text:
Home Rule Sets CLASH of Spears - Hardcover Rulebook
CLASH of Spears - Hardcover Rulebook
CLASH of Spears allows you to play raids and skirmishes in ancient times using small forces of 30 or more figures per side. This fast-paced game demands constant attention from all players as they alternate using their commanders to push units into battle. However, be careful not to drive your soldiers to exhaustion and plan every move since enemies are always ready to interrupt your actions and try to thwart your plans.
This book includes all the rules needed to play the game as well as five encounter scenarios reflecting common types of engagements. The book also includes lists for you to build forces for the different nations engaged at war between the fourth and second century BC in the Western Mediterranean: Republican Romans, Carthaginians, Gauls, Ligurians, Samnites, Etruscans, Campanians, Iberians, Celtiberians, Lusitanians, Sicilian Greeks, Southern Italy Greeks, Numidians and Pyrrhic armies.
CLASH of spears is a fast paced, warband level, wargame to be played in 4' by 4' tables over 1 to 3 hours. It has been designed to be fast and intuitive, while providing realistic feel, tactical depth and high impact decision making.
Gritty and personal, CLASH relies on command points and fatigue management to represent the chaos and desperation of small scale conflict. Skills and traits are used to represent the different specialties and fighting styles of the ancient world.
CLASH moves away from the massed ranks approach of traditional ancients rule sets. It has been thought of specifically for small actions, allowing units to close and open ranks as needed, and according to their preferred fighting style.
Very much a game designed by wargamers for wargamers CLASH is able to achieve tactical depth through the value of combination of arms and terrain management, rather than resorting to board game mechanics and game specific external aids.
Check out the youtube tutorial here

redpajama_set_name: RedPajamaCommonCrawl


PHILADELPHIA (January 11, 2018) – Philadelphia District Attorney Larry Krasner today continued his pledge to transform the Philadelphia District Attorney's Office (DAO) into the most effective prosecutor's office in the nation by appointing Judge Carolyn Engel Temin as his Interim First Assistant District Attorney and restructuring the DAO.
Judge Temin was appointed by the High Representative of Bosnia and Herzegovina as an International Judge on the Court of Bosnia and Herzegovina in Sarajevo. The court tried cases involving organized crime, public corruption, and war crime cases including those from the International Criminal Tribunal for the Former Yugoslavia in the Hague (Sept 2004 through November 2005).
"What District Attorney Krasner has proposed is a complete modernization of the office. I'm honored to be part of the transformation," said Interim First Assistant District Attorney Temin.
After leaving private practice, Judge Temin became the first female lawyer to be hired by the Defender Association of Philadelphia (1964 to 1970) where she was responsible for, including other initiatives, parole and probation hearings. She joined the DAO in 1971, working in the office for nearly ten years, in numerous positions including as a Homicide prosecutor. And in 1983, she was elected to Philadelphia County's Court of Common Pleas where she ruled on thousands of cases until she left the bench in 2003.
Temin has also lectured, taught, and consulted on legal issues and judicial systems in Serbia, Honduras, the Republic of Georgia, Tunisia, Thailand, and Barbados. She has received numerous international, national, and local awards for her work, and participates in a wide range of civic and legal activities and associations. She has a Bachelor of Fine Arts from the University of Pennsylvania (1955) and her Juris Doctorate from the University of Pennsylvania (1958).
This entry was posted on January 11, 2018 at 9:26 am and is filed under Press Release.	You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.


text:
\section{Introduction}

A striking feature common to many disordered systems is that in an out-of-equilibrium phase convergence to equilibrium is very slow, and it gets slower as the system gets `older'. This phenomena is usually called {\it aging} and has been studied extensively in physics community, both experimentally and theoretically. {\it Spin glasses} constitute an important class of disordered systems where aging occurs, and the dynamics of mean field spin glasses is one of the focal points of this article.

\newcommand{\mathcal{G}}{\mathcal{G}}

On the theoretical side, {\it trap models} have become the central paradigm to study aging properties of disordered systems.
They are simple enough to track analytically and many examples exhibit aging that can be established in a mathematically
rigorous way. The general description of trap models is the following. Let $\mathcal{G}=({\mathcal{V}},\mathcal{E})$ be a countable, connected graph where
${\mathcal{V}}$ and $\mathcal{E}$ denote the set of vertices and edges, respectively.
Consider a random collection of positive numbers $\boldsymbol{\tau}=\{\tau_x:x\in {\mathcal{V}}\}$. A {\it trap model}, $(X(t):t\geq 0)$, is a
continuous time Markov process on ${\mathcal{V}}$ whose dynamics is described as follows: at a vertex $x\in{\mathcal{V}}$, it waits for an exponential time
with mean $\tau_x$ and then moves to one of the neighbors of $x$ uniform at random.
Here, one can view a vertex $x$ as a {\it trap} and $\tau_x$ as the {\it depth} of the trap $x$. Let $(Y(i):i\in\mathbb{N}_0)$ denote the discrete time simple random walk on ${\mathcal{V}}$.
Another view of trap models is that $X$ follows $Y$ that gets trapped at vertices, collecting exponential random variables, and hence, is a time change of it.
This time-change process $S$, called the {\it clock-process}, is defined for $k\in\mathbb{N}$ by
\begin{equation}
S(k)=\sum_{i=0}^{k-1}\tau_{Y(i)}e_i,\;\;\;t\geq 0
\end{equation}
where $(e_i:\;i\in\mathbb{N}_0)$ is a collection of i.i.d. exponential mean-one random variables. The clock process and the Markov chain $Y$ completely describes $X$ via
\begin{equation}
	X(t)=Y(S^{\leftarrow}(t))
\end{equation}
where $S^{\leftarrow}$ is the generalized right inverse of $S$.

One central question in the study of the trap models is the analysis of the clock process, more precisely,
 the influence of the random trapping landscape.
If the trapping landscape is very heterogenous on certain large times, one expects that the main contribution to the clock process come from few `deep traps'.
Moreover, if the graph is transient enough on these time scales, the walk, upon leaving a deep trap, will get throughly lost before finding a new deep trap,
and as a result, the jumps of the clock process will be i.i.d. with the completely annealed distribution as the common distribution.
Summation of random variables being dominated by few large ones signifies a stable convergence for the clock process.
In the situations just described, the properly rescaled clock process converges to an $\alpha$-stable subordinator with $0<\alpha<1$.
From this, one can easily deduce that the system exhibits aging, namely, the two-time correlation function
\begin{equation}\label{twotime}
R(t_1,t_2):=\mathbb{P}(X_n(t_1)=X_n(t_2)|\tau)
\end{equation}
can be approximated, for large $t_1$ and $t_2$ corresponding time scales of observation,
by the probability that an $\alpha$-stable subordinator jumps over the interval $[t_1,t_2]$. The latter is a function of the ratio $t_1/t_2$ and is given by the classical arcsine law. The existence of a correlation function that
depends only on the trajectory of $X$ on a time interval $[t_1,t_2]$ whose large time limit is a non-trivial function of the ratio $t_1/t_2$
is usually described as aging both in mathematics and physics literature.

The picture described in the above paragraph has been made rigorous in \cite{BC08} in the context of sequence of diverging finite graphs.
In this article, too, we will study a growing sequence of finite graphs. The reason to study trap models on finite graphs is twofold: Firstly, it
allows one to employ potential theoretical methods strongly. Secondly, our main motivation, the mean field spin glasses, is in this setup.

Our main goal is to understand the one end of the spectrum of the $\alpha$-stable aging, namely, the case $\alpha=0$.
More precisely, we are interested in the situations where as the graphs grow, the heavy-tail index $\alpha$ converges to 0.
In this case, heterogeneity becomes even stronger in the sense that the main contribution to the clock process comes from the `deepest' of
the deep traps the walk can find. Hence, the limiting clock process has a structure of a `record' process, namely, it is an {\it extremal process}. More precisely, after a linear rescaling, contributions from deep traps still grow on an exponential scale. As a result, firstly, the maximal term dominates, and secondly, one has to perform a further non-linear scaling, cohorent with case of sum of i.i.d. random variables with slowly varying tails (see e.g. \cite{D52, KS86}).

In the spirit of \cite{BC08} we will give a set of conditions on the trapping landscape and the potential theoretical properties of the graph which will ensure that the clock process can be approximated, for appropriate large time scales, by an extremal process. Next, we will describe two additional conditions that guarantees that the two-time correlation function in (\ref{twotime}) can be approximated using extremal processes. Here, in order to get a non-trivial limit, one has to slightly enlarge the ratio of the two times with the volume, due to the non-linear scaling of clock process. We have called this type of aging {\it extremal aging}.

\iffalse
The conditions on the trapping landscape is the main difference from the set of conditions in \cite{BC08} while the other conditions are essentially the same. In general, one gets an extremal process for trap models either by observing the system on very short time scales or by increasing an external parameter, e.g. the inverse temperature. In the former case, the walk is even more transient on shorter scales and it is easier to check the potential theoretical results by comparing to the results on the longer time scales.

Let us briefly explain the nature of the limit of the clock process to an extremal process and the type of aging observed in this case that we have called {\it extremal aging}. First, let us give a characterization of $\alpha$-stable subordinators through Poisson Random Measures (PRM). Let $\{(t_i,\xi_i):i\in\mathbb{N}\}$ be the atoms of a PRM on $[0,\infty)\times (0,\infty)$ with intensity measure $dt\times \alpha x^{-2}dx$. Let $V_\alpha$ be an $\alpha$-stable subordinator with the Laplace transform
\begin{equation}
E\left[e^{-\lambda V_{\alpha}(t)}\right]=\exp\left(-ct\lambda^{\alpha}\right)
\end{equation}
where $c>0$. Then $V_\alpha$ can be written as
\begin{equation}\label{imdat}
V_{\alpha}(t)=\frac{1}{\Gamma(1-\alpha)}\sum_{t_i\leq t}(c\xi_i)^{1/\alpha},\;\;t\geq 0.
\end{equation}
Let ${\alpha_n}>0$ be a sequence such that ${\alpha_n}\to 0$. In the situations we are interested, after a proper linear scaling, the clock process can be approximated by $V_{{\alpha_n}}$. Using (\ref{imdat}), we have as $n\to\infty$
\begin{equation}
V_{\alpha_n}(t)=\frac{1}{\Gamma(1-\alpha_n)}\sum_{t_i\leq t}(c\xi_i)^{1/\alpha_n}\approx \max_{t_i\leq t} \;(c\xi_i)^{1/{\alpha_n}}.
\end{equation}
From this, it is clear that one has to perform a further non-linear rescaling to get
\begin{equation}
\big(V_{\alpha_n}(t)\big)^{\alpha_n} \approx c\max_{t_i\leq t} \;\xi_i
\end{equation}
as $n\to\infty$. The object on the right hand side of the above display is an extremal process. Hence, we will have a non-linear normalization and the limiting clock process will be an extremal process. For the aging results, one has to choose a slightly enlargened two-time window, that is, $t_2=t_1 a^{1/{\alpha_n}}$, and in this case, we will see that the limiting aging function is simply $1/a$.
\fi

Let us now discuss the trap model dynamics of spin glass models, also known as Random Hopping Time (RHT) dynamics. Our focus is on mean field models. The simplest mean field spin glass model is Random Energy Model (REM) introduced by Derrida in \cite{Der1}, where there are no correlation between energies. The first aging results for REM were proved in \cite{BBG03a} and \cite{BBG03b} for time scales only slightly shorter than equilibration times, using renewal techniques. Later, \cite{BC08} proved that the dynamics of REM model ages with the arcsine law as the limiting aging function. The time scales where REM exhibits aging were later extended in \cite{CG08}. In the second part of this article we extend these results to include extremal aging. We will let the temperature vary with the volume of the system in order to get a richer picture and prove that the dynamics of REM exhibits extremal aging for a wide range of temperature and time scales. More precisely, extremal aging occurs for fixed positive temperatures and subexponential time scales (in dimension of the system); for temperatures vanishing with the volume and for exponential, subexponential and superexponential time scales; and, for temperatures diverging with the volume and for (very) subexponential time scales. These results also signify that the aging is related to how transient the relaxation of the system is when far from equilibrium and it might have little to do with the actual equilibrium properties of the system. The occurrence of aging even in infinite temperature is a strong demonstration of this fact.

Let us mention the results on correlated mean field spin glass models. The arcsine law as an aging scheme, surprisingly, proved to be true even for some correlated spin glasses. In \cite{BBC08}, authors proved that the same aging mechanism essentially holds true for short enough time scales for the $p$-spin models. Later, finer results on the aging of $p$-spin models were achieved in \cite{BG13}.

Our original motivation to study the $\alpha=0$ case, or rather the dynamics of mean field spin glasses on subexponential time scales, stemmed from the aim of extending the REM universality to the dynamics of the Sherrington Kirkpatrick (SK) model. The results on the statistical properties of the SK model (see \cite{BGK08}) indicated that in order to not feel the correlations in the model, one has to investigate the dynamics on subexponential time scales. The dynamics on such time scales are in the category of extremal aging. In \cite{BG12} we proved that on subexponential time scales the clock process of the SK model and $p$-spin SK models converge to extremal processes and these systems exhibit extremal aging. Finer analysis was carried out later in \cite{BGS13}.

Recently, extremal aging has been observed for a type of model different than spin glass models. In \cite{CFK13}, it was proved that biased random walk on critical Galton-Watson trees conditioned to survive exhibits extremal aging.

The rest of the article is organized as follows: In Section \ref{scClock} we describe precisely the models
we study, give a set of conditions on the trapping landscape and potential theoretical
properties of the graph, and prove that these conditions imply the convergence of the rescaled clock process to an extremal process.
In Section \ref{sc3} we set two additional conditions that leads to extremal aging. In Section \ref{sc4} we prove our results on the dynamics of REM.

