V Éronique Gayrard

AGING IN METROPOLIS DYNAMICS OF THE REM: A PROOF

Keywords: December 1, 2020. 2000 Mathematics Subject Classification. 82C44, 60K35, 60G70 random dynamics, random environments, clock process, Lévy processes, spin glasses, aging, Metropolis dynamics

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The present manuscript is the accepted version of the manuscript officially published by Springer as Gayrard, V. "Aging in Metropolis dynamics of the REM: a proof". Probab. Theory Relat. Fields 174, 501-551 (2019). The final publication is available at link.springer.com: https://link.springer.com/article/10.1007/s00440-018-0873-6

INTRODUCTION

While there is as yet no established theory for the description of glasses, a consensus exists that this amorphous state of matter is intrinsically dynamical in nature [START_REF] Duplantier | Glasses and Grains[END_REF]. Measuring suitable two-time correlation functions indeed reveals that glassy dynamics are history dependent and dominated by ever slower transients: they are aging. The realization in the late 80's that mean-field spin glass dynamics could provide a mathematical formulation for this phenomenon sparked renewed interest in models, such as Derrida's REM and p-spin SK models [START_REF] Derrida | Random-energy model: limit of a family of disordered models[END_REF], [START_REF] Derrida | A generalization of the random energy model which includes correlations between energies[END_REF], whose statics had, until then, been the main focus of attention. Despite this, Bouchaud's phenomenological trap models first took the center stage as they succeeded in predicting the power-law decay of two-time correlation functions observed experimentally, even though they did so at the cost of an ad hoc construction and drastically simplifying assumptions [START_REF] Bouchaud | Aging on Parisi's tree[END_REF].

It was not until 2003 that a trap model dynamics was shown to result for the microscopic Glauber dynamics of a (random) mean-field spin glass Hamiltonian, namely, the REM endowed with the so-called Random Hopping dynamics and observed on time-scales near equilibrium [START_REF] Ben Arous | Aging in the random energy model[END_REF][START_REF] Ben Arous | Glauber dynamics of the random energy model. I. Metastable motion on the extreme states[END_REF][START_REF] Ben Arous | Glauber dynamics of the random energy model. II. Aging below the critical temperature[END_REF]. Quite remarkably, the predicted functional form of two-time correlation functions was recovered. Rapid progress followed over the ensuing decade, beginning with [START_REF] Ben Arous | The arcsine law as a universal aging scheme for trap models[END_REF]. The optimal domain of temperature and time-scales were this prediction applies was obtained in Ref. [START_REF] Gayrard | Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM[END_REF] (almost surely in the random environment except for times scales near equilibrium where the results hold in probability only) and these results were partially extended to the p-spin SK models [START_REF] Ben Arous | Universality of the REM for dynamics of mean-field spin glasses[END_REF], [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF].

The choice of the Random Hopping dynamics, however, clearly favored the emergence of trap models. Just as in trap model constructions, its trajectories are those of a simple random walk on the underlying graph, and thus, do not depend on the random Hamiltonian. This is in sharp contrast with Metropolis dynamics, a choice heralded in the physic's literature as the natural microscopic Glauber dynamics [START_REF] Junier | Microscopic realizations of the trap model[END_REF], whose trajectories are biased against increasing the energy. This dependence on the random Hamiltonian makes the analysis of the two-time correlation functions much harder. This problem was first tackled in [START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF] were a truncated REM is considered, and a natural two-time correlation function is proved to behave as in the Random Hopping dynamics, in the same, optimal range of time-scales and temperatures for which this result holds almost surely in the random environment. In the present paper, we free ourselves of the simplifying truncation assumption and prove that the same result holds true almost surely for the full REM. A partial result was obtained in the recent paper [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF] where its is proved that a certain clock process -a key object in the aging mechanism -converges to a stable subordinator in the M 1 -topology of Skorohod, in probability with respect to the random environment and in a limited domain of the time-scale and temperature parameters (see the discussion below Theorem (1.4) for details). As noted by the authors of ref. [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF], this result and its method of proof did not allow them to deduce aging, namely, convergence of two-time correlation functions.

As explained in detail in the remainder of this introduction, our analysis of two-time correlation functions relies on a scheme that consists in expressing Metropolis dynamics of the REM as an exploration process time-changed by a clock process, and in studying these two (interrelated) processes. Let us briefly discuss a different approach, initiated in [START_REF] Bezerra | Scaling limits and aging for asymmetric trap models on the complete graph and K processes[END_REF][START_REF] Fontes | On the dynamics of trap models in Z d[END_REF] in the simpler context of trap models. In those references, a process defined as "the mean holding time at the currently visited vertex" and known today as the age process was introduced in the hope that this process alone would suffice to establish the aging behavior of any two-time correlation functions. However, even within this very simple framework additional results, including explicit knowledge of the clock process, remained necessary to analyse some classical two-time correlation functions (e.g., (1.8) below). A thorough discussion of the age process in the more complex setting of Metropolis dynamics of the REM can be found in the last section of [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF]. In order to make sense, the age process must now be defined not as the mean holding time at the currently visited vertex, as in [START_REF] Bezerra | Scaling limits and aging for asymmetric trap models on the complete graph and K processes[END_REF], but rather as the mean exit time of the currently visited metastable set containing that vertex, or as some asymptotically equivalent process. This is the idea underlying the generalization of the age process proposed in (8.5) of [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF]. However, the authors could not prove that this process converges and mention the missing proof of statement (8.3) of [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF] as being one of their main obstacles. Statement (8.3) of [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF] is in essence equivalent to Proposition 3.8 of the present paper and, thus, is solved here. The only remaining ingredient needed to prove the desired convergence that we do not provide (nor does [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF]) is the exponentiality of metastable exit times. Addressing this question, which can be done using, for example, the techniques of [START_REF] Bovier | Metastability in stochastic dynamics of disordered mean-field models[END_REF][START_REF] Bovier | Metastability and low lying spectra in reversible Markov chains[END_REF], goes beyond the scope of this paper.

