $\ga$ from $x$ to $y$ of length $\ell: = d(x,y)$. Then
\begin{align}\label{e:avg-perc}
\ol P_p(\{\forall u\in \ga, S_u = 1 \})
&= \prod_{u\in \ga} M(\deg(u), h_u,\be).
\end{align}
As in~\eqref{e:CH}, by the Chernoff bound, with high probablity, most of the $h_u$'s for $u\in \ga$ are large:
\begin{align*}
\P\pa{\abb{\{u\in \ga:|h_u|\ge h_0\}}\ge \fc \ell2} &\ge
1-e^{-\pa{\rc 2 \log \pf{1/2}p + \rc 2 \log\pf{1/2}{1-p}}\ell}\\
&\ge 1-2p^{\rc 2\ell}
\ge 1-\pf{e^{-c_2}}{\De}^\ell
\end{align*}
Under this event,
\begin{align*}
\prod_{u\in \ga} M(\De, h_u, \be) &\le \pf{e^{-2c_2}}{\De^2}^{\ell/2} = \fc{e^{-c_2\ell}}{\De^\ell}
\end{align*}
Hence, by breaking up~\eqref{e:avg-perc} into two bad events,
\begin{align*}
\ol P_p(\{\forall u\in \ga, S_u = 1 \})
&=\P\pa{\abb{\{u\in \ga:|h_u|\ge h_0\}}< \fc \ell2}\\
&\qquad +
\ol P_p\pa{\{\forall u\in \ga, S_u = 1 \} \Big| \abb{\{u\in \ga:|h_u|\ge h_0\}}\ge \fc \ell2}\\
&
\le \fc{e^{-c_2\ell}}{\De^\ell} +  \fc{e^{-c_2\ell}}{\De^\ell}  = \fc{2e^{-c_2\ell}}{\De^\ell}.
\end{align*}
There are at most $\De^j$ paths of length $j$, so taking a union bound over all paths gives
\begin{align*}
\ol P_p(x\lra y)
&\le \sum_{j=\ell}^\iy \De^j \fc{2e^{-c_2j}}{\De^j} \le 4e^{-c_2\ell}.\qedhere
\end{align*}
\end{proof}

Given Lemma~\ref{l:6}, the proof of Theorem~\ref{t:cjn} is exactly the same as in~\cite{camia2018note}, after noting that the neighborhood of a vertex grows at most exponentially.

\begin{proof}[Proof of Theorem~\ref{t:cjn}]
By Lemma~\ref{l:6} applied to $c_2\mapsfrom 2c_2+\log(\De)$,
\begin{align*}
\sum_{y\in V} e^{c_2d(x,y)} \ol P_p(x\lra y) & \le
\sum_{y\in V} e^{c_2d(x,y)} 4 e^{-2c_2d(x,y)} \De^{-d(x,y)} <\iy,
\end{align*}
where we use the fact that the number of vertices at distance $\ell$ from $x$ is at most $\Delta^\ell$.
Expanding $\ol P_p$ as an integral over $h$ and using the Fubini-Tonelli theorem,
\begin{align*}
\int_{\bb R^V} \sum_{y\in V} e^{c_2d(x,y)} P_p(x\lra y) \P(dh)&<\iy,
\end{align*}
which implies
\begin{align*}
\sum_{y\in V}e^{c_2d(x,y)} P_p(x\lra y) &<\iy
\end{align*}
for almost all $h$, and
\begin{align*}
e^{c_2d(x,y)} P_p(x\lra y) &<c_3(x,h) \text{ for all }y\in V
\end{align*}
for almost all $h$.
Using Lemma~\ref{l:bound-by-perc}, we get
\begin{align*}
\sup_{\eta, \xi \in \{\pm 1\}^{B}}
d_{TV}(p^{\tau\wedge \eta}_{\be, h}(\si_A\in \cdot ),p^{\tau\wedge \xi}_{\be, h}(\si_A\in \cdot))
&\le P_p (A\lra B)\\
&\le \sum_{x\in \pl A, y\in \pl B} P_p (x\lra y)\\
&\le \sum_{x\in \pl A, y\in \pl B} c_3(x,h)e^{-c_2d(x,y)}.\qedhere
\end{align*}
\end{proof}