\iffalse

The first version of a trap model was introduced by Bouchaud in \cite{Bou92}. The main idea of Bouchaud is the following caricature that might occur in the dynamics of disordered systems: low energy configurations form form valleys that are far away from each other; the system spends long time in these deep valleys and when it leaves one, since deep valleys are sparse, will move to a new one chosen uniform at random. To describe this in a mathematical language, let ${\mathcal{V}}_n=\{1,\dots,n\},n\in\mathbb{N}$, and to each $x\in{\mathcal{V}}_n$ attach a positive random variable $\tau_x$,
\begin{equation}
P(\tau_x\geq u)=u^{-\alpha},\;\;\;0<\alpha<1
\end{equation}
that are i.i.d. in $x$. The choice of polynomial tails is to reproduce long waiting times seen experimentally. Then, Bouchaud's trap model is a continuous time Markov process, $X_n$, that at each vertex $x$ waits an exponential time with mean $\tau_x$ and than moves to a vertex in ${\mathcal{V}}_n$ chosen uniformly at random. Let $(Y_n(i):i\in\mathbb{N})$ be a collection of i.i.d. random variables on ${\mathcal{V}}_n$ with uniform distribution. It is clear that the trajectory of $X_n$ follows $Y_n$ while it gets trapped at vertices with random mean waiting times. Hence, $X_n$ can be written as a time change of $Y_n$, that is, let $S(0)=0$ and for $k\in\mathbb{N}$ define
\begin{equation}
S_n(k):=\sum_{i=0}^{k-1}\tau_{Y_n(i)}e_i
\end{equation}
where $(e_i:\;i\in\mathbb{N}_0)$ is a collection of i.i.d. mean one exponential random variables. This time change process, that we call {\it clock process}, contains enough information about the system. In this simple model, it is immediate to see that, if one chooses number of jumps short enough so that the whole graph cannot be discovered by the chain $Y_n$, a completely annealed picture would emerge, with a stable convergence for the clock process. Namely, let $r(n)=n^\gamma,\gamma<1$ and $t(n)=n^{\gamma/\alpha}$, then, for a.s. random landscape
\begin{equation}\label{cll}
\frac{S_n(\cdot\;r(n))}{t(n)}\Longrightarrow V_{\alpha}(\cdot\;) \text{ on } D([0,\infty),J_1)
\end{equation}
where $D([0,\infty),J_1)$ denotes weak convergence on the Skorohord topology of c\'adl\'ag functions on $[0,\infty)$ and $V_{\alpha}$ is an $\alpha$-stable subordinator. From this, one can easily deduce that for a.s. random landscape
\begin{equation}\label{cmpage}
P(X_n(t(n))=X_n((1+\theta)t(n)))\longrightarrow {Asl}_{\alpha}(1/1+\theta)
\end{equation}
where $Asl_{\alpha}$ denotes the classical arcsine law. The appearance of $Asl_{\alpha}$ is clear from the relation
\begin{equation}
P(\{V_{\alpha}(t):t\geq 0\}\cap [1,1+\theta]=\emptyset)={Asl}_{\alpha}(1/1+\theta).
\end{equation}
(\ref{cmpage}) is a typical aging result. In order to prove that a system ages, one has to find a correlation function of the system that depends on the trajectory of the dynamics on two time interval $[t_w,t_w+t]$ and prove that it converges to a non-trivial limit, when both $t_w$ and $t$ diverges while their ratio is constant.

Bouchaud's trap models naturally generalize to any graph with any trapping landscape. More precisely, one ingredient is a connected graph with a countable number of vertices and a trapping landscape that assigns a depth to each vertex. Then, trap model is a continuous time Markov process that is time change of the simple symmetric random walk on the graph with the summation of collected waiting times on the trajectory, as in (\ref{cll}). In this article, we focus on the case where we have sequence of finite graphs, just as in BTM.

It is at first not clear that the Bouchaud's picture would be relevant as in more realistic models of disordered system one does not have the crucial heavy tailed distributions. However, the point is that, on certain scales, it is possible to reproduce such a heterogenous environment of traps. The second crucial ingredient of the BTM was that the dynamics follow a simple random walk on a complete graph. This too can be achieved, one needs that the relevant deep traps that form the heterogenous landscape to be sparse enough and the underlying graph have a transient structure so that moving from one deep trap the dynamics gets lost in the graph until and hence, find a new deep trap chosen uniform at random. That is the main idea of \cite{BC08}, where the authors describe a set of condition on the graphs that ensures that on certain depth scales the landscape has a heavy tailed structure with index $\alpha$ and the this density is sparse and the graph is transient enough so that essentially a complete graph picture among deep traps is true. Thus, they prove the convergence of the clock process to an alpha stable subordinator and prove aging.

On the theoretical side, {\it trap models} have become the central paradigm to study aging properties of disordered systems. Main ingredients of a trap model are a graph and a random landscape on the set of vertices. Let $\mathcal{G}=({\mathcal{V}},\mathcal{E})$ be a connected graph with countable number of vertices where ${\mathcal{V}}$ and $\mathcal{E}$ denote the set of vertices and edges, respectively. To each vertex $x\in{\mathcal{V}}$ a positive number, $\tau_x$ is attached; we call the collection $\{\tau_x:x\in {\mathcal{V}}\}$ trapping landscape is chosen random. A {\it trap model} $X$ is a continuous time Markov process on ${\mathcal{V}}$ whose dynamics is described as follows: at a site $x\in{\mathcal{V}}$ it waits for an exponential time with mean $\tau_x$ and than moves to one of the neighbors of $x$ uniform at random. Here, one can view a vertex $x$ as a {\it trap} and $\tau_x$ as the {\it depth} of the trap $x$. More precisely, the trajectory of $X$ follows a simple symmetric random walk on ${\mathcal{V}}$ while it gets trapped at vertices. Let us denote by $Y$ the discrete time simple random walk on ${\mathcal{V}}$, that is, at each time, at each vertex $x$, $Y$ chooses a neighbor of $x$ uniform at random. It is clear that $X$ is a time-change of $Y$. This time-change process, which we denote by $S$, is called {\it clock-process} and described as follows: for $k=0$ set $S(0)=0$ and for $k\in\mathbb{N}$
\begin{equation}
S(k)=\sum_{i=0}^{k-1}\tau_{Y(i)}e_i
\end{equation}
where $(e_i:\;i\in\mathbb{N}_0)$ is a collection of i.i.d. mean one exponential random variables. It is obvious that the clock process and the Markov chain $Y$ completely describes $X$ via
\begin{equation}
	X(t)=Y(k) \;\text{  if  } S(k)\leq t < S(k+1).
\end{equation}

discrete time Markov chain $\{Y(i):i\in\mathbb{N}_0\}$ on a countable space ${\mathcal{V}}$ and a collection of positive numbers $\{\tau_x:x\in {\mathcal{V}}\}$ which will be chosen random. Consider a continuous time Markov process
$\{X(t):t\geq 0\}$ that follows the trajectory of $Y$ while at a vertex $x$ it waits an exponential waiting time with mean $\tau_x$. Here, one can see a vertex $x$ as a {\it trap} and $\tau_x$ as the {\it depth} of the trap $x$. We call $X$ a {\it trap model}. It is clear that $X$ is a time-change of the underlying discrete time Markov chain $Y$. This time-change process, which we denote by $S$, is called {\it clock-process} and described as follows: for $k=0$ set $S(0)=0$ and for $k\in\mathbb{N}$
\begin{equation}
S(k)=\sum_{i=0}^{k-1}\tau_{Y(i)}e_i
\end{equation}
where $(e_i:\;i\in\mathbb{N}_0)$ is a collection of i.i.d. mean one exponential random variables. It is obvious that the clock process and the Markov chain $Y$ completely describes $X$ via
\begin{equation}
	X(t)=Y(k) \;\text{  if  } S(k)\leq t < S(k+1).
\end{equation}

In \cite{BC08}, Ben Arous and \v{C}ern\'y give general conditions on the trapping landscape and the potential theoretical properties of the chain $Y$ that guarantee the convergence of the clock process, on appropriate time scales, to an $\alpha$-stable subordinator. In this article our interest is in the one end of this spectrum, namely, the case `$\alpha=0$'. The $\alpha$-stable convergence occurs in the cases where the clock process is dominated by the terms coming from very few, `deep traps'. In the case where $\alpha\approx 0$, this heterogeneity becomes stronger in the sense that the main contribution to the clock process comes from the `deepest' trap the Markov chain finds. Hence, the limiting clock process has a structure of a `record' process. We will see that the right object is an {\it extremal process} a process that can be seen as the maximum jump of subordinators.

The conditions on the chain $Y$ ensures that a 'completely annealed' picture is possible and the condition on the landscape

Trap models were first introduced by Bouchaud in \cite{Bou92}. The model Bouchaud studied was as follows: ${\mathcal{V}}=\{1,\dots,n\}$ where $n$ is large, $Y$ is the Markov chain where for each jump it chooses a vertex in ${\mathcal{V}}$ uniform at random, and crucially the depths are chosen to be random, i.i.d. and in the domain of attraction of an $\alpha$-stable law with $0<\alpha<1$. For simplicity, we take $P(\tau_x\geq u)=u^{-\alpha},u\geq 1$. It is easy to see that for time scales not very long the properly normalized clock process converges to an $\alpha$-stable subordinator. Let us recall stable subordinators. An $\alpha$-stable subordinator, $(V_{\alpha}(t):t\geq 0)$ is an increasing L\'evy process (independent, homogenous increments) with the Laplace transform
\begin{equation}
E\left[e^{-\lambda V_{\alpha}(t)}\right]=\exp\left(-ct\lambda^{\alpha}\right), \lambda>0
\end{equation}
where $c$ is a positive constant. Another point of view that will be important for us is the following: Let $(t_i,\xi_i)$ be the points of a Poisson point process (PPP) on $[0,\infty)\times (0,\infty)$ with the mean measure $dt\times c' \alpha x^{-1-\alpha}$ with $c'=c/\Gamma(1-\alpha)$, then $V_\alpha$ given by the Laplace transform above is same as
\begin{equation}\label{VaPPP}
V_{\alpha}(t)=\sum_{t_i\leq t}\xi_i.
\end{equation}

The connection of $\alpha$-stable subordinators and aging is through the arcsine law. Let us define
\begin{equation}
R(t_w,t)=P(X(t+t_w)=X(t)|\boldsymbol{\tau}).
\end{equation}
Here, $\boldsymbol{\tau}$ denotes the depths of traps and in the above notation it is meant that the depths are fixed throughout the evolution of $X$. In words, $R(t_w,t)$ is the probability that the system is at the same state at times $t_w$ and $t_w+t$. In most cases it is simple to show that $R(t_w,t)$ is the same as (asymptotically) the probability that the clock process $S$ jumps over the interval $(t_w,t_w+t)$. Now the the classical arcsine law for subordinators states that the probability that $V_\alpha$ jumps over an interval $(t_w,t+t_w)$ is a function of the ratio $t_w/t+t_w$. This, in turn implies that, for the time scales that the clock process converges to $V_\alpha$, $R$ becomes a function of the ratio $t_w/t+t_w$. This signifies that the decorrelation of the system slows down with time, moreover, in this case, it asymptotically becomes the ratio of two times, age of the system $t_w$ and the additional time you observe the system $t$. If one proves a result like this, one say $X$ exhibits aging.

In the important (*use another word) article \cite{BC08}, Ben Arous and \v{C}ern\'y, gave very general conditions on the trapping landscape and the Markov chain that leads to the convergence of clock process to a stable subordinator, and in turn proves the aging of the system.

In this article our interest is the one end of the spectrum described above, namely, we are interested in the case, formally, `$\alpha=0$'. Let us explain what one expects in this case for the convergence of clock process. We should first emphasize that convergence to an $\alpha$-stable subordinator means that the sum of waiting times collected by the trap model $X$ is dominated by a few of them. Moreover, as $\alpha$ gets smaller this domination is done by fewer and fewer very deep traps. In the case $\alpha=0$, finally, the contribution to the clock process comes only from one trap, namely the deepest trap the dynamics can find.

What is then the right normalization and limiting object for clock process? Let us turn to (\ref{VaPPP}) to answer this. As noted in the above paragraph, we expect that the largest jump dominates $V_\alpha$, that is $V_\alpha(t)\approx \max_{t_i\leq t}\xi_i$, as $\alpha\to0$. We have $P(\max_{t_i\leq t} \xi_i \leq a^{1/\alpha})=e^{-ta}$, that is, the logarithm of the largest jump grows like $1/\alpha$. Hence, one has to the a non-linear normalization: $V_\alpha^\alpha(t)\approx \max_{t_i\leq t}\xi_i^\alpha$. Finally note that in distribution, $(t_i,\xi_i^\alpha)$ is the points of PPP on $[0,\infty)\times(0,\infty)$ with mean measure $dt\times x^{-2}dx$. Hence, after a non-linear normalization, the limiting object is a process that gives the maximum of jumps of a PPP. This falls into the category of {\it extremal processes}. In the first part of the this paper, following \cite{BC08} we will describe a set of condition on the trapping landscape and the Markov chain $Y$ that leads to the convergence of the clock process to an extremal process. This, with little extra effort, will lead to a version of aging, which we have called {\it extremal aging}.