1.1. Main result. Let us now specify the model. Denote by V n = {-1, 1} n the ndimensional discrete cube and by E n its edge set. The Hamiltonian (or energy) of the REM is a collection of independent Gaussian random variables, (H n (x), x ∈ V n ), satisfying EH n (x) = 0, EH 2 n (x) = n.

(1.1)

The sequence (H n (x), x ∈ V n ), n > 1, is defined on a common probability space denoted by (Ω, F, P). On V n , we consider the Markov jump process (X n (t), t > 0) with rates

λ n (x, y) = 1 n e -β[Hn(y)-Hn(x)] + , if (x, y) ∈ E n , (1.2)
and λ n (x, y) = 0 else, were a+ = max{a, 0}. This defines the single spin-flip continuous time Metropolis dynamics of the REM at temperature β -1 > 0. Note that the rates are reversible with respect to the measure that assigns to x ∈ V n the mass τ n (x) ≡ exp{-βH n (x)}.

(1.3)

When studying aging the choice of the observation time-scale, c n , is all-important. Given 0 < ε < 1 and 0 < β < ∞, we let c n ≡ c n (β, ε) be the two-parameter sequence defined by 2 εn P(τ n (x) ≥ c n ) = 1.

(1.4) Gaussian tails estimates yield the explicit form

c n = exp nββ c (ε) -(1/2α(ε)) log(β 2 c (ε)n/2) + log 4π + o(1) (1.5)
where

β c (ε) = √ ε2 log 2, (1.6) α(ε) = β c (ε)/β.
(1.7)

A classical choice of two-time correlation function is the probability C n (t, s) to find the process in the same state at the two endpoints of the time interval [c n t, c n (t + s)],

C n (t, s) ≡ P µn (X n (c n t) = X n (c n (t + s))) , t, s > 0.

(1.8)

Here P µn denotes the law of X n conditional on F (i.e. for fixed realizations of the random Hamiltonian) when the initial distribution, µ n , is the uniform measure on V n .

Theorem 1.1. For all 0 < ε < 1 and all β > β c (ε), for all t > 0 and s > 0, P-almost surely,

lim n→∞ P µn (X n (c n t) = X n (c n (t + s))) = sin α(ε)π π t/(t+s) 0 u α(ε)-1 (1-u) -α(ε) du. (1.9)
Remark. We in fact prove the more general statement that (1.9) holds along any n-dependent sequences of the form 0 < ε n ≤ 1-c β n -1 log n+c n -1 log n where 0 < c , c < ∞ are constants, that satisfy lim n→∞ ε n = ε, 0 < ε ≤ 1. Relaxation to stationarity is known to occur, to leading order, on time-scales c n of the form (1.5) with ε n = 1 [START_REF] Fontes | The spectral gap of the REM under Metropolis dynamics[END_REF]. At the other extremity, a behavior known as extremal aging is expected to characterize the process on times scales that are sub-exponential in the volume and defined through sequences ε n that decay to 0 slowly enough [START_REF] Bovier | Convergence to extremal processes in random environments and extremal ageing in SK models[END_REF], [START_REF] Ben Arous | Universality and extremal aging for dynamics of spin glasses on subexponential time scales[END_REF]. This will be the object of a follow up paper.

As in virtually all papers on aging, the proof of Theorem 1.1 relies on a scheme that seeks to isolate the causes of aging by writing the process of interest, X n , as an exploration process time-changed by (the inverse of) a clock process. Aging is then linked to the arcsine law for stable subordinators through the convergence of the suitably rescaled clock process to an α-stable subordinator, 0 < α < 1. This, provided that the two-time correlation function at hand can be brought into a suitable function of the clock.

While this scheme offers the methodological underpinnings of the analysis of aging, two distinct ways of implementing it, through discrete or continuous time objects, respectively, have emerged from the literature (we refer to [START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF], [START_REF] Gayrard | Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on Z d . ALEA[END_REF], and [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF] for in-depth bibliographies). The first arose from the study of models whose exploration process can be chosen as the simple random walk on the underlying graph. As mentioned earlier, this includes all Random Hopping dynamics and several trap models (e.g. on the complete graph or on Z d ). In physically more realistic dynamics the discrete scheme may quickly become intractable. As shown in Ref. [START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF] for Metropolis dynamics of a truncated REM, the associated exploration process is itself an aging process that presents the same complexity as the original dynamics. A similar situation arises when considering asymmetric trap models on Z d . Initiated in that context, the continuous time scheme consists in choosing a (now continuous time) exploration process that mimics the simple random walk.

Since the prescription of the exploration process completely determines the clock process, it is essential to have effective tools to prove that clock processes converge to stable subordinators. Such tools were provided in Ref. [START_REF] Gayrard | Convergence of clock process in random environments and aging in Bouchaud's asymmetric trap model on the complete graph[END_REF] and [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF] for discrete-time clock processes in the general setting of reversible Markov jumps processes in random environment on sequences of finite graphs and, more recently, for both discrete and continuous-time clock processes of similar Markov jumps processes on infinite graphs [START_REF] Gayrard | Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on Z d . ALEA[END_REF]. These tools have allowed to both improve all earlier results on the Random Hopping dynamics of mean-field models [START_REF] Gayrard | Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM[END_REF], [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF], [START_REF] Bovier | Convergence to extremal processes in random environments and extremal ageing in SK models[END_REF], turning statements previously obtained in law into almost sure statements in the random environment, and to obtain the first aging results for several two-time correlation functions of asymmetric trap model on Z d [START_REF] Gayrard | Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on Z d . ALEA[END_REF].

In Section 1.2 below we fill the gap left by continuous-time clock processes in the case of sequences of finite graphs and, thus, extent the results of Ref. [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF] to that setting. This is perhaps no more than an exercise but these results (Theorem 1.2 and Theorem 1.3) are the cornerstone of our approach and, hopefully, of other papers to come. We close this introduction in Section 1.3 by stating a clock process convergence result for Metropolis dynamics of the REM (Theorem 1.4) that is at the heart of the proof of Theorem 1.1.