\section*{Acknowledgements}

This work was undertaken as part of the Phase Transitions and Algorithms working group in the SAMSI Spring 2021 semester program on Combinatorial Probability.  We thank the semester organizers for bringing us together.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
/*
* CODE GENERATED AUTOMATICALLY WITH goexportdefault
* THIS FILE MUST NOT BE EDITED BY HAND
*
* Install goexportdefault with:
* go get github.com/ernesto-jimenez/gogen/cmd/goexportdefault
 */

package testpkg

import ()

// EDUSPMethodVal is a wrapper around ExportedDefaultUnexportedStructPtr.MethodVal
func EDUSPMethodVal() {
	ExportedDefaultUnexportedStructPtr.MethodVal()
}

// EDUSPMethodPtr is a wrapper around ExportedDefaultUnexportedStructPtr.MethodPtr
func EDUSPMethodPtr() {
	ExportedDefaultUnexportedStructPtr.MethodPtr()
}

redpajama_set_name: RedPajamaGithub


If you want to have great interior design and you love the famous art pieces of the Italian Renaissance, but cannot afford to buy original art, we have a great solution for you. Find a sculptor and pay him to make you copies of famous artworks, but not in real marble – in cultured marble. Cultured marble is very similar to classic marble, but is a man made mix that resembles the real thing. This is why cultured marble is cheaper, but still looks as good. Thanks to cultured marble you could have David in your living room and enjoy the beautiful creation of Michelangelo Buonarotti daily, without having to travel to faraway and sunny Tuscany. Actually, with cultured marble artwork copies, you can have your own Tuscany in your house. The only thing is that you need to find someone who agrees to work with this type of marble.
If you want to make your living room resemble a typical Renaissance scene, you should inspire from Michelangelo and Leonardo da Vinci. A great inspiration can be Il Cavallo, even if da Vinci made it in bronze. A great marble inspiration for your own cultured marble artworks can also be Michelagelo's Pieta, the statue that shows Virgin Mary embracing her son, Jesus. The Italian sculptors are famous for their great marble artworks and you can enjoy them day by day if you find a talented sculptor in your area who can remake all you desire in cultured marble.
Of course, this is not the only way to use cultured marble. It can also be used to decorate bathrooms or kitchens or for any interior you would like. Thanks to this great invention, many people benefit of great looking houses without paying a fortune. So, if you dream of having marble on the floor, for example, now you can, because cultured marble will not drain your finances. With cultured marble you can have anything you wanted to have in real marble, but you could not afford. Redesigning your house using cultured marble is great, as well as cheap, so, what are you waiting for?


Pet Cooling Including Dog Cooling Products - Keep All Your Pets Cool!
Pet Cooling Including Dog Cooling Products – Keep All Your Pets Cool This Summer!
Dogs Certainly Feel Hot and Cold Too!
Here's some Low Cost Energy-Efficient Cooling Solutions including Pet Cooling Products for dogs and cats and Alternatives to ACs. Save Money on Your Cooling and Electric Bills. To see all energy-efficient options for your home…click here.
An Alternative to AC…just click to go to Amazon to learn more about this cool product for you…for your pets. The Ice Fan Air Conditioner or Ice Fan Cooler is probably the next best thing to AC. This energy-saving fan is small and portable plus no freon needed. It is also an air humidifier so can be used when conditions are dry. You simply add water or water and ice for a personal cooling experience. And you can even add some essential oil for aromatherapy. So you can cool yourself or your pets. It's a great energy-saving alternative to AC. Just click on the photo image to learn more and see the great reviews.


text:
/* @flow */
import React from 'react';
import Icon from 'mineral-ui/Icon';

import type { IconProps } from 'mineral-ui/Icon/types';

/* eslint-disable prettier/prettier */
export default function IconCheckBoxIndeterminate(props: IconProps) {
  const iconProps = {
    rtl: false,
    ...props
  };

  return (
    <Icon {...iconProps}>
      <g>
        <path d="M4 10h16v4H4z"/>
      </g>
    </Icon>
  );
}