\fi

\section{Convergence of the clock process to an extremal process.}\label{scClock}

We start this section by introducing precisely the type of trap models we will study. Let $\mathcal{G}_n=({\mathcal{V}_n},\mathcal{E}_n)$, $n\in\mathbb{N}$,
 be a sequence of finite, connected graphs with ${\mathcal{V}_n}$ and $\mathcal{E}_n$ denoting the set of vertices and edges, respectively.
We use the notation $x\sim y$ for an edge $(x,y)\in\mathcal{E}_n$. For each $n\in\mathbb{N}$ and vertex $x\in{\mathcal{V}_n}$ we attach a positive number $\tau_x$
which constitutes the "depth" of the trap at site $x$. We denote the collection of depths by $\boldsymbol{\tau}_{n}=\{{\tau_x}:\;x\in{\mathcal{V}_n}\}$. We will
call $\boldsymbol{\tau}_{n}$ the `trapping landscape' or `environment' and later we will choose $\boldsymbol{\tau}_{n}$ random. Given the trapping landscape $\boldsymbol{\tau}_{n}$ we define a continuous time Markov process $\{X_n(t):t\geq 0\}$ on ${\mathcal{V}_n}$ whose transition rates are given by
$w_{xy}^{\boldsymbol{\tau}_{n}}=\inc{x\sim y}/(d_x \tau_x)$ where $d_x=\#\{y\in{\mathcal{V}_n}:x\sim y\}$ is the degree of the vertex $x$.
In other words, at a vertex $x$, the Markov process $X_n$ waits an exponential time with mean ${\tau_x}$ and than it moves to one
of its neighbors chosen uniformly at random. We denote by $\mathbb{P}_x$ and $\mathbb{E}_x$ the distribution and expectation of $X_n$ starting from
$x\in{\mathcal{V}_n}$. We will always start $X_n$ from an arbitrary but fixed vertex we denote by $0$ that does not depend on $\boldsymbol{\tau}_{n}$ and
write for simplicity $\mathbb{P}=\mathbb{P}_{0}$ and $\mathbb{E}=\mathbb{E}_0$. Note that, although $X_n$ depends on $\boldsymbol{\tau}_{n}$ we surpressed it in the notation.

For each $n\in\mathbb{N}$ we take the trapping landscapes $\boldsymbol{\tau}_{n}\in (0,\infty)^{{\mathcal{V}_n}}$ random with $P_n$ and $E_n$
denoting its distribution and expectation, respectively. We embed all the random variables $\boldsymbol{\tau}_{n}$, $n\in\mathbb{N}$,
independently into a common probability space and $P$ and $E$ stands for the distribution and expectation of this probability space.
For the events that happens $P$ almost surely, we will simply say `for a.s. random environment $\boldsymbol{\tau}$'.

For any $n\in\mathbb{N}$, let $\{Y_n(i):i\in\mathbb{N}_0\}$ be the simple random walk on the graph $\mathcal{G}_n$, that is,
 the discrete time Markov chain on ${\mathcal{V}_n}$ with the transition probabilities $p_{xy}=\inc{x\sim y}/d_x$ and we
set the starting point $Y_n(0)=0$. We assume that the distribution of $Y_n$ is defined in the probability space $\mathbb{P}_0$.
  We define the clock process $S_n$ by setting $S_n(u)=0$ for $u\in[0,1)$ and for $u\in[1,\infty)$
\begin{equation}
S_n(u):=\sum_{i=0}^{\lfloor u\rfloor -1}\tau_{Y_n(i)}e_i
\end{equation}
where $\{e_i:i\in\mathbb{N}\}$ is an i.i.d. collection of exponential mean-one random variables independent from anything else. In words, for $k\in\mathbb{N}$, $S_n(k)$ is the time it takes for $X_n$ to make $k$ jumps. Clearly we have,
\begin{equation}
X_n(t)=Y_n(k) \;\;\text{ if   }S_n(k)\leq t < S_n(k+1).
\end{equation}

For $\boldsymbol{\tau}_{n}$ fixed, $S_n$ is a random variable taking values in $D([0,\infty))$, the space of c\'adl\'ag function on $[0,\infty)$.
Similarly, for any fixed $T>0$ the restriction of $S_n$ to $[0,T]$ is a random variable on $D([0,T])$,  the space of c\'adl\'ag function on $[0,T]$.

We need now further notation. Let $T_n$ be a stopping time for the chain $Y_n$. We define $G_{T_n}^n(x,y),x,y\in{\mathcal{V}_n}$ to be the Green's function of $Y_n$, that is, the mean number of times $Y_n$ visits $y$ before $T_n$, started from $x$:
\begin{equation}
 G_{T_n}^n(x,y):= \mathbb{E}_x\left[\;\sum_{i=0}^{T_n-1}\inc{Y_n(i)=y}\right].
\end{equation}
For $A\subseteq {\mathcal{V}_n}$ we define the first hitting time of $A$ by
\begin{equation}
H_n(A):=\inf\{i\geq 0:\;Y_n(i)\in A\}.
\end{equation}
For ease of notation we write $G_A^n$ for $G_{H_n(A)}^n$. Finally, we say that a random subset $A\subseteq {\mathcal{V}}_n$ is a percolation cloud with density $\rho\in(0,1)$ if $x\in A$ with probability $\rho$, independently of all other vertices.

Presently, we set 4 conditions that are about the trapping landscape and the properties of the walk $Y_n$ on ${\mathcal{V}}_n$. This set of very general, potential theoretical conditions will be used to prove our main result.

The first condition tells that a certain density of traps exceed a depth scale in a very heterogeneous way.

\newcommand{g_n}{g_n}
\newcommand{b_n}{b_n}
\noindent{\bf Condition A:} For any $n\in\mathbb{N}$ let $\boldsymbol{\tau}_{n}$ be i.i.d. in $x\in{\mathcal{V}}_n$. There exists a {\it depth rate scale} $\alpha_n$, a {\it depth scale} $g_n$ and a {\it density scale} $b_n$ with $g_n\to\infty$ and ${\alpha_n},b_n\to 0$ as $n\to\infty$ such that
\begin{equation}\label{condA1}
b_n^{-1}P_n\left(\tau_x\geq u^{1/{\alpha_n}}g_n\right)\overset{n\to\infty}\longrightarrow {1}/{u}
\end{equation}
uniformly in $u$ on all compact subsets of $(0,\infty)$. Moreover, there exists a constant $C$ such that for all $u>0$ and $d>0$, for all $n\in\mathbb{N}$
\begin{equation}\label{condA2}
P_n\left(\tau_x\geq u^{1/{\alpha_n}}dg_n\right) \leq \frac{Cb_n}{ud^{{\alpha_n}}}.
\end{equation}

\vspace{0.12in}

The next two conditions concern the potential theoretical properties of the graph.

\newcommand{f_n}{f_n}
\newcommand{t_n}{t_n}

\noindent{\bf Condition B:} Let $b_n$ be as in Condition A. Let $A_n,n\in\mathbb{N}$ be a sequence of percolation clouds
on ${\mathcal{V}_n}$ with densities $\rhob_n$ where $\rho\in(0,\infty)$. There exists a scale $f_n$ with $f_n\to\infty$ as $n\to\infty$ and a constant $\mathcal{K}_G$ independent of $\rho$ such that for a.s. sequence $A_n$
\begin{equation}
\max_{x\in A_n}\left|f_n^{-1}G_{A_n\setminus \{x\}}^n(x,x)-\mathcal{K}_G\right|\overset{n\to\infty}\longrightarrow 0.
\end{equation}

\newcommand{r_n}{r_n}

\noindent{\bf Condition C:} Let $A_n$ be as in Condition B. There exists a constant $\mathcal{K}_r>0$ such that for all $s>0$ and a.s. sequence $A_n$
\begin{equation}
\max_{x\in A_n\cup \{0\}}\left|\mathbb{E}_x\left[\exp(-\frac{s}{r_n} H_n(A_n\setminus \{x\}))\right]-\frac{\mathcal{K}_r\rho}{s+\mathcal{K}_r \rho}\right|\overset{n\to\infty}\longrightarrow 0,
\end{equation}
where $r_n=f_n/b_n$.
\vspace{0.12in}

The last condition contains technical restrictions.

\noindent{\bf Condition D:}
\begin{itemize}
\item[(i)] There exists a sequence of positive numbers $\lambda_n$ and a positive constant $\mathcal{K}_s$ such that for all $T>0$ and $n$ large enough
\begin{equation}
\sum_{x\in{\mathcal{V}}_n}\big(e^{\lambda_n G_{Tr_n}(0,x)}-1\big)\leq \mathcal{K}_s \lambda_n Tr_n
\end{equation}
and $\sum_{n=1}^\infty \exp(-c \lambda_n f_n)<\infty$ for some $c>0$.

\vspace{0.12in}

\item[(ii)] ${\alpha_n}\log(f_n)\longrightarrow 0$ as $n\to\infty$.
\end{itemize}

\vspace{0.12in}

Now we introduce formally the extremal processes. Let $F$ be a probability distribution function on $(-\infty,\infty)$.
For $l\in\mathbb{N}$ and $0\leq t_1\leq\cdots\leq t_l$ define the finite dimensional distributions
\begin{equation}\label{fddext}
F_{t_1,\dots,t_l}(x_1,\dots,x_l)=F^{t_1}\left(\bigwedge_{i=1}^l x_i\right) F^{t_2-t_1}\left(\bigwedge_{i=2}^l x_i\right)\cdots F^{t_l-t_{l-1}}\left(\bigwedge_{i=l}^l x_l\right)
\end{equation}
where $\bigwedge$ stands for the minimum. The family of finite dimensional distributions defined by (\ref{fddext}) is consistent and thus, by Kolmogorov extension theorem there exists a continuous time stochastic process $W=(W(t):t\geq 0)$ with these finite dimensional distributions. $W$ is called an extremal process generated by $F$ or $F$-extremal process.

We give another description of an extremal process. Let $F$ be as in the previous paragraph. Assume that $F$ is a continuous distribution with $supp(F)=\mathbb{R}$. Let $N$ denote a Poisson Random Measure (PRM) on  $[0,\infty)\times(0,\infty)$ with mean measure $dt\times \nu(dx)$ where $\nu(x,\infty)=-\log F(x)$. Let us denote by $(t_j,\xi_j)$ the marks of $N$. Then if we define
\begin{equation}
W(t)=\max_{t_i\leq t} \xi_i,\;\;\;t\geq 0
\end{equation}
$W$ is an $F$-extremal process. It is enough to check that $W$ satisfies (\ref{fddext}) for any $0\leq t_1\leq\cdots\leq t_l$ and for
any $x_1\leq x_2\leq \cdots\leq x_l$. By the continuity of $F$ and the independence properties of a PRM we get
\begin{equation}\label{poschr}
\begin{aligned}
&P(W(t_1)\leq x_1,\dots W(t_l)\leq x_l)\\&=P\big(N([0,t_1]\times [x_1,\infty))=0\big)\cdots P\big(N([t_{l-1},t_l]\times [x_l,\infty))=0\big)
\\&= e^{-t_1 \nu(x_1,\infty)}\cdots e^{-(t_l-t_{l-1})\nu(x_l,\infty)}
\\&= F^{t_1}(x_1)\cdots F^{t_l-t_{l-1}}(x_l).
\end{aligned}
\end{equation}
For details about extremal processes we refer readers to \cite{Res}.

For our convergence results of clock processes we use two different topologies on $D([0,T])$, namely, $M_1$ and $J_1$ topologies.
We will indicate the topology of weak convergence by simply writing `in $D([0,T],J_1)$' and `in $D([0,T],M_1)$'. The essential difference between
these two topologies is that while $M_1$ topology allows approximating processes make several jumps in short intervals of time to produce one bigger jump of the limiting process, while $J_1$ does not. See \cite{Whitt} for detailed
information on these topologies.

Below is our main result about the convergence of clock processes to an extremal process.

\begin{theorem}\label{thrm1} Assume Conditions A-D are satisfied. Set $t_n=\fng_n$. Then for a.s.~random environment $\boldsymbol{\tau}$, for any $T>0$
\begin{equation}\label{eqnthrm1}
\left(\frac{S_n(\cdot\;r_n)}{t_n}\right)^{\alpha_n}\overset{n\to\infty}\longrightarrow W(\cdot) \text{   in   } D([0,T];M_1)
\end{equation}
where $W$ is the extremal process generated by the distribution function $F(x)=e^{-1/x}$. Moreover, the above convergence holds in the stronger topology $D([0,T];J_1)$ if $f(n)=1$ and $\mathcal{K}_G=1$.
\end{theorem}

\begin{remark}
 Theorem \ref{thrm1} can be stated in a more general way. Namely, Condition A can be generalized such that (\ref{condA1}) is replaced by
\begin{equation}
 b_nP_n(\tau_x\geq L^{\leftarrow}(u h_n))\overset{n\to\infty}\longrightarrow \frac{1}{u}
\end{equation}
and (\ref{condA2}) by
\begin{equation}
 P_n(\tau_x\geq L_n^{\leftarrow}(u d^{\alpha_n} h_n))\leq \frac{Cb_n}{u d^{\alpha_n}}
\end{equation}
where $h_n$ is a diverging scale. Here, for each $n\in\mathbb{N}$, $L_n$ is positive, as $u\to\infty$, $L_n(u)\to\infty$ and for any $\lambda>0$
\begin{equation}
 \frac{L_n(\lambda u)}{L_n(u)}\longrightarrow \lambda^{{\alpha_n}}
\end{equation}
in a mildly uniform way in $n$. Also, in this setup one can have ${\alpha_n}=0$ which would mean that $L_n$ is a slowly varying function for such an $n$.
Then (\ref{eqnthrm1}) in Theorem
\ref{thrm1} becomes
\begin{equation}
 \frac{L_n\left(S_n(\cdot \;r_n )/f_n\right)}{h_n}\Longrightarrow W(\cdot).
\end{equation}
Since such a setup makes the notation very difficult and does not bring a new conceptual insight we preferred to use the current setup. Finally,
note that, choosing $L_n(u)=u^{{\alpha_n}}$ and $g_n=h_n^{1/{\alpha_n}}$ gives Theorem \ref{thrm1}.