1.2.

Convergence of continuous-time clock processes. We now enlarge our focus to the following abstract setting. Let G n (V n , E n ) be a sequence of loop-free graphs with set of vertices V n and set of edges E n . A random environment is a family of possibly dependent positive random variables, (τ n (x), x ∈ V n ). The sequence (τ n (x), x ∈ V n ), n > 1, is defined on a common probability space denoted by (Ω, F, P). On V n we consider a Markov jump process, (X n (t), t > 0), with initial distribution µ n and jump rates (λ n (x, y)) x,y∈Vn satisfying λ n (y, x) = 0 if (x, y) / ∈ E n and

τ n (x)λ n (x, y) = τ n (y)λ n (y, x) if (x, y) ∈ E n , x = y. (1.10)
Thus X n is reversible with respect to the (random measure) that assigns to x ∈ V n the mass τ n (x). To X n we associate an exploration process Y n . This is any Markov jump process, (Y n (t), t > 0), with state space V n , initial distribution µ n , and jump rates ( λ n (x, y)) x,y∈Vn chosen such that X n and Y n have the same trajectories, that is to say,

λ n (x, y) λ n (x) = λ n (x, y) λ n (x) ∀(x, y) ∈ E n , (1.11)
where λ -1 n (x) and λ -1 n (x) are, respectively, the mean holding times at x of Y n and X n :

λ n (x) ≡ y:(x,y)∈En λ n (x, y), (1.12
)

λ n (x) ≡ y:(x,y)∈En λ n (x, y). (1.13)
Then X n and Y n are related to each other through the time change

X n (t) = Y n ( S ← n (t)), t ≥ 0, (1.14)
where S ← n denotes the generalized right continuous inverse of S n , and S n , the so-called continuous-time clock process, is given by

S n (t) = t 0 λ -1 n (Y n (s)) λ n (Y n (s))ds, t ≥ 0. (1.15)
Note that there is considerable freedom in the choice of the exploration process Y n . We come back to this issue at the end of this subsection and focus, for the time being, on the analysis of the asymptotic behavior of the general clock process (1.15).

For future reference, we denote by F Y the σ-algebra generated by the processes Y n . We write P for the law of the process Y n conditional on the σ-algebra F, i.e. for fixed realizations of the random environment. Likewise we call P the law of X n conditional on F. If the initial distribution, µ n , has to be specified we write P µn and P µn . Expectation with respect to P, P µn , and P µn are denoted by E, E µn , and E µn , respectively.

Our main aim is to obtain simple and robust criteria for the convergence of the (suitably rescaled) clock process (1.15) to a stable subordinator. More precisely, we will ask whether there exist sequences a n and c n that make the rescaled clock process

S n (t) = c -1 n S n (a n t) , t ≥ 0, (1.16)
converge weakly, as n ↑ ∞, as a sequence of random elements in Skorokhod's space D((0, ∞]), and strive to obtain P-almost sure results in the random environment since such results (also referred to as quenched) contain the most useful information from the point of view of physics.

As for discrete-time clock processes [START_REF] Gayrard | Convergence of clock process in random environments and aging in Bouchaud's asymmetric trap model on the complete graph[END_REF], [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF], the driving force behind our approach is a powerful method developed by Durrett and Resnick [START_REF] Durrett | Functional limit theorems for dependent variables[END_REF] to prove functional limit theorems for sums of dependent variables. Clearly this method does not cover the case of our continuous-time clock processes. The simple idea (already present in [START_REF] Gayrard | Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on Z d . ALEA[END_REF]) is to introduce a suitable "blocking" that turns the rescaled clock process (1.16) into a partial sum process to which Durrett and Resnick method can now be applied. For this we introduce a new scale, θ n , and set k n (t) ≡ a n t/θ n .

(1.17)

The blocked clock process, S b n (t), is defined through

S b n (t) = kn(t) i=1 Z n,i (1.18)
where, for each i ≥ 1,

Z n,i ≡ c -1 n x∈Vn λ -1 n (x) λ n (x) [ x n (θ n i) -x n (θ n (i -1))], (1.19)
and where, for each x ∈ V n ,

x n (t) = t 0 1 {Yn(s)=x} ds (1.20)
is the local time at x. The next theorem gives sufficient conditions for S b n to converge. These conditions are expressed in terms of a small number of objects. For each t > 0, let

π Y,t n (y) = k -1 n (t) kn(t)-1 i=1 1 {Yn(iθ)=y} (1.21)
be the empirical measure on V n constructed from the sequence (Y n (iθ), i ∈ N). For y ∈ V n and u > 0, denote by Q u n (y) ≡ P y (Z n,1 > u) (1.22) the tail distribution of the aggregated jumps when X n (equivalently, Y n ) starts in y. Using these quantities, define the functions

ν Y,t n (u, ∞) ≡ k n (t) y∈Vn π Y,t n (y)Q u n (y), (1.23)
σ Y,t n (u, ∞) ≡ k n (t) y∈Vn π Y,t n (y) [Q u n (y)] 2 . (1.24)
Observe that the sequence of measures π Y,t n as well as the sequence of functions Q u n (y), y ∈ V n , are random variables on the probability space (Ω, F, P) of the random environment. Thus, the functions ν Y,t n and σ Y,t n also are random variables on that space. We now formulate four conditions for the sequence S b n to converge to a subordinator. These conditions refer to a given sequence of initial distributions µ n , given sequences of numbers a n , c n , and θ n as well as a given realization of the random environment. Condition (A0). For all u > 0,

lim n→∞ P µn (Z n,1 > u) = 0.
(1.25)

Condition (A1).