IconCheckBoxIndeterminate.displayName = 'IconCheckBoxIndeterminate';
IconCheckBoxIndeterminate.category = 'toggle';


text:
Charlotte Pearson
Dendroarchaeological Research
September 5, 2016 pbrewer
I'm interested in developing new methodologies for the improvement of archaeological chronology and the synchronization of interdisciplinary evidence. Specifically the use of tree-rings and other proxies in combination with diverse cultural records to study global / regional 'events' of significance for past human societies. In particular sudden onset climatic changes and / or volcanic events and the chronologies associated with these in the ancient Mediterranean. I'm also interested in developing new applications in dendrochemistry and geoarchaeological research.
Interdisciplinary Chronology of Civilizations Project (ICCP)
The ICCP will combine tree-ring chronologies and annual resolution radiocarbon analysis to accelerate chronological research in the wider Aegean region through an archaeologically informed, multi-proxy approach. A primary objective will be the creation of an annually-resolved radiocarbon calibration curve from c. 2500-1000 BC which will be used address key chronological issues for Aegean and Near Eastern archaeology. This work will represent the first stage of what we hope will be a longer duration project contributing to global efforts to enhance the precision and accuracy of radiocarbon analysis for a wide range of users.
Enhancing Shared Facilities for Archaeological Science at the University of Arizona
This is an NSF-funded project with collaborators in Anthropology, Arizona State Museum and LTRR. It centers on the purchase of new XRF instrumentation and the development of cross-disciplinary applications in archaeological and conservation science, tree-ring research and outreach.
Mediterranean Dendroarchaeology Project
This is an ongoing continuation of the work of Peter Kuniholm's "Aegean Dendrochronology Project". Building on nearly 40 years research, our team's aim is to construct a continuous, absolutely dated Mediterranean tree-ring sequence spanning more than 10,000 years. We are working on wood samples from a range of Archaeological sites to expand existing chronologies, provide new dates for excavators and synchronize regional tree-ring connections. We are also constructing chronologies of sub-fossil material from Bosnia and Croatia with a view to bridging gaps in the record and linking with longer, continuous sequences from Norther Europe.
The Center for Mediterranean Archaeology and the Environment (CMATE)
I'm Associate Director of CMATE – a UA Center designed to further interdisciplinary collaboration between leading scholars in anthropology, dendrochronology, radiocarbon analysis and geosciences at the University of Arizona. By combining our different perspectives and methodologies for reconstructing the past, we aim to gain a greater and more highly resolved picture of past human experience in the cradle of Old World civilizations.
Rm 404 Bannister Tree-Ring Laboratory 1215 E. Lowell Street Box 210045 Tucson, AZ 85721-0045 USA
c.pearson@ltrr.arizona.edu

redpajama_set_name: RedPajamaCommonCrawl


I succesfully run MM2 on a macmini late 2009. Currently im testing some modules and so far so good except for one module Motion detector. I was looking on the code on node_helper.js and found the line with the command "vcgencmd display_power 0".
In OSX not work but "pmset displaysleepnow" to put the screen on sleep does the trick and " caffeinate -u -t 1" to wake up.
Its looks like work but im getting this message on console log "pmset displaysleepnow blocked by app electron.js" and the screen keeps alive.
Or maybe toggling the body to hide to get the black screen?
Its the correct place ?
Sorry but im totally new on node.
Finally figured it out, this module now can put the screen off and on (osx el capitan).


Stepping closer to the next edition of Jacket Required, as the premium trade show upholds its promise to present a distinctive combination of international leading labels and new brands to discover for autumn/winter 2018.
Portuguese garments, lovingly and locally produced, inspired by land and sea. Handcrafted Japanese bags and accessories that demonstrate the finest utilitarian design. Contemporary knitwear incorporating the German tradition for high quality. European clothing, made for the custom culture composed for the ride, road trip and adventures. Considered collections, refined and uniquely crafted, that draw upon rich heritage and core values of craft and British manufacture.
A sartorial selection, suited to all walks of winter – city, country or coast.
P O R T E R Y O S H I D A & C O .


Two children walking with luggage trolley in the airport.
Mother with two children walking in autumn park.
Behind walking mother with children in autumn park.
Two children with present bags.
Woman with bags. Walking people. Legs. Subway.


Text categorization is usually performed by supervised algorithms on the large amount of hand-labelled documents which are labor-intensive and often not available. To avoid this drawback, this paper proposes a text categorization approach that is designed to fully exploiting semantic resources. It employs the ontological knowledge not only as lexical support for disambiguating terms and deriving their sense inventory, but also to classify documents in topic categories. Specifically, our work relates to apply two corpus-based thesauri (i.e. WordNet and WordNet Domains) for selecting the correct sense of words in a document while utilizing domain names for classification purposes. Experiments presented show how our approach performs well in classifying a large corpus of documents. A key part of the paper is the discussion of important aspects related to the use of surrounding words and different methods for word sense disambiguation.