\end{remark}
\newcommand{\dn}[1]{{d_n({#1})}}
\newcommand{\un}[1]{{U_n({#1})}}
\newcommand{\sn}[1]{{s_n({#1})}}
\newcommand{\mn}[1]{{m_n({#1})}}

In the rest of the current section we prove Theorem \ref{thrm1}.
We start by defining set of deep traps, very deep traps and shallow traps determined by the depth scale and the
depth rate scale ${\alpha_n}$ and $g_n$: for $0<\epsilon<M$
\begin{equation}
\begin{array}{lll}
T_{\epsilon}^{M}(n):=\{x\in{\mathcal{V}}:\epsilon^{1/{\alpha_n}} g_n \leq \tau_x\leq M^{1/{\alpha_n}} g_n\;\}&\textbf{deep traps},\\
\;T_M(n):=\{x\in{\mathcal{V}}:\tau_x> M^{1/{\alpha_n}}g_n\}&\textbf{very deep traps},\\
\;\;T^{\epsilon}(n):=\{x\in{\mathcal{V}}:\tau_x<\epsilon^{1/{\alpha_n}}g_n\}&\textbf{shallow traps.}
\end{array}
\end{equation}
Let $\dn{j}$ be sequence of times where a deep trap different from the last deep trap visited is found. We set $\dn{0}=0$ and for $j\in\mathbb{N}$ define recursively
\begin{equation}
\dn{j}:=\min\left\{i>\dn{j-1}:Y_n(i)\in T_{\epsilon}^{M}(n)\setminus\{Y_n(\dn{j-1})\}\right\}.
\end{equation}
We define the process $(U_n(j):j\in\mathbb{N}_0)$
\begin{equation}
U_n(j):=Y_n(\dn{j}),
\end{equation}
and $\zeta_n$ as the last time the random walk finds a deep trap before $Tr_n$:
\begin{equation}
{\zeta_n(T)}:=\max\{j:\;\dn{j}\leq Tr_n\}.
\end{equation}
Let $\sn{j}$ be the time $X_n$ spends at $\un{j}$ between the first time it visits $\un{j}$ until it finds a different deep trap, that is,
\begin{equation}
\sn{j}=\sum_{i=\dn{j}}^{\dn{j+1}}e_i \tau_{Y_n(i)}\inc{Y_n(i)=\un{j}}.
\end{equation}
We call $\sn{j}$ the score of $\un{j}$. Note that, for given environment $\boldsymbol{\tau}_{n}$ and $U_n(j)$, the expectation of $\sn{j}$ over all the other random sources is
\begin{equation}\label{meanscore}
\tau_{\un{j}}G_{T_{\epsilon}^{M}(n)\setminus \{\un{j}\}}^n(\un{j},\un{j}).
\end{equation}
Finally, we define $m_n$, the record process of $s_n$, for we expect it to be a good approximation of the clock process. For technical convenience we set $m_n(0)=0$ and $m_n(1)=S_n(\dn{1})$, and define for $j\geq 2$
\begin{equation}\label{defmn}
\mn{j}:=\max_{i=1,\dots,j-1}s_n(i).
\end{equation}
As a first step of the proof, we need to control the distribution of depths of deep traps visited. We introduce the following notation:
\begin{equation}
\rho_a^b:=a^{-1}-b^{-1},\;\;\;\;\;\;0<a<b.
\end{equation}
We need the following lemma which is from \cite{BC08}, stated in a slightly different way.
\begin{lemma}\label{lebc}(Lemma 2.5 on page 304 in \cite{BC08})
Recall that $A_n$ is a percolation cloud with density $\rhob_n$ and assume Condition C. Let $A_n^1$ and $A_n^2$ be such that $A_n^1\cup A_n^2 =A_n$ and $A_n^1\cap A_n^2=\emptyset$ and
\begin{equation}
\lim_{n\to\infty}\frac{|A_n^1|}{|A_n|}=\frac{\rho_1}{\rho},\;\;\;\lim_{n\to\infty}\frac{|A_n^2|}{|A_n|}=\frac{\rho_2}{\rho} \;\;\;\text{       with      }\;\;\;\rho_1+\rho_2=1.
\end{equation}
Then
\begin{equation}
\lim_{n\to\infty}\max_{x\in A_n\cup \{0\}}\left|\mathbb{P}_x\Big(H_n(A_n^1\setminus \{x\})<H_n(A_n^2\setminus \{x\})\Big)-\frac{\rho_1}{\rho}\right|=0.
\end{equation}
\end{lemma}
Let $(\sigma_\epsilon^M(j):j\in\mathbb{N})$ be an i.i.d. sequence of random variables with distribution
\begin{equation}\label{distsig}
P(\sigma_{\epsilon}^M(j)\geq u)=\frac{\rho_u^M}{\rho_\epsilon^M}.
\end{equation}
\begin{proposition}\label{asympindp}
The collection
$\left(\left(\tau_{\un{j}}/g(n)\right)^{\alpha_n}:j\in\mathbb{N}\right)$ converges weakly to $(\sigma_\epsilon^M(j):j\in\mathbb{N})$ as $n\to\infty$.
\end{proposition}
\begin{proof}
Since $(\un{j}:j\in\mathbb{N})$ is a Markov sequence, it is enough to show that for any $j\in\mathbb{N}$, as $n\to\infty$
\begin{equation}\label{arat}
\mathbb{P}\left(\left(\tau_{\un{j}}/g(n)\right)^{\alpha_n}\geq u|\un{j-1}\right)\longrightarrow \frac{\rho_u^\epsilon}{\rho_\epsilon^M}.
\end{equation}
We have
\begin{equation}\label{yetti}
\mathbb{P}\left(\left(\tau_{\un{j}}/g(n)\right)^{\alpha_n}\geq u|\un{j-1}\right)=\mathbb{P}_x\Big(H_n(T_u^M(n)\setminus \{x\})<H_n(T_\epsilon^u(n)\setminus \{x\})\Big)
\end{equation}
where $x=\un{j-1}$. Note that $T_{\epsilon}^{M}(n)=T_u^M(n)\cup T_\epsilon^u(n)$ and $T_u^M(n)\cap T_\epsilon^u(n)=0$. Moreover, by Condition A
\begin{equation}
\lim_{n\to\infty}\frac{|T_u^M(n)|}{|T_{\epsilon}^{M}(n)|}=\frac{\rho_u^M}{\rho_{\epsilon}^M}\;\;\;\text{ and }\;\;\;\lim_{n\to\infty}\frac{|T_\epsilon^u(n)|}{|T_{\epsilon}^{M}(n)|}=\frac{\rho_\epsilon^u}{\rho_{\epsilon}^M}.
\end{equation}
Hence, (\ref{yetti}), Condition A and Lemma \ref{lebc} finish the proof.
\end{proof}
\begin{lemma}\label{asindm}
The collection $\left(\left(\sn{j}/t_n\right)^{\alpha_n}:j\in\mathbb{N}\right)$ converges weakly to $(\sigma_\epsilon^M(j):j\in\mathbb{N})$ as $n\to\infty$.
\end{lemma}
\begin{proof}
Using the Markov property of the random walk we see that
\begin{equation}\label{arasifir}
\begin{aligned}
&\;\;\mathbb{P}\left((s_n(j)/t_n)^{\alpha_n}\geq u \big| s_n(1),\dots,s_n(j-1)\right)\\&=\int_\epsilon^M \mathbb{P}\left(s_n(j)\geq u^{1/{\alpha_n}}t_n\big|\tau_{\un{j}}=v^{1/{\alpha_n}}g_n,s_n(1),\dots,s_n(j-1)\right)
\\&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times \mathbb{P}\left((\tau_{\un{j}}/g_n)^{\alpha_n}\in dv \big| s_n(1),\dots,s_n(j-1)\right).
\end{aligned}
\end{equation}
Note that the first term inside the integral in the above display is equal to
\begin{equation}
\mathbb{P}\left(s_n(j)\geq u^{1/{\alpha_n}}t_n\big|\tau_{\un{j}}=v^{1/{\alpha_n}}g_n\right)\leq \left(\frac{u}{v}\right)^{1/{\alpha_n}} \frac{g_n}{t_n}\;G_{T_{\epsilon}^{M}(n)\setminus\{\un{j}\}}^n(\un{j},\un{j})
\end{equation}
where we use Chebyshev's inequality and (\ref{meanscore}) for the upper bound. By Condition B we have $G_{T_{\epsilon}^{M}(n)\setminus\{x\}}^n(x,x)=\mathcal{K}_{G}f_n(1+o(1))$ as $n\to\infty$ and the error term is uniformly bounded in $x\inT_{\epsilon}^{M}(n)$. Hence, recalling that $\fng_n=t_n$, if $u>v$ we have
\begin{equation}\label{arabir}
\mathbb{P}\left(s_n(j)\geq u^{1/{\alpha_n}}t_n\big|\tau_{\un{j}}=v^{1/{\alpha_n}}g_n\right)\longrightarrow 0.
\end{equation}
Since $\sum_{i=\dn{j}}^{\dn{j+1}}\inc{Y_n(i)=\un{j}}$ is always at least one and $t_n=\fng_n$, using Condition D part $(ii)$ one we have for $u<v$
\begin{equation}\label{araiki}
 \mathbb{P}\left(s_n(j)\geq u^{1/{\alpha_n}}t_n\big|\tau_{\un{j}}=v^{1/{\alpha_n}}g_n\right)\geq \mathbb{P}(e_1v^{1/{\alpha_n}}\geq u^{1/{\alpha_n}}f_n)\longrightarrow 1.
\end{equation}
Using (\ref{arabir}) and (\ref{araiki}) in (\ref{arasifir}), than applying Proposition \ref{asympindp} finishes the proof.
\end{proof}

We next show that the contribution from the shallow traps are negligible.
\begin{proposition}\label{shallowt}
For any $a,d>0$ and $T>0$, for any $\delta>0$ given, $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}
\mathbb{E}\left[\sum_{i=0}^{Tr_n} \tau_{\yn{i}}e_i \inc{\yn{i}\in T^a(n)}\Big|\;\boldsymbol{\tau}\right]\leq (a+d)^{1/{\alpha_n}}t_n.
\end{equation}
\end{proposition}
\begin{proof}
We first prove that $\boldsymbol{\tau}$ a.s. for $n$ large enough, for all $j\in\mathbb{N}$
\begin{equation}\label{aug}
\mathbb{E}\left[\sum_{i=1}^{Tr_n}e_i\tau_{Y_n(i)}\inc{Y_n(i)\in T_{a2^{-j{\alpha_n}}}^{a 2^{-j{\alpha_n}+{\alpha_n}}}(n)}\Big|\;\boldsymbol{\tau}\right]\leq (a+d/2)^{1/{\alpha_n}}t_n 2^{j{\alpha_n}-j+1}.
\end{equation}
By Condition A we have for all $n\in\mathbb{N}$ and $j\in\mathbb{N}$
\begin{equation}\label{dumonde}
p_{n,j}:=P\Big(\inc{x\in T_{a2^{-j{\alpha_n}}}^{a 2^{-j{\alpha_n}+{\alpha_n}}}(n)}\Big)\leq \frac{C 2^{j{\alpha_n}}}{a}b_n.
\end{equation}
We have
\begin{align}
\nonumber&\;\;P\left(\mathbb{E}\left[\sum_{i=1}^{Tr_n}e_i\tau_{Y_n(i)}\inc{Y_n(i)\in T_{a2^{-j{\alpha_n}}}^{a 2^{-j{\alpha_n}+{\alpha_n}}}(n)}\Big|\;\boldsymbol{\tau}\right]\geq (a+d/2)^{1/{\alpha_n}}t_n 2^{j{\alpha_n}-j+1}\right)\\
\nonumber&=P\left(\sum_{x\in{\mathcal{V}}_n}G_{Tr_n}(0,x)\tau_x \inc{x\in T_{a2^{-j{\alpha_n}}}^{a 2^{-j{\alpha_n}+{\alpha_n}}}(n)}\geq (a+d/2)^{1/{\alpha_n}}t_n 2^{j{\alpha_n}-j+1}\right)\\
\label{cello}&\leq P\left(\sum_{x\in{\mathcal{V}}_n}G_{Tr_n}(0,x)\inc{x\in T_{a2^{-j{\alpha_n}}}^{a 2^{-j{\alpha_n}+{\alpha_n}}}(n)}\geq \left(\frac{a+d/2}{a}\right)^{1/{\alpha_n}}2^{j{\alpha_n}}f_n\right)
\end{align}
Using exponential Chebyshev inequality with $\lambda_n$ from Condition D, (\ref{dumonde}) and the bound $\log(1+x)\leq x,$ for $x\geq 0$; we conclude that (\ref{cello}) is bounded above by
\begin{equation}\label{tout}
\exp\left(-\lambda_n \left(\frac{a+d/2}{a}\right)^{1/{\alpha_n}} 2^{j{\alpha_n}}f_n+\frac{C 2^{j{\alpha_n}}}{a}b_n \sum_{x\in{\mathcal{V}}_n}\big(e^{\lambda_n G_{Tr_n}(0,x)}-1\big)\right).
\end{equation}
Now using Condition D and that $b_nr_n=f_n$, we arrive at that (\ref{tout}) is bounded above by
\begin{equation}
\exp\left(-2^{j{\alpha_n}}\lambda_n f_n D_n\right)
\end{equation}
for all $n$ large enough, for all $j\in\mathbb{N}$, where
\begin{equation}
D_n:=\left\{\left(\frac{a+d/2}{a}\right)^{1/{\alpha_n}} -\frac{C\mathcal{K}_s T}{a}\right\}.
\end{equation}
We have for all $n$
\begin{equation}\label{matin}
\sum_{j=1}^\infty \exp\left(-2^{j{\alpha_n}}\lambda_n f_n D_n\right)\leq \exp\left(-2^{{\alpha_n}}\lambda_n f_n D_n\right)+\frac{\exp\left(-2^{{\alpha_n}}\lambda_n f_n D_n\right)}{{\alpha_n}(\log 2)2^{{\alpha_n}}\lambda_n f_n D_n}
\end{equation}
Since both ${\alpha_n} D_n\to\infty$ and $D_n\to\infty$ as $n\to\infty$; by the part of Condition D that states that $\sum_{n=1}^\infty \exp(-c \lambda_n f(n))<\infty$ for some $c>0$, we can conclude that the right hand side of (\ref{matin}) is summable in $n$. Hence, Borel-Cantelli lemma yields (\ref{aug}). Finally, summing over $j$ finishes the proof of Lemma \ref{shallowt}.
\iffalse
We first estimate $E[\tau_x\inc{x\in T^a(n)}]$. By Condition A
\begin{equation}
\begin{aligned}
E[\tau_x\inc{x\in T^a(n)}]&=\int_0^{a^{1/{\alpha_n}}g_n}P(\tau_x\geq u)du-a^{1/{\alpha_n}}g_n P(\tau_x\geq a^{1/{\alpha_n}}g_n)
\\&\leq C a^{1/{\alpha_n}-1}g_nb_n(1/{\alpha_n}-1).
\end{aligned}
\end{equation}
Using this and the Chebyshev's inequality we have for some positive constant $C'$, as $n\to\infty$
\begin{equation}
\begin{aligned}
P\left(\mathbb{E}\left[\sum_{i=0}^{tr_n}\tau_{\yn{i}}e_i\inc{\yn{i}\in T^a(n)}\Big|\boldsymbol{\tau}\right]\geq (a+d)^{1/{\alpha_n}}t_n\right)&\leq \frac{Ca^{1/{\alpha_n}-1}g_nb_n(1/{\alpha_n}-1)}{(a+d)^{1/{\alpha_n}}}\\&\leq C'\left(\frac{a}{a+d}\right)^{1/{\alpha_n}}.
\end{aligned}
\end{equation}
In the last step above we used the relations $r_n=f_n/b_n$ and $t_n=g_nf_n$. By Condition D part $(ii)$ the right hand side of the the last inequality above is summable. Thus, Borel-Cantelli lemma finishes the proof.
\fi
\end{proof}
The next proposition shows that the sum of scores is dominated by the largest score.
\begin{proposition}\label{comp}
For any $\delta>0$ given, for any $\epsilon$ small and $M$ large enough, $\boldsymbol{\tau}$ a.s. there exists a constant $K\geq 1$ such that for $n$ large enough
\begin{equation}
\mathbb{P}\big(S_n(\dn{j})\leq K m_n(j),\;\;\forall j \text{  s.t.  }\dn{j}\leq Tr_n\big)\geq 1-\delta.
\end{equation}
\end{proposition}
\begin{proof}