There exists a σ-finite measure ν on (0, ∞) satisfying ∞ 0 (x∧1)ν(dx) < ∞ and such that for all continuity points x of the distribution function of ν, for all t > 0 and all u > 0,

P µn ν Y,t n (u, ∞) -tν(u, ∞) < = 1 -o(1) , ∀ > 0 . (1.26)
Condition (A2). For all u > 0 and all t > 0,

P µn σ Y,t n (u, ∞) < = 1 -o(1) , ∀ > 0 . (1.27)
Condition (A3). For all t > 0, lim

↓0 lim sup n↑∞ k n (t) y∈Vn E µn (π Y,t n (y))E y (Z n,1 1 {Z n,1 ≤ } ) = 0.
(1.28) Theorem 1.2. For all sequences of initial distributions µ n and all sequences a n , c n , and 1 ≤ θ n a n for which Conditions (A0), (A1), (A2), and (A3) are verified, either P-almost surely or in P-probability, the following holds w.r.t. the same convergence mode:

S b n ⇒ J 1 S ν , (1.29)
where S ν is the Lévy subordinator with Lévy measure ν and zero drift. Convergence holds weakly on the space D([0, ∞)) equipped with the Skorokhod J 1 -topology.

Remark. Note that the theorem is stated for the blocked process S b n rather than the original process S n of (1.16). This may falsely appear as an undesirable consequence of our techniques. We stress that for applications to correlation functions, one needs statements that are valid in the strong J 1 topology whereas forming blocks is needed in order to make sense of writing J 1 convergence statements in the setting of continuous-time clocks.

As for the discrete-time clocks of Ref. [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF], our next step consists in reducing Conditions (A1) and (A2) of Theorem 1.2 to (i) a mixing condition for the chain Y n , and (ii) a law of large numbers for the random variables Q n . Again we formulate three conditions for a given sequence of initial distributions µ n , given sequences a n , c n , and θ n , and a given realization of the random environment. Condition (B0). Denote by π n the invariant measure of Y n . There exists a sequence κ n ∈ N and a positive decreasing sequence ρ n , satisfying ρ n ↓ 0 as n ↑ ∞, such that, for all pairs x, y ∈ V n , and all t ≥ 0,

|P x (Y n (t + κ n ) = y) -π n (y)| ≤ ρ n π n (y).
(1.30)

Condition (B1).

There exists a measure ν as in Condition (A1) such that, for all t > 0 and all u > 0,

ν t n (u, ∞) ≡ k n (t) y∈Vn π n (y)Q u n (y) → tν(u, ∞), (1.31)
Condition (B2). For all t > 0 and all u > 0,

σ t n (u, ∞) ≡ k n (t) y∈Vn π n (y) [Q u n (y)] 2 → 0. (1.32) Condition (B3). For all t > 0, lim ↓0 lim sup n↑∞ k n (t) y∈Vn π n (y)E y (Z n,1 1 {Z n,1 ≤ } ) = 0.
(1.33) Theorem 1.3. Assume that for all sequences of initial distributions µ n and all sequences a n , c n , κ n , and κ n ≤ θ n a n , Conditions (A0), (B0), (B1), (B2), and (B3) hold P-almost surely, respectively in P-probability. Then, as in (1.29), S b n ⇒ J 1 S ν , P-almost surely, respectively in P-probability.

Theorem 1.3 is our key tool for proving convergence of blocked clock processes to subordinators. It is of course essential for the success of our strategy that the convergence criteria we obtained be tractable. Going back to (1.11) we thus now ask, in this light, how best to choose the exploration process Y n .

A tentative answer to this question is to mimic the exploration process of the Random Hopping dynamics, which means choose Y n such that its invariant measure, π n , is "close" to the uniform measure and its mixing time, κ n , is short compared to that of the process X n . The following class of jump rates, inspired from an ingenious choice made in Ref. [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF], is intended to favor the emergence of these properties. Given a fresh sequence η n ≥ 0, set λ n (x, y) = max(η n , τ n (x))λ n (x, y).

(1.34)

One easily checks that (1.11) is verified, that Y n is reversible with respect to the measure

π n (x) = min η n , τ n (x)
x∈Vn min η n , τ n (x)

1 {ηn>0} + |V n | -1 1 {ηn=0} , x ∈ V n , (1.35)
and that the clock (1.15) becomes

S n (t) = t 0 max η n , τ n (Y n (s)) ds.
(1.36)

We will see in Section 3.1 that in Metropolis dynamics of REM the parameter η n has a capping effect on the mixing time of the exploration process, namely, κ n in (1.30) can be made as small as needed by taking η n large enough, while on the other hand π n can be kept as close as desired to the uniform measure (obtained when choosing η n = 0 in (1.35)) by keeping η n small enough. This still gives us plenty of freedom to choose η n .

Let us finally stress that the sole convergence statement (1.29) does not suffice to deduce aging, namely, the specific power law decay of the two-time correlation function of (1.9). One still has to show that the correlation function can be reduced, asymptotically, to the arcsine law for stable subordinators, and this typically requires extra information on the behavior of the exploration process within the blocks of S b n and in between given blocks.

1.3. Application to Metropolis dynamics of the REM. From that point onwards we focus on Metropolis dynamics of the REM (see (1.1)-(1.2)) started in the uniform measure on V n . Applying the abstract results of Section 1.2 enables us to prove P-almost sure convergence of the blocked clock process S b n (t), defined in (1.18), when the continuoustime clock process S n (t), given by (1.15), is chosen as in (1.36).