text:
Ancient British Ship Found After 168 Years in the Canadian Arctic
By Sky Selrach
Sep 14, 2016 04:29 AM EDT
The Ancient British Ship, HMS Terror, has finally been found after 168 years in the Canadian Arctic.
(Photo : Illustrated London News/Getty Images)
In hopes of discovering the Northern Passage, Sir John Franklin had set off on an expedition in 1845 with two ships, the HMS Erebus and Terror. The expedition failed with what was left of the HMS Erebus found in 2014. It was only recent that the other ship, HMS Terror, was discovered 168 years after in the Canadian Arctic.
The Arctic Northern Passage is said to be a sea route that connects the Atlantic and the Pacific Ocean. This route would greatly help British traders save money and time by not having to go around South America or Africa.
Sir Franklin was hopeful his ships would make the expedition, but both ships had vanished. Historians believed that the ships got stuck in ice and that its crew attempted to head back to Britain on foot.
Finally in 2014, the HMS Erebus was found off the coast of Nunavut. On the other hand, HMS Terror was only discovered by the Arctic Research Foundation much recently. The ships -- or what was left of them -- were found approximately 60 miles from where historians claimed the ships had sunk.
According to reports by The Guardian, HMS Terror was located in 24 meters of water. The three masts were broken but were still standing. Also, the hatches were closed, glass panes in the cabin were still intact and almost everything was stowed.
"We have successfully entered the mess hall, worked our way into a few cabins and found the food storage room with plates and one can on the shelves. We spotted two wine bottles, tables and empty shelving. Found a desk with open drawers with something in the back corner of the drawer," stated Adrian Schimnowski, the operations director of the foundation.
The condition of the ship and its contents certainly changes written history. History suggests that the crew of the HMS Terror met a tragic fate at sea. However, with the condition of the HMS Terror, it is most likely that the crew had shut down the ship and had boarded the HMS Erebus.
"It's almost certain that HMS Terror was operationally closed down by the remaining crew who then re-boarded HMS Erebus and sailed south where they met their ultimate tragic fate," stated Jim Balsillie, who had helped in establishing the Arctic Research Foundation.
TagsAncient British Ship, Canadian Arctic, HMS Terror, shipwreck, ancient british ship in canada, Canadian Arctic, ancient ship in canadian arctic
DNA of Long-Dead 19th Century Sailors Could Help Identify Doomed Crew of the Franklin Expedition
Astronaut Reveals Secret Treasure Map from Space, Sparking Treasure Hunt in the Caribbean
Legendary Orichalcum Metal From Atlantis Discovered in 2,600-Year-Old Shipwreck
The True Story Behind 'Moby-Dick' is Scarier than a Cannibal Horror Film
Scientists Explore Sunken WWII Shipwreck in California


Join us in March for our 'Made In' event series, coming to Glasgow.
Bringing together an unmissable collection of wines made with craft and passion including new wines from England, France, Germany, Spain and New Zealand.
This is a private On Trade event. Current customers please contact your Account Manager for information about attending. Non-customers please contact events@bibendum-wine.co.uk for more information.


text:
Information published on 27 June 2017 in the UIC electronic newsletter "UIC eNews" Nr 554.
Germany: DB Schenker transports over 100,000 tons of power station components to Egypt
The final push for a gigantic order: Until September, DB Schenker will be transporting a total of 100,000 tons with 400,000 cbm of freight to Egypt for Siemens AG, which is building one of the largest gas-fired power stations in the world. Burullus is the site of one of three Siemens power stations. Powered by natural gas, each plant has a capacity of 4.8 gigawatts, which will be generated by eight Siemens gas turbines and four steam turbines. This technology has been selected for its high efficiency.
The individual components include turbines, generators, transformers, and boilers, with single weights up to almost 500 tons. Each of the three power stations requires 248 of these massive components in total.
The heavy components from Europe and the USA are shipped straight to the Burullus fishing port. Parts from Asia arrive first at the port of Alexandria by ocean-going ship. There, they are reloaded onto smaller coasters which can access the port at Burullus, some 80 kilometers away, thanks to their smaller draft.
DB Schenker undertook the necessary preparation work to ensure the massive components could be unloaded at Burullus harbor: Its depth had to be increased to 5.50 meters and a pier needed reinforcement, while access routes and facilities for temporary storage had to be constructed. A 600 ton crane was installed to discharge the individual components to low-bed trailers. The port at Burullus is just 16 kilometers from the construction site.
All three power stations are going on stream successively already since the end of 2016.
(Source: DB)
23rd Meeting of the UIC Regional Assembly for Asia-Pacific (APRA) was held in Beijing Appointment: Jochen Eickholt, CEO of Siemens Mobility, has been elected new Chairman of UNIFE Czech Republic: SŽDC will present new rolling stock at the Czech Raildays Fair Finland: Seaborne trade between Finland and foreign countries increased by almost 6 million tonnes in 2016 Israel Railways launches the International Level Crossing Awareness Day (ILCAD) 2017 Italy: Prime Minister Paolo Gentiloni opens the new Napoli Afragola station Montenegro: New board appointments within rail freight operator Montecargo Project Group Security of Stations has launched an online survey – Please take part! TopRail meeting held from 20 – 21 June 2017 in Barcelona