Recall that ${\zeta_n(T)}$ is the last time the random walk finds a deep trap before $Tr_n$. Note that for $j=0,1$ we have by definition $S_n(\dn{j})=\mn{j}$. Hence, we assume that ${\zeta_n(T)}\geq 2$ and prove that
\begin{equation}
\mathbb{P}\big(S_n(\dn{j})\leq K m_n(j),\;\;\forall j=2,\dots,{\zeta_n(T)}|\boldsymbol{\tau}\big)\geq 1-\delta.
\end{equation}
We define the sequence of events that the random walk cannot find a very deep trap before time $Tr_n$:
\begin{equation}
I_n^1:=\{H_n(T_M(n))\geq T r_n \}.
\end{equation}
We can use Condition C with $\rho=1/M$ to conclude that for $M$ large enough
\begin{equation}
\mathbb{P}(I_n^1\;|\boldsymbol{\tau})\geq 1-\delta/4.
\end{equation}
For $d>0$ define the sequence of events
\begin{equation}
I_n^2:=\{s_n(1)\geq (\epsilon+d)^{1/{\alpha_n}}t_n\}.
\end{equation}
Using Proposition \ref{asympindp}, for $d$ small enough, $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}\label{firstscore}
\mathbb{P}(I_n^2\;|\boldsymbol{\tau})\geq 1-\delta/4.
\end{equation}
We define another sequence of events
\begin{equation}\label{smalls}
I_n^3:=\left\{\sum_{i=0}^{Tr_n}e_i \tau_{\yn{i}}\inc{\yn{i}\in T^{\epsilon}(n)}\leq (\epsilon+d/2)^{1/{\alpha_n}}t_n\right\}.
\end{equation}
By Proposition \ref{shallowt} we have $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}
\mathbb{P}(I_n^3\;|\boldsymbol{\tau})\geq 1-\delta/4.
\end{equation}
Finally, defining
\begin{equation}
I_n^4:=\{{\zeta_n(T)}\leq K-1\},
\end{equation}
using Condition C, for $K$ large enough, $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}
\mathbb{P}(I_n^4\;|\boldsymbol{\tau})\geq 1-\delta/4.
\end{equation}
For $d$ and $M$ chosen as above, for $j=2,\dots,{\zeta_n(T)}$, we partition the sum $S_n(\dn{j})$ as follows
\begin{equation}\label{sonsum}
S_n(\dn{j})=\sum_{i=0}^{\dn{j}-1}e_i\tau_{\yn{i}}\inc{\yn{i}\inT^{\epsilon}(n)}+ \sum_{i=0}^{\dn{j}-1}e_i\tau_{\yn{i}}\inc{\yn{i}\inT_M(n)}+\sum_{k=1}^j s_n(k).
\end{equation}
Recall that $\mn{j}=\max_{i=1,\dots,j-1}s_n(i)
$. Let $I_n:=I_n^1\cap I_n^2\cap I_n^3\cap I_n^4$. We have $\boldsymbol{\tau}$ a.s. for $n$ large enough $\mathbb{P}(I_n)\geq 1-\delta$. On the event $I_n$, the first term on the right hand side of is bounded above by $m_n(j)$ for any $j\geq 1$ using (\ref{firstscore}) and (\ref{smalls}); the second term is 0 since on $I_n^1$ no deep trap has been found; the third term is bounded by $K-1$ since on $I_n^4$ ${\zeta_n(T)}\leq K-1$. Hence, on $I_n$, $S_n(\dn{j})$ is bounded $Km_n(j)$ and we are finished with the proof.
\end{proof}
\begin{proposition}\label{keyp}
For any $\delta>0$ given, for $\epsilon$ small and $M$ large enough, $\boldsymbol{\tau}$ a.s. for large enough $n$
\begin{equation}
\mathbb{P}\left(\max_{j\leq {\zeta_n(T)}} \left|\left(\frac{S_n(\dn{j})}{t_n}\right)^{\alpha_n}-\left(\frac{\mn{j}}{t_n}\right)^{\alpha_n}\right|\geq \delta\Big|\boldsymbol{\tau}\right)\leq \delta.
\end{equation}
\end{proposition}
\begin{proof}
By definition we have $\mn{j}\leq S_n(\dn{j})$ for any $j$. Using Proposition \ref{comp} we can find a positive constant $K$ s.t. $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}\label{dnn1}
\mathbb{P}\big(S_n(\dn{j})\leq K\mn{j},\;\forall j=1,\dots,{\zeta_n(T)}|\boldsymbol{\tau}\big)\geq 1-\delta/4.
\end{equation}
For $K$ as above, using Lemma \ref{asindm} we can choose a small enough $d$ so that $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}\label{dnn2}
\mathbb{P}\big(\mn{j}\in [(\epsilon-d)^{1/{\alpha_n}}t_n,(M+d)^{1/{\alpha_n}}t_n],\forall j=1,\dots,K\big)\geq 1-\delta/4.
\end{equation}
Finally, for $K$ and $d$ as above, for $n$ large enough
\begin{equation}\label{dnn3}
|K^{\alpha_n}-1|(M+2d)\leq \delta/2.
\end{equation}
We denote by $I_n$ the intersection of the events inside the probability displays in (\ref{dnn1}) and (\ref{dnn2}). Then using (\ref{dnn1}), (\ref{dnn2}) and (\ref{dnn3}) we get
\begin{equation}
\begin{aligned}
&\;\;\;\;\mathbb{P}\left(\max_{j\leq {\zeta_n(T)}} \left|\left(\frac{S_n(\dn{j})}{t_n}\right)^{\alpha_n}-\left(\frac{\mn{j}}{t_n}\right)^{\alpha_n}\right|\geq \delta\Big|\boldsymbol{\tau}\right)
\\&\leq \delta/2+\mathbb{P}\left(\max_{j\leq {\zeta_n(T)}} \left|\left(\frac{K\mn{j}}{t_n}\right)^{\alpha_n}-\left(\frac{\mn{j}}{t_n}\right)^{\alpha_n}\right|\geq \delta\Big|I_n, \boldsymbol{\tau}\right)= \delta/2.
\end{aligned}
\end{equation}
\end{proof}
\begin{proof}[Proof of Theorem \ref{thrm1}]
\newcommand{{\zeta_n(t_i)}}{{\zeta_n(t_i)}}
We start the proof with the proof of convergence of the finite dimensional distributions. Let $0=t_0\leq t_1\leq t_2\leq\cdots\leq t_k\leq T$ and $0\leq x_1\leq x_2\leq \cdots\leq x_k$ be given. Consider the random variables ${\zeta_n(t_i)},i=1,\dots,k$. For convenience, we denote by $N(\lambda)$, a Poisson random variable with mean $\lambda \rho_\epsilon^M$. Let $\{N(t_i-t_{i-1}):i=1,\dots,k\}$ be an independent collection. By Condition C, we have as $\boldsymbol{\tau}$ a.s. $n\to\infty$
\begin{equation}\label{indpois}
\left\{\zeta_n(t_i)-\zeta_n(t_{i-1}):i=1,\dots,k\right\}\Longrightarrow \left\{N(t_i-t_{i-1}):i=1,\dots,k\right\}
\end{equation}
where $\Longrightarrow$ stands for convergence in distribution. For ease of notation we defined the rescaled clock process
\begin{equation}
\bar{S}_n(t):=\left(\frac{S_n(t r_n)}{t_n}\right)^{\alpha_n}.
\end{equation}

\newcommand{\bsn}[1]{\bar{S}_n(#1)}
\newcommand{\bar{S}_n}{\bar{S}_n}

By Proposition \ref{keyp} and using Proposition \ref{shallowt} as in the proof of Proposition \ref{keyp}, we can conclude that $\boldsymbol{\tau}$ a.s. the sequence of events that for all $i=0,1,\dots,k$
\begin{equation}
\left(\frac{\mn{{\zeta_n(t_i)}}}{t_n}\right)^{\alpha_n}-\delta \leq \bsn{t_i} \leq \left(\frac{\mn{{\zeta_n(t_i)}}}{t_n}\right)^{\alpha_n}+\delta
\end{equation}
has probability larger than $1-\delta$ for all $n$ large enough. Hence, the sequence of finite dimensional distributions
\begin{equation}
\mathbb{P}\left(\bsn{t_1}\leq x_1,\dots,\bsn{t_k}\leq x_k\Big|\boldsymbol{\tau}\right)
\end{equation}
is bounded above by
\begin{equation}\label{extup}
\mathbb{P}\left(\left(\frac{m_n(\zeta_n(t_1))}{t_n}\right)^{\alpha_n}\leq x_1+\delta,\dots,\left(\frac{m_n(\zeta_n(t_k))}{t_n}\right)^{\alpha_n}\leq x_k+\delta\Big|\boldsymbol{\tau}\right)
\end{equation}
and below by
\begin{equation}
\mathbb{P}\left(\left(\frac{m_n(\zeta_n(t_1))}{t_n}\right)^{\alpha_n}\leq x_1-\delta,\dots,\left(\frac{m_n(\zeta_n(t_k))}{t_n}\right)^{\alpha_n}\leq x_k-\delta\Big|\boldsymbol{\tau}\right).
\end{equation}
We prove only the upper bound for the lower bound can be achieved similarly. By Lemma \ref{asindm} and (\ref{indpois}) (also recall (\ref{distsig})), $\boldsymbol{\tau}$ a.s. as $n\to\infty$, the sequence of probability terms in (\ref{extup}) converges to
\begin{equation}\label{istan}
\mathbb{E}\left[\left(\rho_{\epsilon}^{x_1+\delta}/\rho_{\epsilon}^M\right)^{N(t_1)}\right]\mathbb{E}\left[\left(\rho_{\epsilon}^{x_2+\delta}/\rho_{\epsilon}^M\right)^{N(t_2-t_1)}\right]\cdots \mathbb{E}\left[\left(\rho_{\epsilon}^{x_k+\delta}/\rho_{\epsilon}^M\right)^{N(t_k-t_{k-1})}\right].
\end{equation}
A simple calculation yields that for any $x,\lambda\geq 0$
\begin{equation}
\mathbb{E}\left[\left(\rho_\epsilon^x/\rho_\epsilon^M\right)^{N(\lambda)}\right]=\exp(-\lambda \rho_{x}^M).
\end{equation}
Hence, (\ref{istan}) is equal to
\begin{equation}
\exp\left(-\frac{t_1}{x_1+\delta}\right)\exp\left(-\frac{t_2-t_1}{x_2+\delta}\right)\cdots \exp\left(-\frac{t_k-t_{k-1}}{x_k+\delta}\right).
\end{equation}
Finally, letting $\delta\to 0$ finishes the proof of the convergence of finite dimensional distributions.