To sate this result we must specify several quantities: the parameter η n , the time-scales, a n and c n , and the block length, θ n , entering the definitions of S n (t) and S b n (t). We begin by defining a sequence, r n , that is ubiquitous throughout the rest of the paper: given β > 0 and a constant c > 1 + log 4, we let r n ≡ r n (β, c ) be the solution of

n c P(τ n (x) ≥ r n ) = 1.
(1.37)

In explicit form

r n = exp β 2c n log n 1 -log log n 8c log n (1 + o(1)) . (1.38)
We now take η n ≡ (r n ) -1 in (1.34) which, combined with (1.2), yields

λ n (x, y) = 1 nr n min(τ n (y), τ n (x)) min 1 r n , τ n (x) , if (x, y) ∈ E n , (1.39)
and λ n (x, y) = 0 else. The observation time-scale, c n , is chosen as in (1.4). It is naturally the same as in the Random Hopping dynamics. On the contrary, the definition of the auxiliary time-scale, a n , contrasts sharply with the simple choice a n = 2 εn made in the Random Hopping dynamics. We here must take

a n = 2 εn /b n (1.40)
where the sequence b n is defined as follows. Recalling (1.6) and (1.7), define

F β,ε,n (x) ≡ x αn(ε)-log x 2nβ 2 1 -log x nββc(ε) -1 , x > 0, (1.41)
where

α n (ε) ≡ (nβ 2 ) -1 log c n , that is, in view of (1.5), α n (ε) = α(ε)(1 -o(1)
). Further introduce the random set

T n ≡ x ∈ V n | τ n (x) ≥ c n (n 2 θ n ) -1 . (1.42)
Then, for x n as in (1.20), we set

b n ≡ (θ n π n (T n )) -1 x∈Tn E πn [F β,ε,n, ( x n (θ n ))] . (1.43)
This somewhat daunting definition is discussed below. One of the strengths of the method, however, is that it does not require a deep understanding of b n whose fine properties ultimately do not matter. It now only remains to choose the block length θ n . (The notation x n y n means that the sequences x n > 0 and y n > 0 satisfy x n /y n → 0 as n → ∞.) Theorem 1.4. Given 0 < ε < 1 let θ n be any sequence such that

4 1-α(ε) log r n < log θ n n (1.44)
and let c n and a n be as in (1.4) and (1.40)-(1.43), respectively. Then, for all 0 < ε < 1 and all β > β c (ε), P-almost surely,

S b n ⇒ J 1 V α(ε) (1.45)
where V α(ε) is a stable subordinator with zero drift and Lévy measure ν defined through

ν(u, ∞) = u -α(ε) , u > 0, (1.46)
and where ⇒ J 1 denotes weak convergence in the space D([0, ∞)) of càdlàg functions equipped with the Skorokhod J 1 -topology.

We again emphasize (see the remark below Theorem (1.2)) that the J 1 convergence statement of Theorem 1.4 is a necessary ingredient of the proof of the convergence of the correlation function. Of course, Theorem 1.4 immediately implies that the original (non blocked) clock process (1.16) converges to the same limit in the M 1 topology of Skorokhod, but this strictly weaker result does not allow to retrieve information on the correlation function. Such a result was proved in Ref. [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF] (for the clock obtained by taking η n = 1 in (1.36)) albeit only in P-probability and in the restricted domain of parameters β > β c (ε) and 1/2 < ε < 1. When 1/2 < ε < 1 the graph structure of the set T n reduces, as shown in [START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF] (see lemma 2.1), to a collection a collection of isolated vertices, namely, no element of T n has a neighbor in T n . This feature of the REM's random landscape leads to drastic simplifications. In particular, it has the remarkable implication that given T n , the law of the exploration process Y n becomes independent of the random environment in T n , as can easily be seen from (1.39).

Let us now examine the sequence b n introduced in (1.40) and defined in (1.43). We do not have much intuition to offer for this complicated definition except that it emerges in a straightforward way from the verification of Condition (B1) of Theorem 1.3. One sees that b n is a priori random in the random environment and depends on a sequence, θ n , that can itself be chosen within the two widely different bounds of (1.44). The next proposition provides deterministic upper and lower bounds on b n that are not affected by the choice of θ n and are valid P-almost surely.

Proposition 1.5. Given 0 < ε < 1, let c n and θ n be as in Theorem 1.4. Then, there exists a subset Ω ⊆ Ω with P(Ω ) = 1 such that on Ω , for all but a finite number of indices n

n c -(r n ) 1+αn(ε)+o(1) -1 ≤ b n ≤ n c + (r n ) 1+αn(ε) (1.47) where 0 < c -, c + ≤ ∞ are numerical constants. Thus lim n→∞ n -1 log a n = ε P-a.s..
Remark. The form of (1.43) naturally prompts us to ask whether b n converges as n diverges and, if so, whether the limit remains random or not. We have not been able to answer these questions. Indeed, the randomness of b n enters mainly through the local times which depend on the fine details of the random environment locally, in some vicinity of the set T n , and are delicate to control. However, as already mentioned, a strong side of the method is that no knowledge of the fine asymptotic properties of b n is needed. Deterministic bounds suffice.

Remark. The precision of Theorem 1.1 does depend on the precision of the bounds on b n through the domain of validity of the parameters ε and β (bad bounds would have affected this domain) but not the nature of the aging result itself: b n is an auxiliary time-scale whose properties have no impact on aging.

Remark. The definition (1.40)-(1.43) of a n and that of the sequence R N in (2.10) of Ref. [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF] bear a distinct resemblance. Our control of a n through Proposition 1.5, which is sharp up to error terms of order e ±cst √ n log n ,must be compared to the bounds on R N of Lemma 4.4 of [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF] that differ by multiplicative error terms of order e ± n , > 0.

Remark. One may wonder whether the lower bound of (1.44) can be improved. The main obstacle to doing so is the lower bound on mean hitting times of Lemma 3.4. In particular, trying to improve the bound (3.3) on the spectral gap by choosing η n larger, say as large as 1 as in Ref. [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF], can at best improve the constant 4 1-α(ε) in front of log r n in (1.44).

The rest of the paper is organized as follows. Section 2 is concerned with the properties of the REM's landscape: several level sets that play an important role in our analysis are introduced and their properties collected. Section 3 gathers all needed results on the exploration process Y n . The proof of Theorem 1.4 can then begin. Section 4, 5, and 6 are devoted, respectively, to the verification of Condition (B1), (B2), and (B3) of Theorem 1.3. The proof of Proposition 1.5 is given at the end of Subsection 4.2. The proof of Theorem 1.4 is completed in Section 7. Also in Section 7, the link between the blocked clock process of (1.45) and the two-time correlation function (1.8) is made, and the proof of Theorem 1.1 is concluded. An appendix (Section 8) contains the proof of the results of Section 1.2.