(ANSA) - Milan, November 25 - Italian journalists and assorted entertainment stars have been flocking to the Albanian capital of Tirana to take part in Italian entrepreneur Francesco Becchetti's new broadcaster, Agon Channel Italia.
Becchetti, who became a media mogul after successfully patenting a technique to generate energy from trash, said he is proud to have created employment in Albania.
"Entrepreneurs can either stand by and watch or they can take the plunge and invest," he said. "They might lose a little sleep but they feel fulfilled, as I do after creating jobs for 500 young people in a year".
Becchetti, who invested 40 million euros in Agon Channel Italia, said he expects to attract a 1% share and 20 million euros in advertising within the first year of programming. Broadcasts begin December 1, to be preceded by a preview on Wednesday with a gala featuring Oscar-winning Australian actress Nicole Kidman.


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Jun 16, 2019 | HIPAA, IT, Security
Avoiding and Preventing Ransomware Attacks
Recently Health & Human Services (HHS) provided some guidance on avoiding ransomware attacks in its "Cybersecurity Best Practices" report.
HHS statistics showed that in the first six months of 2016, there were an average of 4,000 ransomware attacks every day, a 300 percent increase from 2015, when the daily average was 1,000 attacks. While this figure was startling, the scariest part of the report was HHS' position that
"[w]hen [ePHI] is encrypted as a result of a ransomware attack, a breach has occurred because the ePHI encrypted by the ransomware was acquired[,] and thus is a 'disclosure' not permitted under the HIPAA Privacy Rule."
In other words, you are required to presume a ransomware incident is a breach unless the evidence demonstrates a low probability of compromise based on the HIPAA breach risk assessment factors, including specifically that the PHI was not actually viewed or acquired. The guidance expanded the four-factor risk assessment under HIPAA when ransomware is involved to include consideration of the availability and integrity of the data.
The Report defines ransomware as a type of malware that:
"attempts to deny access to a user's data, usually by encrypting the data with a key known only to the hacker who deployed the malware, until a ransom is paid. After the user's data is encrypted, the ransomware directs the user to pay the ransom to the hacker (usually in a cryptocurrency, such as Bitcoin) in order to receive a decryption key."
Organizations will often discover that their systems have been impacted by ransomware in one of two ways:
Employees arrive to work one morning and report that they are unable to log in to their computers, which an IT investigation reveals is due to the encryption of files and data on workstations and/or servers, or
an employee reports receiving an email or pop-up on his or her screen that looks something like the picture above
Once they discover ransomware, most victims check to determine whether sufficient backups exist to restore the encrypted files and avoid paying the ransom. But be careful, IT assistance should be sought, before any systems are restored, to perform an analysis of how the ransomware accessed the system. For healthcare providers, it is imperative to evaluate the ransomware and the method of to determine whether there the data was actually compromised. If the server(s) or workstation(s) on which the ransomware was found is formatted clean before a forensic image is made, there may be insufficient evidence to make that determination, and you will then be required to notify of all individuals whose information was on the affected computer.
To avoid ransomware in the first place, make sure to train your staff about email precautions. Employees should be trained to ask themselves the following questions before clicking on a link within an email or downloading any attachment, whether it is from an outside or inside source:
Do you know the sender?
Are there any spelling or grammatical errors in the body of the email?