\newcommand{\varepsilon}{\varepsilon}

For tightness characterizations we need the following definitions:
\begin{align*}
&w_f(\delta)=\sup\left\{\min\big(|f(t)-f(t_1)|,|f(t_2)-f(t)|\big):t_1\leq t\leq t_2\leq T,t_2-t_1\leq \delta\right\},\\
&w'_f(\delta)=\sup\{\inf_{\alpha \in [0,1]}|f(t)-(\alpha f(t_1)+(1-\alpha)f(t_2))|:t_1\leq t \leq t_2\leq T, t_2-t_1\leq \delta\},\\
& v_f(t,\delta)=\sup \{|f(t_1)-f(t_2)|:t_1,t_2\in [0,T]\cap (t-\delta,t+\delta)\}.
\end{align*}
Following is from Theorem 12.12.3 of \cite{Whitt} and Theorem 15.3 of \cite{bill}.
\begin{theorem}\label{kiki}
 The sequence of probability measures $\{P_n\}$ on $D([0,T])$ is tight in the $M_1$-topology if

(i) For each positive $\varepsilon$ there exists a $c$ such that
\begin{equation}
 P_n[h:||h||_\infty \geq c]\leq \varepsilon, \hspace{0.4in} n\geq 1
\end{equation}

(ii) For each $\varepsilon>0$ and $\eta >0$, there exists a $\delta$, $0<\delta<T$, and an integer $n_0$ such that
\begin{equation}\label{budane}
 P_n[h: w'_h(\delta)\geq \eta]\leq \varepsilon, \hspace{0.4in} n\geq n_0
\end{equation}
and
\begin{equation}\label{ends}
 P_n[h:v_h(0,\delta)\geq \eta]\leq \varepsilon \; \text{and} \; P_n[h:v_h(T,\delta)\geq \eta]\leq \varepsilon, \;\; n\geq n_0
\end{equation}
Moreover, the same is true for $J_1$ topology with $w'_h(\delta)$ in (\ref{budane}) replaced by $w_h(\delta)$.
\end{theorem}

For the first claim in Theorem \ref{thrm1}, we check tightness in $M_1$ topology, using Theorem \ref{kiki}. Since $\bsn{t}$ is non-decreasing in $t$, to check condition (i) it is enough to check that $\bsn{T}$ is tight. In this case, the convergence of fixed time distribution gives the desired result.

Since for monotone functions $w'_{h}(\delta)$ is 0 in part (ii) of Theorem \ref{kiki} we only need to control $v_{\bar{S}_n}(0,\delta)$ and $v_{\bar{S}_n}(T,\delta)$. Again, using the monotonicity, controlling $v_{\bar{S}_n}(0,\delta)$ boils down to check that $\mathbb{P}[\bar{S}_n(\delta)\geq \eta]\leq \varepsilon$ for small enough $\delta$ and large enough $n$. By convergence of the fixed time distribution it is enough to check $\mathbb{P}(W(\delta)\geq \eta/2)\leq \varepsilon/2$. Since $\mathbb{P}(W(\delta)\geq \eta/2)=1-e^{-2\delta/\eta}$, this claim is true for small enough $\delta$. Similarly, to control $v_{\bar{S}_n}(T,\delta)$ it is enough to find $\eta$ small enough so that
\begin{equation}\label{fosfos}
 \mathbb{P}[W(T)-W( T-\delta)\geq \eta]\leq \varepsilon/2.
\end{equation}
Observe that by (\ref{poschr})
\begin{equation}
 \mathbb{P}[W(T)-W(T-\delta)=0]=\frac{T-\delta}{T},
\end{equation}
then
\begin{equation*}
 \mathbb{P}[W(T)-W( T-\delta)\geq \eta]\leq 1-\mathbb{P}[W(T)-W(T-\delta)=0]=\frac{\delta}{T}.
\end{equation*}
Hence, (\ref{fosfos}) follows by taking $\delta\leq T\varepsilon/4$. Hence, we are finished with the proof of the first part of Theorem \ref{thrm1}.

Now we assume that $f_n=1$ and $\mathcal{K}_G=1$ and prove the tightness of $\bar{S}_n$ in $J_1$ topology. We only need to check that (\ref{budane}) with $w_{\bar{S}_n}$. It is enough to show that $\boldsymbol{\tau}$ a.s. as $n\to\infty$
\begin{equation}\label{esss}
\max_{x\inT_{\epsilon}^{M}(n)}\mathbb{P}_x\left(Y_n(i)\notin T_{\epsilon}^{M}(n),\; i=1,\dots \delta r_n\right)\leq \varepsilon
\end{equation}
for $\delta$ small enough. Since Condition B is satisfied with $f_n=\mathcal{K}_G=1$ we have
\begin{equation}
\max_{x\inT_{\epsilon}^{M}(n)}\mathbb{P}_x(H'_n(x)<H_n(T_{\epsilon}^{M}(n)\setminus\{x\}))\longrightarrow 0
\end{equation}
where
\begin{equation}
H_n'(x):=\min\{i\geq 1: Y_n(i)=x\}
\end{equation}
Combining this with Condition C yields (\ref{esss}). Hence, we have proved the $J_1$ convergence.

\end{proof}

\begin{section}{Extremal Aging}\label{sc3}
\newcommand{\qn}[1]{{q_n({#1})}}
\newcommand{\kn}[1]{{k_n({#1})}}
\newcommand{\Vn}[1]{{V_n({#1})}}

Next, we give two extra conditions that ensure extremal aging occurs. For this purpose we need to control the random walk between record sites. First, we define the sequence of times when a new record site is found. We define $q_n(1)=1$ and for $j\geq 2$
\begin{equation}
\qn{j}=\min\{i>\qn{j-1}:\;\mn{i}>\mn{\qn{j-1}}\}.
\end{equation}
Hence, $\qn{j}$th score is greater than all the scores before it. To keep track of the times when the random walk visits a record site we introduce
\begin{equation}
k_n(j)=\dn{\qn{j}},\;\;\; j\in\mathbb{N}.
\end{equation}
Finally, we define the process $V_n(j)$ that records the trajectory of $Y_n$ restricted to deep traps whose score is a record:
\begin{equation}
\Vn{j}=U_n(\qn{j}).
\end{equation}
We fix numbers $a$ and $b$ with $0<a<b$. We set $\xi_n=Tr_n$, the number of steps of the random walk that we observe. Later we will choose $T$ large enough so that the clock process reaches the level $a^{1/{\alpha_n}}t_n$. The first condition is that in the time between the record site $\Vn{j}$ is found and the next record site $\Vn{j+1}$ is found, with a high probability the trap model is at $\Vn{j}$. More precisely, let $t_n$ be a deterministic sequence of times satisfying $(a/2)^{1/{\alpha_n}}t_n\leq t'_n\leq b^{1/{\alpha_n}}t_n$ and let $\delta>0$. We define $j_n\in\mathbb{N}$ by
\begin{equation}\label{tnsqn}
\left(\frac{S_n(\kn{j_n})}{t_n}\right) +\delta \leq \left(\frac{t'_n}{t_n}\right)^{\alpha_n} \leq \left(\frac{S_n(\kn{j_n+1})}{t_n}\right) -\delta
\end{equation}
and $j_n=\infty$ if the above inequalities are not satisfied by an integer. Let $A_n(\delta)$ be the event
\begin{equation}\label{ansqn}
A_n(\delta):=\{0<j_n<{\zeta_n(T)}\}.
\end{equation}

\noindent{\bf Condition 1:} For any $\delta>0$ it is possible to choose $\epsilon$ small and $M$ large enough so that $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}
\mathbb{P}(X_n(t'_n)=V_n(j_n)|A_n(\delta),\boldsymbol{\tau})\geq 1-\delta.
\end{equation}
The second condition states that there are no repetitions among record sites.

\noindent{\bf Condition 2:} For any $\epsilon$ and $M$, $\boldsymbol{\tau}$ a.s.
\begin{equation}
\lim_{n\to\infty}\mathbb{P}(\exists i,j \text{ s.t. } i\not=j,\qn{i},\qn{j}\leq {\zeta_n(T)}, \Vn{i}=\Vn{j}|\boldsymbol{\tau})=0.
\end{equation}
As discussed before, our choice of he two-time correlation function $R_n$ is
\begin{equation}
R_n(t_1,t_2):=\mathbb{P}(X_n(t_1)=X_n(t_2)|\boldsymbol{\tau}_{n})
\end{equation}
Now we are ready to state our extremal aging result.
\begin{theorem}\label{thrm2}
Assume that Conditions A-D and 1-2 hold and let $0<a<b$. Then $\boldsymbol{\tau}$ a.s.
\begin{equation}
\lim_{n\to\infty}R_n(a^{1/{\alpha_n}}t_n,b^{1/{\alpha_n}}t_n|\boldsymbol{\tau})=\frac{a}{b}.
\end{equation}
\end{theorem}

\begin{proof}[Proof of Theorem \ref{thrm2}]
We first calculate the probability that the record process jumps over an interval. We define the range of $m_n$ by
\begin{equation}
\mathcal{R}(m_n):=\left\{\left(\frac{\mn{j}}{t_n}\right)^{\alpha_n}:\;j\in\mathbb{N}\right\}.
\end{equation}
Note that, $m_n$ depends on $\epsilon$ and $M$ for the choice of $T_{\epsilon}^{M}(n)$.
\begin{lemma}\label{extjump} For $\epsilon<a<b<M$, $\boldsymbol{\tau}$ a.s.
\begin{equation}
\lim_{n\to\infty}\mathbb{P}\Big(\mathcal{R}(m_n)\cap [a,b]=\emptyset\Big)=\frac{\rho_b^M}{\rho_a^M}.
\end{equation}
\end{lemma}
\begin{proof}
By definition we have $m_n(1)=S_n(\dn{1})$. Since $\epsilon<a$, by Lemma \ref{shallowt}, as $n\to\infty$ the probability that $\mn{1} \geq a^{1/{\alpha_n}} t_n$ vanishes. Hence, using the weak convergence result in Lemma \ref{asindm} and the fact that the distribution of $\sigma_\epsilon^M$ has no atoms
\begin{equation}\label{extj}
\lim_{n\to\infty} \mathbb{P}\Big(\mathcal{R}(m_n)\cap [a,b]=\emptyset\Big)=\mathbb{P}\Big(\big\{\max_{i=1,\dots,j}\sigma_{\epsilon}^M(i):\;j\in\mathbb{N}\big\}\cap [a,b]=\emptyset\Big)
\end{equation}
Since $(\sigma_\epsilon^M(i):i\in\mathbb{N})$ is an i.i.d. sequence we have (\ref{extj}) equal to
\begin{equation}
\begin{aligned}
&\mathbb{P}\Big(\sigma_\epsilon^M(1)\geq b\Big)+\sum_{j=0}^\infty \mathbb{P}\Big(\max_{i=1,\dots,j}\sigma_{\epsilon}^M(i)\leq a\Big)\mathbb{P}\Big(\sigma_\epsilon^M(j+1)\geq b\Big)
\\&= \sum_{j=0}^\infty \mathbb{P}(\sigma_\epsilon^M(1)\leq a)^j \mathbb{P}(\sigma_\epsilon^M(1)\geq b)=\frac{\mathbb{P}(\sigma_\epsilon^M(1)\geq b)}{\mathbb{P}(\sigma_\epsilon^M(1)\geq a)}=\frac{\rho_b^M}{\rho_{a}^M}.
\end{aligned}
\end{equation}
\end{proof}

\newcommand{\text{dist}}{\text{dist}}

Let $\delta>0$ be small enough so that $\epsilon<a-2\delta<b+\delta<M$. Define the sequence of events $I_n^1,I_n^2$ and $I_n^3$ as follows:
\begin{equation}
\begin{aligned}
&I_n^1:=\Big\{\text{dist}(a,\mathcal{R}(m_n))\leq \delta\;\;\text{ or } \text{dist}(b,\mathcal{R}(m_n))\leq\delta\Big\},
\\& I_n^2:=\Big\{\text{dist}(a,\mathcal{R}{m_n})\geq \delta,\;\text{dist}(b,\mathcal{R}(m_n))\geq \delta \text{ and } [a+\delta,b-\delta]\cap \mathcal{R}(m_n)\not=\emptyset\Big\},
\\&I_n^3:=\Big\{[a-\delta,b+\delta]\cap \mathcal{R}(m_n)=\emptyset\Big\}
\end{aligned}
\end{equation}
We also define $F_n:=\big\{m_n({\zeta_n(T)})\geq b^{1/{\alpha_n}}t_n\big\}$. Since a.s. $W(t)\to\infty$ as $t\to\infty$, using Theorem \ref{thrm1} we can choose $T>0$ large enough so that $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}\label{ufak}
\mathbb{P}(F_n|\boldsymbol{\tau})\geq 1-\delta/4.
\end{equation}
Finally, let us define the sequence of events we want to approximate
\begin{equation}
G_n:=\Big\{X_n(a^{1/{\alpha_n}}t_n)=X_n(b^{1/{\alpha_n}}t_n)|\boldsymbol{\tau}\Big\}.
\end{equation}
We want to show that $G_n$ can be well approximated by $I_n^3$. Since the jumps of $m_n$ have continuous distribution we have $(I_n^3)^c=I_n^1\cup I_n^2$ and $(I_n^2)^c\subset I_n^1\cup I_n^3$. This yields to
\begin{equation}
\mathbb{P}(G_n\cap I_n^3)\leq \mathbb{P}(G_n)\leq \mathbb{P}(I_n^1)+\mathbb{P}(I_n^3)+\mathbb{P}(G_n \cap I_n^2).
\end{equation}
On $F_n$ and $I_n^2$ there exist $j_1$ and $j_2$ with $0<j_1<j_2$ and $q_n(j_1),q_n(j_2)<{\zeta_n(T)}$ such that
\begin{equation}
\begin{aligned}
&\left(\frac{m_n(q_n(j_1))}{t_n}\right)^{\alpha_n}+\delta \leq a \leq \left(\frac{m_n(q_n(j_1+1))}{t_n}\right)^{\alpha_n} -\delta \\
&\left(\frac{m_n(q_n(j_2))}{t_n}\right)^{\alpha_n}+\delta \leq a \leq \left(\frac{m_n(q_n(j_2+1))}{t_n}\right)^{\alpha_n} -\delta.
\end{aligned}
\end{equation}
Using Proposition \ref{keyp} this yields to
\begin{equation}
\begin{aligned}
&\left(\frac{S_n(q_n(j_1))}{t_n}\right)^{\alpha_n}+\delta/2 \leq a \leq \left(\frac{S_n(q_n(j_1+1))}{t_n}\right)^{\alpha_n} -\delta/2 \\
&\left(\frac{S_n(q_n(j_2))}{t_n}\right)^{\alpha_n}+\delta/2 \leq a \leq \left(\frac{S_n(q_n(j_2+1))}{t_n}\right)^{\alpha_n} -\delta/2.
\end{aligned}
\end{equation}
Hence, by Condition 1 we have $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}
\mathbb{P}(X_n(a^{1/{\alpha_n}})=V_n(j_1),X_n(b^{1/{\alpha_n}})=V_n(j_2)|I_n^2,F_n,\boldsymbol{\tau})\geq 1-\delta.
\end{equation}
Combining this with Condition 2 and (\ref{ufak}) we get
\begin{equation}
\mathbb{P}(G_n\cap I_n^2)\leq \delta.
\end{equation}
Similarly on $F_n\cap I_n^3$ there exists a $j_1$ such that
\begin{equation}
\left(\frac{m_n(q_n(j_1))}{t_n}\right)^{\alpha_n}+\delta \leq a \leq b \leq \left(\frac{m_n(q_n(j_1+1))}{t_n}\right)^{\alpha_n} -\delta.
\end{equation}
Hence, by Condition 1 we have $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}
\mathbb{P}(X_n(a^{1/{\alpha_n}}t_n)=X_n(b^{1/{\alpha_n}}t_n)=V_n(j_1)|\boldsymbol{\tau})\geq 1-\delta,
\end{equation}
and since $\mathbb{P}(G_n)\geq \mathbb{P}(X_n(a^{1/{\alpha_n}}t_n)=X_n(b^{1/{\alpha_n}}t_n)=V_n(j_1)|\boldsymbol{\tau})$ we get
\begin{equation}
\mathbb{P}(I_n^3)-\delta\leq \mathbb{P}(I_n^3\cap G_n).
\end{equation}
By Lemma \ref{extjump} we have $\mathbb{P}(I_n^1)\to 0$ and $\mathbb{P}(I_n^3)\to \rho_{b}^M/\rho_a^M$ as $\delta\to 0$. Finally, taking $M\to\infty$ and $\delta\to 0$ finishes the proof.
\end{proof}
\end{section}