LEVEL SETS OF THE REM'S LANDSCAPE: THE TOP AND OTHER SETS

Given V ⊆ V n we denote by G ≡ G(V ) the undirected graph which has vertex set V and edge set E(G(V )) ⊆ E n consisting of pairs of vertices {x, y} in V with dist(x, y) = 1, where dist(x, x ) ≡ 1 2 n i=1 |x i -x i | is the graph distance on V n . When dist(x, y) = 1 we simply write x ∼ y. We now introduce several sets that play key roles in our analysis: they are level sets of the form

V n (ρ) = {x ∈ V n | τ n (x) ≥ r n (ρ)} (2.1)
where, for different values of ρ > 0, the threshold level r n (ρ) is the sequence defined by

2 ρn P(τ n (x) ≥ r n (ρ)) = 1. (2.2)
• The sets V n and V n (of local valleys and hills). Set set

V n ≡ V n (ρ n ) where ρ n ≡ c log n n log 2 (2.3)
for c as in (1.37). V n can uniquely be decomposed into a collection of subsets

V n = ∪ L l=1 C n,l , C n,l ∩ C n,k ∀l = k, L ≡ L n (ρ n ), (2.4)
such that each graph G(C n,l ) is connected but any two distinct graphs G(C n,l ) and G(C n,k ) are disconnected. With a little abuse of terminology we call the sets C n,l the connected components of the graph G(V n ). From now on we write r n ≡ r n (ρ n ). Let

V n ≡ V n (ρ n ) = x ∈ V n | τ -1 n (x) ≥ r n (2.5) be the set obtained from V n (ρ n ) by substituting -H n (x) for H n (x) in (1.3). Since H n (x)
is symmetrical V n has the same random graph properties as V n . Note that the form of the rates (1.39) depend on the set V n , namely,

λ n (x, y) = 1 n e -β max(Hn(y),Hn(x)) , if x / ∈ V n , 1 nr n e -β[Hn(y)-Hn(x)] + , if x ∈ V n .
(2.6) Key properties of these rates are gathered at the end on this section. As shown later in Lemma 2.1, the sets V n and V n contain only "small" connected components and their complement, V n \ (V n ∪ V n ), forms a totally connected "giant" component (see [START_REF] Fontes | The spectral gap of the REM under Metropolis dynamics[END_REF]). We may thus think of the connected components of V n and V n as containing, respectively, the local "valleys" and "hills" of the random energy landscape, H n , whereas in their complement, or "horizon level", H n has only small fluctuations.

• Immersions in V n . Given any subset A ⊂ V n we call immersion of A in V n and denote by A the set

A ≡ ∪ L l=1 A n,l , A n,l = C n,l , if C n,l ∩ A = ∅, ∅, else. (2.7)
Thus the sets A n,l are the valleys C n,l that contain at least one element of A. Clearly,

V n ∩ V n = ∅.
Hence by (2.6), immersed sets have the property that

λ n (x, y) ≤ n -1 r n for all x ∼ y such that x ∈ A , y / ∈ A or y ∈ A , x / ∈ A . (2.8)
• The top, T n , and the associated sets T n , T • n and I n . Given a sequence δ n ↓ 0 as n ↑ ∞, set ε n ≡ ε -δ n and let the top be the set

T n ≡ V n (ε n ) (2.9)
obtained by taking ρ = ε n in (2.1). (δ n will later be chosen so that the definitions (2.9) and (1.42) coincide.) Clearly, T n contains the top of the order statistics of -H n (that is, the deepest valleys of the random landscape). Since ρ n ε n , T n ⊂ V n , so that T n can be immersed in V n . According to (2.7) we write

T n ≡ ∪ L l=1 T n,l . (2.10) To each x ∈ T n corresponds a unique index 1 ≤ l ≡ l(x) ≤ L such that x ∈ T n,l(x) .
Of course a given valley T n,l may contain several vertices of T n . A set that is of special importance in the sequel is the subset T • n of vertices of T n that are alone in their valley,

T • n ≡ x ∈ T n | T n,l(x) ∩ T n = {x} . (2.11)
Finally, define

I n ≡ {x ∈ V n | τ n (x) ≥ r n (ε n ), ∀ y∼x (r n ) -1 < τ n (y) < r n } ⊆ T • n .
(2.12)

The content of the next three lemmata is taken from [START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF]: the first one gives estimates on the size of various sets, the second one expresses the function r n (ρ) defined through (2.2) and the last one states needed bounds, in particular, on the maximal jump rate.

Lemma 2.1. There exists Ω ⊂ Ω with P (Ω ) = 1 such that on Ω , for all but a finite number of indices n,

1 ≤ |C n,l | ≤ {ρ n [1 -2c -1 (1 + O(log n/n))]} -1 , 1 ≤ l ≤ L .
(2.13) Furthermore,

|V n | = 2 n n -c (1 + o(n -c )) and |V n | = 2 n n -c (1 + o(n -c )), (2.14)
|T n | = 2 n(1-εn) (1 + O(n2 -nεn/2 )), (2.15)
|T • n | = 2 n(1-εn) (1 + O(n2 -nεn/2 )), (2.16
)

|T n \ T • n | ≤ n 4 2 n(1-2εn) (1 + o(1)), (2.17
)

|I n | = 2 n(1-εn) (1 -2n -c +1 (1 + o(1)), (2.18
) ). For all ρ > 0, possibly depending on n, and such that ρn ↑ ∞ as n ↑ ∞,

|T • n \ I n | = 2n -c +1 2 n(1-εn) (1 + o(1)). ( 2
r n (ρ) = exp nββ c (ρ) -(β/2β c (ρ)) log(β 2 c (ρ)n/2) + log 4π + o(β/β c (ρ)) . (2.20) Lemma 2.3 (Lemma 2.

of [24]