Did you hover over a link to see the URL destination, ensuring it does not mask a malicious site?
Is the "from" address a legitimate email address, without misspellings?
Is this email too good to be true?
Most importantly, users should ask themselves if they were expecting to receive a document or link from this user. To confirm, users should call the sender – not email – to confirm whether an email is legitimate, since unauthorized actors often remain in the email box, responding affirmatively to questions of legitimacy.
In addition, the HHS Report suggests that small to medium practices do the following:
Implement proven and tested response procedures when employees click on phishing emails. This can be accomplished by conducting phishing simulations, as described in the technical volume for small entities. These tests will provide entities with an understanding of how likely their workforce is to click on potentially malicious links (i.e., how many people actually click the link), and they can help organizations identify attacks that bypass established email security protections.
Establish cyber threat information sharing with other healthcare providers. Sharing information about how an organization has been attacked may seem scary, but the Report points out that pursuant to Executive Order 13691,
"when a member organization provides an information sharing and analysis organization (ISAO) with information about cyber-related breaches, interference, compromise, or incapacitation, the ISAO must: protect the individuals' privacy and civil liberties, preserve business confidentiality, and safeguard the information being shared."
In other words, the ISAO shares information about the attack, not about the entity that reported the information. Receiving cyber intelligence through ISAO sharing can help entities increase safety precautions around emerging threats they otherwise would not be aware of.
Ransomware is becoming a significant threat to small providers and taking steps to prevent and avoid attacks are important especially in light of the more aggressive pursual of HIPAA violations.
If you have questions, feel free to contact me:
Peter J. Cass, O.D.
VP Development, Practice Compliance Solutions
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TRENDS ANALYSIS IN GIG ECONOMY
Dragan Milosevic University Union - Nikola Tesla, Serbia http://orcid.org/0000-0002-5979-2562
Dejan Ilic University Union- Nikola Tesla http://orcid.org/0000-0001-8966-9955
Jovanka Popovic University Union - Nikola Tesla http://orcid.org/0000-0003-0263-0295
https://doi.org/10.26458/jedep.v10i3.702
Gig economy, Digital economy, work environment, freelancers, outsourcing
This paper aims to research the main characteristics and trends of the freelancer's market in temporary and occasional jobs in the Digital economy. The research was conducted by analyzing the content of reports of companies that hire freelancers, consulting companies and blogs of experts. The Digital economy is one of the results of the widespread application of information and communication technologies (ICT) that have influenced the transformation of the business model. Widespread application of ICT increased the flexibility of the work environment from the perspective of the place and time of work, as well as the way of engagement of the workforce. The application of ICT has expanded the scope and range of services available in today's market of temporary and occasional jobs. Contemporary companies in the Digital economy are increasingly focusing on their core competencies, leaving other jobs to external associates. The Gig economy refers to the engagement of labour in temporary and occasional jobs. Of the total number of employees in the USA during 2020, 36% worked as freelancers. Freelancers, during 2020, generated total revenue of US$ 1.2 trillion. Also, it is important to point out that a certain percentage of freelancers, currently, realize additional jobs in this way, but that there is a noticeable increase in the number of freelancers who accept this model of employment as the only way of permanent employment. Also, further growth in the number of freelancers in the world is predicted.