\begin{section}{Extremal Aging for Random Energy Model}\label{sc4}
The Random Energy Model was first introduced by Bernard Derrida in \cite{Der1} as an exactly solvable mean field spin glass model. The state space is ${\mathcal{V}}_n=\{-1,+1\}^n$, the $n$ dimensional hypercube. To each configuration $\sigma\in{\mathcal{V}}_n$ is attached a random Hamiltonian $H_n(\sigma)$. The choice for the energy landscape in REM is that of i.i.d. Gaussians with mean zero and variance $n$. More precisely, $H_n(\sigma)=-\sqrt{n}Z_\sigma$ where
\begin{equation}\label{seqgauss}
\Big\{Z_\sigma:\sigma\in{\mathcal{V}}_n\Big\}
\end{equation}
is an i.i.d. sequence of standard Gaussian random variables. Let $\beta>0$ be the inverse temperature (later we will let $\beta$ vary with the dimension $n$), then the Gibbs measure at inverse temperature $\beta$ is given by
\begin{equation}
\mu_n(\sigma)=\frac{1}{\mathcal{Z}_{n,\beta}}e^{-\beta H_n(\sigma)}=\frac{1}{\mathcal{Z}_{n,\beta}} e^{\beta\sqrt{n} Z_\sigma},\;\;\;\sigma\in{\mathcal{V}}_n
\end{equation}
where $\mathcal{Z}_{n,\beta}$ is the usual partition function.

\newcommand{\mathcal{G}}{\mathcal{G}}
\newcommand{\mathcal{E}_n}{\mathcal{E}_n}

We will study the trap model dynamics of REM. In our setting, the graph is $\mathcal{G}_n=({\mathcal{V}}_n,\mathcal{E}_n)$ where the set of edges $\mathcal{E}_n$ is given by
\begin{equation}
\mathcal{E}_n=\Big\{(\sigma,\sigma'):\frac{1}{2}\sum_{i=1}^n|\sigma_i-\sigma_i'|=1\Big\}.
\end{equation}
In other words, two configurations in ${\mathcal{V}}_n$ are neighbors if they differ only at one spin. For the trapping landscape $\{\tau_\sigma:\sigma\in{\mathcal{V}}_n\}$ we choose the Gibbs waits
\begin{equation}
\tau_\sigma:=\exp(-\beta H_n(\sigma))=\exp(\beta\sqrt{n} Z_\sigma),\;\;\;\sigma\in {\mathcal{V}}_n.
\end{equation}
As before we denote the corresponding trap model by $(X_n(t):\;t\geq 0)$. It is trivial that the Gibbs measure of REM is the unique invariant measure of $X_n$. We want to note there that in the literature this type of dynamics is sometimes called Random Hopping Time dynamics.

We will study the dynamics of REM model in the following context. We let the temperature vary by the volume of the system, hence, here after we parameterize the inverse temperature by $n$ and replace $\beta$ with $\beta_n$ in the above equations. Let us rewrite the trapping landscape in this new setup
\begin{equation}\label{trgas}
\boldsymbol{\tau}:=\Big\{\exp(\beta_n \sqrt{n} Z_\sigma):\;\sigma\in{\mathcal{V}}_n\Big\}
\end{equation}
where the sequence $\{Z_\sigma:\sigma\in{\mathcal{V}}_n\}$ is as in (\ref{seqgauss}). As before, we denote by $Y_n$ the simple random walk on the hypercube ${\mathcal{V}}_n$ and the clock process by $S_n$.

\newcommand{c_n}{c_n}
\newcommand{\alpha_n^2}{\alpha_n^2}
Next we choose the time scales that we will observe the dynamics at. We set first our depth rate scale ${\alpha_n}$; as in Section \ref{scClock}, ${\alpha_n}\to 0$ as $n\to\infty$. We consider the following scales:
\begin{equation}\label{scales}
g_n=t_n=\exp({\alpha_n}\beta_n^2 n),\;\;b_n^{-1}=r_n={\alpha_n}\beta_n\sqrt{2\pi n}\exp(\alpha_n^2\beta_n^2 n/2),\;\;f_n=1.
\end{equation}
We will always assume that
\begin{equation}\label{assum0}
\limsup_{n\to\infty}{\alpha_n}\beta_n<\sqrt{2\log 2}.
\end{equation}
Furthermore, we will assume that ${\alpha_n}$ and $\beta_n$ are chosen such that
\begin{equation}\label{assum}
 n\log n \ll r_n\;\;\text{ as }\;n\to\infty.
\end{equation}
The following theorem describes the extremal aging for the dynamics of REM.
\begin{theorem}\label{thrm3}
For a.s. random environment $\boldsymbol{\tau}$
\begin{itemize}
\item[(i)] for any $T>0$, as $n\to\infty$
\begin{equation}
\hspace{0.99in}\left(\frac{S_n\big(\cdot\;r_n\big)}{t_n}\right)^{\alpha_n} \Longrightarrow W(\cdot)\;\;\;\;\;\;\;\; \text{    in  }\;\; D([0,T],J_1)
\end{equation}
where $W$ is the extremal process generated by the distribution function $F(x)=e^{-1/x}$,

\item[(ii)] for any $0<a<b$ as $n\to\infty$
\begin{equation}
R_n\Big(a^{1/{\alpha_n}}t_n,b^{1/{\alpha_n}}t_n\big|\boldsymbol{\tau}\Big)\longrightarrow\frac{a}{b}.
\end{equation}
\end{itemize}
\end{theorem}

\vspace{0.1in}

\begin{remark}
Note that the ratio of the two times $t_n a^{1/{\alpha_n}}$ and $t_n b^{1/{\alpha_n}}$ diverges with $n$ but  the logarithmic ratio $\log (t_n a^{1/{\alpha_n}})/\log (t_n b^{1/{\alpha_n}})$ converges to 1 as $n\to\infty$, due to (\ref{assum}).
Hence, we can think of the extremal aging result of Theorem \ref{thrm3} as ``just before aging".
\end{remark}
\begin{remark}
Let us describe the cases that Theorem \ref{thrm3} covers. For constant temperature case $\beta_n\equiv \beta>0$, the time scales in Theorem \ref{thrm3} consist of subexponential time scales (in $n$). For the case that $1\ll\beta_n$, that is, zero temperature dynamics, depending on $\beta_n$ and ${\alpha_n}$, it covers superexponential, exponential and subexponential time scales. Finally, for $\beta_n\ll 1$, time scales covered are subexponential.
\end{remark}
\begin{proof}[Proof of Theorem \ref{thrm3}]
In order to prove Theorem \ref{thrm3} we will check Conditions A-D and Conditions 1-2; and to achieve convergence in $J_1$ we will check Condition B with $\mathcal{K}_G=1$.

\vspace{0.05in}

\noindent{\bf Condition A:} It is well-known for a standard Gaussian random variable $Z$ that
\begin{equation}\label{e1}
\;\;\;\;P(Z\geq u)=\frac{1}{u\sqrt{2\pi}}e^{-u^2/2}(1+o(1))\;\;\;\text{ as  }u\to\infty,
\end{equation}
and
\begin{equation}\label{e2}
P(Z\geq u)\leq \frac{1}{u\sqrt{2\pi}}e^{-u^2/2},\;\;\;\forall u>0.
\end{equation}
By (\ref{assum}) we have $1\ll {\alpha_n}\beta_n\sqrt{n}$. Hence, using (\ref{trgas}) and (\ref{e1})
\begin{equation}
\begin{aligned}
\Big({\alpha_n}&\beta_n\sqrt{2\pi n}\Big) e^{\alpha_n^2\beta_n^2 n/2}P(\tau_x\geq u^{1/{\alpha_n}}e^{{\alpha_n}\beta_n^2 n})\\&=\Big({\alpha_n}\beta_n\sqrt{2\pi n}\Big) e^{\alpha_n^2\beta_n^2 n/2} P\Big(Z\geq {\alpha_n}\beta_n\sqrt{n} +\frac{\log u}{{\alpha_n}\beta_n\sqrt{n}}\Big)\underset{n\to\infty}\longrightarrow \frac{1}{u}.
\end{aligned}
\end{equation}
This proves the first part of Condition A. The second part of Condition A follows trivially from (\ref{e2}) and a calculation similar to the above.

\vspace{0.05in}

\noindent{\bf Condition C:}
For proving Condition $C$ we use the following theorem from \cite{CG08}:

\begin{theorem} (Theorem 1 in \cite{CG08}) Let $\bar{m}(n)$ be such that
\begin{equation}
 n\log n \ll \bar{m}(n) \ll 2^n (\log n)^{-1},
\end{equation}
and let $A_n$ be a sequence of percolation clouds on ${\mathcal{V}}_n$ with densities $\bar{m}(n)^{-1}$. Then, for all $a>0$,
\begin{equation}
 \lim_{n\to\infty}\max_{x\in {\mathcal{V}_n}}\left|P_x\Big(H_n(A_n\setminus \{x\})\geq a\bar{m}(n)\Big)-\exp(-a)\right|=0.
\end{equation}
\end{theorem}

In our case, Condition $C$ is equivalent to the result of the above theorem when $\bar{m}(n)=(\rhob_n)^{-1}$, $\rho>0$. We have in (\ref{scales}) $b_n^{-1}={\alpha_n}\beta_n\sqrt{2\pi n}\exp(\alpha_n^2\beta_n^2 n/2)$ and, thus, (\ref{assum0}) and (\ref{assum}) yield $n\log n \ll (\rhob_n)^{-1}\ll 2^n (\log n)^{-1}$. Hence, we can apply the theorem and verify Condition $C$.

\vspace{0.05in}

\noindent{\bf Condition B:} Recall the notation $\xi_n=Tr_n$. It is trivial that for any $k_1,k_2$ with $k_1\leq k_2$, and for all $x,y\in{\mathcal{V}}_n$, $G^n_{k_1}(x,y)\leq G^n_{k_2}(x,y)$. Also, for any $k\in\mathbb{N}$ and $x\in{\mathcal{V}}_n$, $G^n_k(x,x)\geq 1$ by definition. Hence, using part (a) of Lemma 3.9 in \cite{BC08} we conclude that
\begin{equation}\label{cbass}
\limsup_{n\to\infty} G_{\xi_n\log \xi_n}^n(0,0)=1.
\end{equation}
Now we claim that uniformly for $x\in T_{\epsilon}^{M}(n)$ that
\begin{equation}\label{cellop}
\lim_{n\to\infty}G_{T_{\epsilon}^{M}(n)\setminus \{x\}}^n(x,x)=1,
\end{equation}
which is enough to check Condition B with $f_n=1$ and $\mathcal{K}_G=1$. Recalling that $H_n'(x)=\min\{i\geq 1: Y_n(i)=x\}$ we have
\begin{equation}
G_{\xi_n\log \xi_n}^n(0,0)=\big(P_x(H_n(T_{\epsilon}^{M}(n)\setminus\{x\})<H_n'(x))\big)^{-1}.
\end{equation}
We have
\begin{equation}
P_x(H_n(T_{\epsilon}^{M}(n)\setminus\{x\})\geq H_n'(x))\leq P_x(H_n'(x)\leq \xi_n\log \xi_n)+P_x(H_n(T_{\epsilon}^{M}(n)\setminus \{x\})\geq \xi_n\log\xi_n).
\end{equation}
The first term on the right hand side of the above display is obviously independent of $x$ and converges to 0 by (\ref{cbass}); using Condition C, that is, the fact that uniformly for $x\inT_{\epsilon}^{M}(n)$, $H_n(T_{\epsilon}^{M}(n)\setminus\{x\})/\xi_n$ is asymptotically an exponential random variables, we can conclude that the second term also vanishes uniformly for $x\in{\mathcal{V}}_n$. Hence, we have proved (\ref{cellop}).