). There exists a subset Ω 0 ⊆ Ω with P Ω 0 = 1 such that on Ω 0 , for all but a finite number of indices n the following holds: (

e -β min{max(Hn(y),Hn(x)) | (x,y)∈En} ≤ e βn √ log 2(1+2 log n/n log 2) ≡ ν n , (2.21

2.22)

To close this section let us collect some elementary but key properties of the rates. First note that by (2.6) and (1.12), denoting by ∂A = {x ∈ V n | dist(x, A) = 1} the outer boundary of A ⊂ V n , we have that for all x ∈ ∂V n ,

λ n (x) = y∈(V n ) c λ n (x, y) + (nr n ) -1 1 {x∈V n } + τ n (x)n -1 1 {x∈(V n ) c } |∂x ∩ V n |. (2.23)
Hence, given V n , the mean holding time at x ∈ (V n ) c does not depend on the variables {τ n (y), y ∈ V n } but only depends on the variables {τ n (y), y

∈ (V n ) c }. Next, introduce the set M n ≡ {x ∈ V n | τ n (x) > τ n (y) for all y ∼ x} (2.24)
of local minima of H n and observe that by (2.6), for all x ∈ M n ∩ V n and all y ∼ x,

λ n (x, y) = n -1 τ n (y) and λ n (y, x) = n -1 τ n (y), if y / ∈ V n , n -1 , if y ∈ V n , (2.25)
Hence, given M n ∩ V n , the generator of the process Y n does not depend on the variables {τ n (x), x ∈ M n ∩ V n }. (One in fact may show that on a set of full measure, for all large enough n, it does not depend on the variables {τ n (x), x ∈ M n }.) Since

T • n ⊆ {x ∈ V n | τ n (x) ≥ r n (ε n ), ∀ y∼x τ n (y) < r n (ε n )} ⊆ M n , (2.26)
the generator of the process Y n does not depend on the variables {τ n (x), x ∈ T • n }.

PROPERTIES OF THE EXPLORATION PROCESS Y n

In this Section we establish the properties of the exploration process needed in the rest of the paper. By (1.35) with η n ≡ (r n ) -1 and (2.5), the invariant measure π n of Y n can be written as

π n (x) = 1 {x / ∈V n } + r n τ n (x)1 {x∈V n } Z -1 β,n , x ∈ V n (3.1)
where

Z β,n ≡ |V n \ V n | + x∈V n r n τ n (x).
Lemma 3.1. On Ω , for all but a finite number of indices n, for all subset

A ⊆ V n such that A ∩ V n = ∅ π n (A) = |A|2 -n (1 + o(1)) (3.
2) whereas for arbitrary A, (3.2) holds with equality replaced by "less than or equal". In both cases o(1) is independent of A.

Proof. Since {x ∈ V n } = {r n τ n (x) ≤ 1}, |V n \ V n | ≤ Z β,n ≤ |V n \ V n | + |V n | ≤ 2 n . Eq. (2.14) of Lemma 2.1 then yields 2 n (1 -n -c (1 + o(n -c )) ≤ Z β,n ≤ 2 n .
The claim of the lemma directly follows.

3.1. Spectral gap and mixing condition. Denote by L n the Markov generator matrix of Y n (that is, the matrix with off-diagonal entries λ n (x, y) and diagonal entries -λ n (x)), and by

0 = ϑ n,0 < ϑ n,1 ≤ • • • ≤ ϑ n,2 n -1 the eigenvalues of -L n .
Proposition 3.2. If c > 1 + log 4 then for all β > 0, there exists a subset Ω 1 ⊂ Ω with P (Ω 1 ) = 1 such that, on Ω 1 , for all but a finite number of indices n,

1/ϑ n,1 ≤ 5 2 n 2 r n (1 + o(1)) ≡ κn (3.3)
As a direct consequence on Proposition 3.2, Condition (B0) of Theorem 1.3 is satisfied P-almost surely with e.g.

κ n ≡ n 4 r n .

(3.4) Proposition 3.3. On Ω 1 , for all but a finite number of indices n, for all pairs x, y ∈ V n and all t ≥ 0,

|P x (Y n (t + κ n ) = y) -π n (y)| ≤ ρ n π n (y), (3.5
) where κ n is given by (3.4) and ρ n < e -n .

Proof of Proposition 3.2. This is a simple adaptation of the proof of [START_REF] Fontes | The spectral gap of the REM under Metropolis dynamics[END_REF] (or [START_REF] Černý | Aging of the metropolis dynamics on the random energy model[END_REF], albeit with other constants).

Proof of Proposition 3.3. Using (3.1), the bound Z β,n ≤ 2 n and (2.22) of Lemma 2.3 to bound sup z∈Vn π -1 n (z) from above, the claim of Proposition 3.3 readily follows from the bound (1.10) of Proposition 3 of Ref. [START_REF] Diaconis | Geometric bounds for eigenvalues of Markov chains[END_REF] and Proposition 3.2, choosing κ n as in (3.4).

3.2.

Hitting time for the stationary chain. Drawing heavily on Aldous and Brown's work [START_REF] Aldous | Inequalities for rare events in time-reversible Markov chains. I. In Stochastic inequalities[END_REF], this section collects results on hitting times for the process Y n at stationarity. Let

H(A) = inf{t ≥ 0 | Y n (t) ∈ A} (3.6)
be the hitting time of A ⊆ V n . We begin with bounds on the mean value of H(A).