Dragan Milosevic, University Union - Nikola Tesla, Serbia
Dejan Ilic, University Union- Nikola Tesla
Faculty of Business Studies and Law
Jovanka Popovic, University Union - Nikola Tesla
And.co. (2021). The Slash Workers. Downloaded from And.co: https://www.and.co/slash-workers.html
Burning Glass Technologies. (2018). The New Foundational Skills of the Digital Economy: Developing the Professionals of the Future. The Business-Higher Education Forum. Burning Glass / BHEF.
CBRE. (2014). Fast Foward 2030: The Future of Work and the Workplace.CBRE
CBRE. (2021). The Future of Work: Reseting Asia Pacific Office Strategy. CBRE.
Downey L. (2021). Freelancer. Downloaded from Investopedia: https://www.investopedia.com/terms/f/freelancer.asp
European Economic and Social Committee. (2017). Impact of digitalisation and the on-demand economy on labour markets and the consequences for employment and industrial relations. European Economic and Social Committee.
Euroufound. (2020). New forms of emplozment: 2020 update. Euroufound.
Freelancer. (2021, February 17). Press - Freelancer.com Hits 50 Million Users. Downloaded from Freelancer: https://s3.amazonaws.com/press.freelancer.com/50%20Million%20Media%20Release.pdf
IDC. (2018a). IDC FutureScape: Worldwide IT Industry 2018 Predictions. IDC.
IDC. (2018b). Shaping the Future of Work in a Digital Era. IDC.
IDC. (2020a). Future of Work: Device Convergence Empowering Next-Generation Mobile Workers. IDC.
IDC. (2020b). Worldwide Digital Transformation Spending Guide. Downloaded from IDC: https://www.idc.com/getdoc.jsp?containerId=prUS46377220
Investopedia. (2021). Gig Economy. Downloaded from Investopedia: https://www.investopedia.com/terms/g/gig-economy.asp
McKinsey Global Institute. (2021). The future of work after COVID 19. McKinsey Global Institute.
Nedelkoska, L., & Quintini, G. (2018). OECD Social, Employment and Migration Working Papers No 202: Automation, skills use. Downloaded from OECD: https://www.oecd-ilibrary.org/employment/automation-skills-use-and-training_2e2f4eea-en
NOVO. (2021). A Freelancer's Quick Guide to Paying Taxes in 2020. Downloaded from Tools to Help You Build an Awesome Business: https://resources.banknovo.com/a-freelancers-quick-guide-to-paying-taxes-in-2020/
OECD. (2020a). A Roadmap Toward a Common Framework for Measuring the Digital Economy. OECD.
OECD. (2020b). OECD Employment Outlook 2020: Worker Security and the COVID-19 Crisis. OECD.
Statista. (2020). Leading reasons for doing freelance work in the United States in 2020. Downloaded from Statista: https://www.statista.com/statistics/530887/top-ten-reasons-for-freelance-work-us/
STATISTA. (2021). The number of freelancers in the United States from 2017 to 2028. Downloaded from STATISTA: https://www.statista.com/statistics/921593/gig-economy-number-of-freelancers-us/
Twin A. (2021). Outsourcing. Downloaded from Investopedia: https://www.investopedia.com/terms/o/outsourcing.asp
Tyler B. (2020). Gig economy. Salem Press Encyclopedia.
Upwork. (2020). Freelance Forward 2020. Upwork.
Upwork. (2021a). Remote Freelancing Across Industries. Upwork.
Upwork. (2021b). The 30 Most In-Demand Jobs and Skills [2019 - 2021]. Downloaded from Upwork: https://www.upwork.com/resources/in-demand-jobs-and-skills
Wired & Deloitte Digital. (2019). Digital transformation: A Primer. Downloaded from Wired: https://www.wired.com/brandlab/2019/10/deloittedigital-connecting-digital-age/
Wolfe L. (2021). The Difference Between Freelancers and Contractors. Downloaded from The Balance Small Business: https://www.thebalancesmb.com/are-freelancers-contractors-or-independent-contractors-3515143
World Economic Forum. (2020). The Future of Jobs Report 2020. World Economic Forum.
Journal of Economic Development, Environment and People by Authors is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at http://ojs.spiruharet.ro/index.php/jedep/issue/archive.