\vspace{0.05in}

\noindent{\bf Condition D:} By Lemma 3.9 in \cite{BC08} we have for all $x\not= 0$, for all $n$ large enough
\begin{equation}\label{pcello}
G_{\xi_n}^n(0,x)\leq C/n
\end{equation}
for some positive constant $C>0$. By (\ref{assum}) we can choose a sequence $\lambda_n$ such that
\begin{equation}
\lambda_n\ll n,\;\;\;\lambda_n\ll \alpha_n^2\beta_n^2 n
\end{equation}
and $\sum_n\exp(-\lambda_n)<\infty$ (recall that $f_n=1$). For such $\lambda_n$, using (\ref{pcello}) we have $\lambda_n G_{\xi_n(0,x)}\ll 1$ uniformly for all $x\in{\mathcal{V}}_n\setminus\{x\}$. Combining this with (\ref{cbass}) we get that for some positive constant $K$, for all $n$ large enough
\begin{equation}
\sum_{x\in {\mathcal{V}}_n}(e^{\lambda_n G_{\xi_n}^n(0,x)}-1)\leq e^{2\lambda_n}+\sum_{x\in {\mathcal{V}}_n\setminus\{x\}}(e^{\lambda_n G_{\xi_n}^n(0,x)}-1)\leq e^{2\lambda_n}+\lambda_n Tr_n\leq K T\lambda_nr_n
\end{equation}
where in the last step we used that $\lambda_n\ll\alpha_n^2\beta_n^2 n$. Hence, we have checked the first part of Condition D. The second part of Condition D is trivial since $f_n=1$.

\vspace{0.05in}

\noindent{\bf Condition 1:} Let $t'_n$ be a deterministic sequence of times satisfying (\ref{tnsqn}) and $A_n(\delta)$ be defined as (\ref{ansqn}). We check that Condition 1 is satisfies uniformly for $0<a<b$, conditioned on $m_n(q_n(j_n))=a^{1/{\alpha_n}}t_n$ and $m_n(q_n(j_n+1))=b^{1/{\alpha_n}}t_n$. For $\epsilon>0$ small enough so that $a+2\epsilon<b-2\epsilon$ we define the sequence of events:
\begin{equation}
 B_n:=\left\{\tau_{Y_n(i)}\leq (b-2\epsilon)^{1/{\alpha_n}} t_n:\;\forall i=k_n(j_n),\dots,k_n(j_n+1)-1\right\}.
\end{equation}
By Condition C and Proposition \ref{asympindp} we can choose $\epsilon$ small enough so that $\boldsymbol{\tau}$ a.s. for $n$ large enough we have
\begin{equation}
\mathbb{P}(B_n|\boldsymbol{\tau})\geq 1-\delta/4.
\end{equation}
By Proposition \ref{keyp}, $\boldsymbol{\tau}$ a.s. for $n$ large enough
\begin{equation}
\mathbb{P}\left(S_n(k_n(j_n+1))-S_n(k_n(j_n))\geq (b-\epsilon/2)^{1/{\alpha_n}}t_n-(a+\epsilon)^{1/{\alpha_n}}t_n\big|\boldsymbol{\tau}\right)\geq 1-\delta/4.
\end{equation}
Note that $(b-\epsilon/2)^{1/{\alpha_n}}t_n-(a+\epsilon)^{1/{\alpha_n}}t_n \gg (b-2\epsilon)^{1/{\alpha_n}}t_n$ as $n\to\infty$. Also, by Proposition \ref{shallowt}, on $I_n$, the contribution from traps between $k_n(j_n)$ and $k_n(j_n+1)$ is smaller than $(b-\epsilon)^{1/{\alpha_n}}t_n$ with a probability larger than $1-\delta/4$. Hence, conditioned on $I_n$, $A_n(\delta)$ and the sequence of events inside the probability term in the last display, the only way $X_n(t'_n)\not= V_n(j_n)$ is if the random walk comes back to $V_n(j_n)$ before visiting $V_n(j_n+1)$. However, by (\ref{cbass}) this probability goes to 0. Thus, Condition 1 is satisfied.

\vspace{0.05in}

\noindent{\bf Condition 2:} By Condition C, the time for $Y_n$ to visit a deep trap divided by $r_n$ is approximately exponentially distributed. Hence, by (\ref{cbass}), we have that the probability of $Y_n$ revisits a deep trap converges to 0. Thus, Condition 2 follows.

\end{proof}

\end{section}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Hear here
27 podcasts to make you smarter
Want to feel like a genius? These mind-expanding podcasts will give you everything you need to do just that
Nell Frizzell
Wed 7 Sep 2016 07.55 EDT Last modified on Wed 20 Sep 2017 04.17 EDT
Learn from the best … Melvyn Bragg, Helen Zaltzman, and Prof Brian Cox. Composite: Getty Images; Teri Pengilley; BBC
As the cold rustle of conkers start to hit the pavements and a new generation of pencil cases bulge under the weight of novelty felt-tips, it must be the start of a new term. Sadly, we can't all be going back to school. But that doesn't mean we need to let our brains fester like an opened tub of yoghurt on a hot day. The audio wonderland of modern podcasts is ripe with all the insight, analysis, facts, statistics and research you need to at least blag a GCSE. Here is a rundown of some of the best podcasts to make you feel (if not actually sound) like a genius:
The weekly stories from This American Life often spread beyond traditional state borders – their recent episodes from Greece's refugee camps were brilliant, with reports on romance, wild pigs, women doing their laundry in a baseball stadium locker room and what it's like to live in a former psychiatric hospital. You can also trawl the archive for stories from Haiti, the lie that saved Brazil's economy and even the odd look at Europe (yes, they too covered Brexit).
A weekly listen to From Our Own Correspondent will do more for your understanding of global news than my curriculum ever managed, while The Documentary podcast from BBC World Service can cover anything from protest in Putin's Russia to Syria's secret library and incubator babies on display in Coney Island.
The New Yorker Fiction Podcast is like the greatest book group, English seminar and public lecture you never joined. Each month the magazine's fiction editor, Deborah Treisman, invites a different New Yorker author to choose and read aloud a story from the magazine's archive then discuss it. I started with AM Homes reading Shirley Jackson's short story The Lottery, set in a sinister village, and it haunts me still.
Is English language your bag? Then Helen Zaltzman's The Allusionist is the etymology party you've been waiting for. After all, who knew a simple word like please could have such completely different meanings and uses on either side of the Atlantic?
If you like your journalism lengthy, perhaps the Longform interviews with journalists such as Malcolm Gladwell and Kathryn Schulz, speechwriters like Jon Favreau who worked with President Obama and editors like David Remnick will tickle your pickle, or the Guardian's own excellent Long Reads audio series. After all, reading's for squares, I heard.
There are some amazing science stories in the RadioLab archive, from a genuinely terrifying exploration of the howling, spit-thickening spread of rabies, to the woman put in a coma as an experiment and a look at the day the dinosaurs died.
You can go a long way at a dinner party with a couple of the BBC's Inside Science episodes under your belt, and first dates will whizz by with the science anecdotes afforded by the last season of Invisibilia. Not to mention the unending popularity of Prof Brian Cox and Robin Ince's Infinite Monkey Cage.
If Melvyn Bragg had been my history teacher, rather than a woman who decorated her classroom with warnings about the dangers of ragwort, I may have ended up with quite a different degree. In Our Time will teach you more about world history than most museums – and you don't even need to pack a lunch. I'm particularly enjoying the latest series on the history of The North.
Elsewhere, Stuff You Missed in History Class will give you a good grounding in everything from China's Great Leap Forward to the Matchgirls Strike and the Anglo-Cherokee war.
For those who prefer their history a little more specific, I'm a huge fan of Great Lives, in which a notable modern figure looks at a person from the past who has inspired or excited them – try Sara Pascoe on Virginia Woolf or Anthony Horowitz on Alfred Hitchcock.
The 50 podcasts you need to hear
If The Essay from BBC Radio 3 was a country, I'd move there. There are amazing features on everything from British film comedians to Dadaism, great sonnets, and the artistic impact of the fall of the Berlin Wall. Adventures in Design is a topical (if now behind-paywall) look at modern graphic design, while Song Exploder invites musicians and songwriters to pick apart their compositions in forensic detail. And if you want to delve into the world buildings-first, then 99% Invisible is far more than just an architecture podcast.
What I know about sport could fill the sweaty confines of a Slazenger Classic Abdo Guard, but even I have been known to laugh at the ArseBlog podcast. Particularly when Irish writer and presenter, Andrew Mangan, described striker Olivier Giroud as "a big fucking ride".
The murky subjects of toxic debt, trading in oil, the economy of housing, Brexit and how to count your bitcoins are made more comprehensible thanks to NPR's Planet Money. It won't teach you how to do long division, but it will give you some prime number chat. Also worth a listen is the BBC's mathematics show More or Less, hosted by the Undercover Economist Tim Harford, which takes a look at the real stories behind the statistics found in the news.
Whether you're dating, starting a new job, or just want something new to say to your pets, you can learn a lot from the The Inquiry, which has programmes on everything from coral reefs to putting solar panels in the Sahara desert. The Middle East Week podcast will give you a great grounding on some of the world's most controversial issues, More Perfect is a fascinating look at the American justice system and you can get to grips with some serious international relations via Global Dispatches. Little Atoms, from Resonance FM, will help you scrub up on politics, literature, science, art and comedy, and I have learned pretty much everything I know about farming, food, technology, growing vegetables and the rural economy thanks to The Archers (not to mention the rigours of baking ginger scones).


Glen Weyl, principal researcher at Microsoft and visiting senior research scholar in economics and law at Yale University, will discuss the book he co-authored with Eric A. Posner, Radical Markets: Uprooting Capitalism and Democracy for a Just Society.
Many blame today's economic inequality, stagnation, and political instability on the free market. The solution is to rein in the market, right? Radical Markets turns this thinking—and pretty much all conventional thinking about markets, both for and against — on its head. The book reveals bold new ways to organise markets for the good of everyone. It shows how the emancipatory force of genuinely open, free, and competitive markets can reawaken the dormant nineteenth-century spirit of liberal reform and lead to greater equality, prosperity, and cooperation.


Color of stamp: The Horn on Stamp.
The Spain Postal Administration issued the aero-phones musical instrument series on stamp but not in once time. On 1st of July 2010, one of the above stamp series issued is the horn, a wind-brass instrument belonging to the family of the aero-phones. The horn depicted in the stamp is a 20th century piece belonging to the musical instrument collection of the Museo Interactivo de la Música de Málaga (MIMMA).
The horn is a circular shaped instrument with a 35cm diameter. There are a mouthpiece and valves in the upper part and a conical bore that gradually becomes wider. The three or four valves, usually rotary, are operated with the left hand, to route the air into extra tubing to change the pitch. In its central part, there are also additional screws. The sound of the horn is produced by the vibration of the lips in the mouthpiece and by blowing air into the instrument. More lip tension and faster air produces higher notes.


Ubisoft announced the first brand in the Tom Clancy's franchise developed exclusively for mobile devices. The game is currently only avalible in Canada.
Next Post NVIDIA Announces TITAN Xp, The World's Most Powerful GPU.


text:
Q: Flip tooltips depending on quadrant I have some simple tooltips which appear on hover and follow the mouse cursor.
Each trigger is situated in a different quadrant of the viewport.

// The following will find the tooltips, and then adds an event on mousemove that allows the position of the tooltip to follow the cursor's location

var tooltip = document.querySelectorAll('.z-tooltip');

  document.addEventListener('mousemove', fn, false);

  function fn(e) {
      for (var i=tooltip.length; i--;) {
          tooltip[i].style.left = e.clientX + 10 + 'px';
          tooltip[i].style.top = e.clientY + 10 + 'px';
      }
  }
/* Hide and show the tooltip */

.z-trigger:hover .z-tooltip {
    display: block;
}

.z-tooltip {
    display: none;
    position: fixed;
    z-index: 1;
    border: 1px solid;
}

/* Position the links in different quadrants of the viewport for example's sake */

.z-top-left {
position: absolute;
top: 25%;
left: 25%
}

.z-top-right {
position: absolute;
top: 25%;
right: 25%
}

.z-bottom-left {
position: absolute;
bottom: 25%;
left: 25%
}

.z-bottom-right {
position: absolute;
bottom: 25%;
right: 25%
}

  Top left quadrant

      I want this to display to the bottom right of the cursor - it does :)

  Top right quadrant

      I want this to display to the bottom left of the cursor - it doesn't :(

  Bottom left quadrant

      I want this to display to the top right of the cursor - it doesn't :(

  Bottom right quadrant

      I want this to display to the top left of the cursor - it doesn't :(

What I would like javascript (or jQuery) to do is to:

*

*Detect which quadrant of the screen the 'trigger' is situated within

*Make the resulting hover tooltip appear next to the cursor - but pointing away from the quadrant in which the trigger lies

For example if I hover over one of these trigger elements and it happens to be anywhere within the top right quadrant of the viewport, I want the tooltip to appear to the bottom and to the left of the cursor position.
Any help appreciated :)

redpajama_set_name: RedPajamaStackExchange


If you want to be a tourist in your own town, the Crab Shack is the place to do it. Big steamer dishes of crab legs, oysters, corn, shrimp, sausage, and potatoes are fun. When local oysters are in season, they can be found here too. Naturally, the menu offers plenty of fried and grilled options. Cold beer and cocktails are abundant as well, and the bartenders at the downtown location know how to entertain visitors and treat the locals right at the same time. Acoustic performers weekly.
Seafood by the bucket, beer by the bushel, and free peanuts by the barrel.
• $2 domestic beers, $3 house liquors, $3.99 house wine, $1 off all call and premium liquors.


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We have something we want to tell Cris.
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Sally complained about the weather.
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Tell Hans to quit doing that.
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Sigurd wanted to make a mix CD of music for Jitendra, but all he had were DVD blanks.
Dennis was smart about the way he set all of this up.
Yvonne was trying to reach out to her.