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Buhari Shelves N5000 Monthly Stipend; Urges Nigerians Living Abroad To Be More Focus
Posted by African Examiner Featured, Latest News, News Monday, February 29th, 2016
By Niyi Adeyi
President Muhammadu Buhari has shelved the plan by his party to pay N5000 monthly stipend, to the vulnerable and unemployed youths as well as urged Nigerians living in diaspora to be more focus and diligent on their means of livelihood.
Buhari said these over the weekend at two different interactive for in Saudi Arabia and Qatar, as part of his activities in the one week visit to the two Gulf countries.
Mr. President indicated that there was a slight difference in his priority and his party on the payment of N5,000, he considered as "largesse." to unemployed youths.
Instead, Buhari who is expected back in Nigeria today, held in lieu of paying unemployed youths the monthly token, his administration would build infrastructure; the school; correct them and empower agriculture, mining so that every able-bodied person could go and get work and be productively engaged.
While addressing the Nigerian community Sunday in Doha, President Buhari charged Nigerians to rekindle the spirit of patriotism and selfless service to encourage more foreign investment in the country.
"Like I tell other Nigerians living in other countries, you must be good ambassadors of the country because globally things are quite difficult in terms of security and the economy.
"If you have something doing, please hold it tight to your heart and try to improve on it because things are quite difficult all over.
"For us in Nigeria, we are trying to do our best to reduce unemployment and we realised this quite early enough, that was why we campaigned on the diversification of the economy.
"We have to go back to agriculture, like in the good old days of cocoa, palm oil, palm kernel, cotton, groundnuts, hides and skins'' the President advised.
On the fight against terrorism, the President told Nigerians in the oil reach State that the Military have made remarkable progress in dislodging Boko Haram terrorists from their previous strongholds in the North eastern part of the country.
"When we came in May 2015, the Boko Haram were effectively holding 14 local governments out of 774 local governments in Nigeria but today they are not holding a single local government,'' he said.
He said the military and law enforcement agencies were working very hard to prevent and counter Boko Haram attacks on soft targets in the country.
Mr. President affirmed: "We have abundant natural and human resources in the country, including huge oil and gas deposit.
"But the challenge is securing those installations and to persuade some of our countrymen to be more patriotic and stop sabotaging the installations.
"When people are being abducted and some are being murdered, when installations are being blown up now and then, the incentive for people to invest in our infrastructure is quite slim''.
President Buhari also promised to look into the complaints of alleged blanket denial of visas to families and relatives of Nigerians intending to visit the State of Qatar.
Judge Withdraws From Suit Seeking Tambuwal's Removal
Dickson Orders Payment of March Salaries Through BVN
9 People Killed in Ebonyi, Cross River Renewed Communal Clash
BREAKING: APGA Candidate, Willy Obiano Cruising to Victory in Anambra Election
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I decided a few months ago that I really needed to start teaching my soon to be four year old some basic skills. Things like learning to be on a schedule and things that should be routine. Not just something we do when I think about it or when we remember...or more likely, when she is bored and I'm scrambling to be that oh so creative Momma, with my Thirtyone bag of arts, crafts, and games to thrill and educate her on a whim!
There are some basics that she does well and I think I focused too much on those things and sort of...got lazy on other things she should or could be learning.
She does surprisingly well with bathing and brushing her teeth with supervision.
So, what else could I be teaching her now?
The last two weeks we have been learning together.
She still wants me hovering over each"worksheet" but it is a start. If I move or try to leave her to do it herself, she immediately is "done". Unless, it is art or painting.
I have started by using free printable sites and found numerous pages for her to learn the alphabet, trace letters, find letters or objects, decorate or design her own coloring page, count and much more.
We are starting with the letter E this week. This letter is obviously familiar to her as her name begins with E, so I found her tons of pages all about that letter.
She loves that she knows "Her" letter and gets super excited every time she finds it...ANYWHERE!
I admit, I was totally impressed at how well she did!
We also did a little art work this week. She named each color as she asked to have it opened. I can't believe how well and carefully she finished this...on her own!
This girl can stay in the lines like a four year old BOSS!
The weather has been amazing this week, so we took our "class" outside to have some cousin, play, and learning time!
Hopscotch is always a WIN!
They had sooo much fun and we mixed it up a bit.
They could tip toe, big jump, hop or run on the numbers. They could call out a number and the other had to "jump" to find it.
There are so many ways to add learning to a simple chalk graph of numbers. They even got a little P.E. time and lots of laughs.
School and learning shouldn't have to feel like a battle between you and the "student".
If you love to learn and show them how to do it creatively, it is a win/win!
CoffeeWastedMomma and Little E = week #1 done!
We all need a little Momma love!


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Pillars of Black Media, Once Vibrant, Now Fighting for Survival
Posted on July 3, 2016 by The Soul Brother
By Sydney Ember and Nicholas Fandos
For the black community in Chicago and elsewhere, Johnson Publishing Company represented a certain kind of hope.The company's magazines, most notably Ebony and Jet, gained prominence during the struggle for civil rights — Jet published graphic photos of the murdered black teenager Emmett Till that helped intensify the movement — and made it their mission to chronicle African-American life.At a time when much of the media was ignoring black people, or showing them primarily in the context of poverty or crime, Ebony and Jet celebrated their success, featuring stars like Muhammad Ali and Aretha Franklin on their covers. When Barack Obama was elected president in 2008, the first print publication he granted an interview to was Ebony.So when Johnson Publishing, which is based in Chicago, announced a little more than two weeks ago that it had sold Ebony and Jet to a private equity firm in Texas, there was a sense of loss."It was a very heartbreaking day," said Melody Spann-Cooper, the chairwoman of Midway Broadcasting Corporation, which owns a Chicago radio station, WVON, aimed at a black audience. "Ebony gave to African-Americans what Life didn't."
Read More: Pillars of Black Media, Once Vibrant, Now Fighting for Survival – The New York Times
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