Lemma 3.4. On Ω 1 , for all but a finite number of indices n, for all A ⊆ V n ,

(1 -nπ n (A)) 2 r n nπ n (A)(1 -π n (A)) ≤ E πn H(A) 1 -π n (A) ≤ κn π n (A) . (3.7)
The next lemma gives bounds on the density function h n,A (t), t > 0, of H(A) when Y n starts in its invariant measure, π n . Lemma 3.5. On Ω 1 , for all but a finite number of indices n, for all A ⊆ V n and all t > 0,

1 E πn H(A) 1 - κn E πn H(A) 2 1 - t E πn H(A) ≤ h n,A (t) ≤ 1 E πn H(A) 1 + κn 2t .
The bounds of Lemma 3.5 imply that h n,A (t) ≈

1

Eπ n H(A) when κn t E πn H(A). Complementing this, Lemma 3.6 is well suited to dealing with "small" values of t. Lemma 3.6. On Ω , for all but a finite number of indices n, for all A ⊆ V n and all t > 0,

P πn (H(A) > t) ≥ (1 -nπ n (A)) exp -t r n nπ n (A) 1 -nπ n (A) . (3.8)
In particular, for any A and any sequence t n such that t n r n nπ n (A) → 0 as n → ∞,

P πn (H(A) ≤ t n ) < t n r n nπ n (A) (1 + t n r n nπ n (A)) (3.9)
for all large enough n.

The next Corollary is stated for later convenience.

Corollary 3.7. Under the assumptions of Lemma 3.6 the following holds: For all 0 < ε < 1, for any sequence t n such that t n r n n2 -nεn → 0 as n → ∞

P πn (H(T n \ T • n ) ≤ t n ) ≤ t n r n n 5 2 -2nεn (1 + o(1)), (3.10)
P πn (H(T • n ) ≤ t n ) ≤ t n r n n2 -nεn (1 + o(1)). (3.11)
We now prove these results, beginning with Lemma 3.6.

Proof of Lemma 3.6.

Write A = B ∪ B c where B = A ∩ V n and B c = A \ B. Let B be the immersion of B in V n (see (2.7)). Since A ⊆ B ∪ B c , H(A) ≥ H(B ∪ B c
), and

P πn (H(A) > t) ≥ P πn (H(B ∪ B c ) > t). (3.12)
To bound the right-hand side of (3.12), we use a well know lower bound on hitting times for stationary reversible chains taken from Ref. [START_REF] Aldous | Inequalities for rare events in time-reversible Markov chains. I. In Stochastic inequalities[END_REF] (combine Theorem 3 and Lemma 2 therein) that states that for all C ⊆ V n and all t > 0,

P πn (H(C) > t) ≥ (1 -π n (C)) exp -t q n (C, C c ) 1 -π n (C) (3.13)
where, for for any two sets C and C such that C ∩ C = ∅,

q n (C, C) ≡ x∈C y∈ C π n (x) λ n (x, y). (3.14)
Let us thus evaluate (3.14

) with C = B ∪ B c . Clearly q n (B ∪ B c , (B ∪ B c ) c ) ≤ q n (B , (B ∪ B c ) c ) + q n (B c , (B ∪ B c ) c ).
Clearly also, by (2.6), λ n (x, y) ≤ n -1 r n for any x ∈ B c and any y ∼ x. Thus q n (B c ,

(B ∪ B c ) c ) ≤ r n π n (B c ). Next, by (2.8), q n (B , (B ∪ B c ) c ) ≤ r n π n (B ). Thus q n (B ∪ B c , (B ∪ B c ) c ) ≤ r n [π n (B ) + π n (B c )]. ( 3
(B ) = Z -1 β,n |B | ≤ nZ -1 β,n |B| = nπ n (B). Therefore, π n (B ∪B c ) ≤ π n (B )+π n (B c ) ≤ nπ n (B)+π n (B c ) ≤ nπ n (B∪B c ) = nπ n (A). (3.16)
Using (3.16) in the right-and side of (3.15) and plugging the result in (3.13) finally yields (3.8).

Proof of Corollary 3.7. This follows from (3.2) of Lemma 3.1, (2.16), and (2.17).

Proof of Lemma 3.5. Proceed as in Lemma 13 of Ref. [START_REF] Aldous | Inequalities for rare events in time-reversible Markov chains. I. In Stochastic inequalities[END_REF] and use Proposition 3.2.

Proof of Lemma 3.4. The rightmost inequality is that of Lemma 2 of Ref. [START_REF] Aldous | Inequalities for rare events in time-reversible Markov chains. I. In Stochastic inequalities[END_REF] combined with Proposition 3.2. Lemma 2 of Ref. [START_REF] Aldous | Inequalities for rare events in time-reversible Markov chains. I. In Stochastic inequalities[END_REF] also states that for C ⊆ V n and q n (C, C c ) defined as in (3.14),

E πn H(C) 1 -π n (C) ≥ 1 -π n (C) q n (C, C c ) . ( 3

.17)

Given A ⊆ V n let B and B c be defined as in the first line of the proof of Lemma 3.6. Since 3.3. On hitting the top starting in the top. Let T • n and I n be as in (2.11) and (2.12). Proposition 3.8. Given > 0 there exists a subset Ω • ⊂ Ω with P (Ω • ) = 1 such that on Ω • , for all but a finite number of indices n, for all s > 0

H(A) ≥ H(B ∪ B c ), E πn H(A) ≥ E πn H(B ∪ B c ). Using (3.17) with C = B ∪ B c , ( 3
|T • n | -1 x∈T • n P x (H(T • n \ x) ≤ s) ≤ sn c +3 r n π n (T • n ). (3.18)
The next proposition is a variant of Proposition 3.8 that we state for later convenience.

Proposition 3.9. Under the assumptions and with the notation of Proposition 3.8, on Ω • , for all but a finite number of indices n, for all s > 0

|T • n \ I n | -1 x∈T • n \I n P x (H(I n ) ≤ s) ≤ sn 2 r n π n (I n )(1 + o(1)). (3.19)
Proof of Proposition 3.8. A key ingredient of the proof is an explicit expression of the density function h x n,A (t), t ≥ 0, of the hitting time H(A) when Y n starts in x ∈ A c ≡ V n \A that we take from [START_REF] Keilson | Markov chain models-rarity and exponentiality[END_REF] (see Section 6.2, p. 83). Consider the matrix P n = ( p n (x, y)) defined by P n = I + ν -1 n L n where I denotes the identity matrix, L n the Markov generator matrix of Y n and ν n is defined in (2.22). By Lemma 2.3, on Ω 0 ,

0 < max (x,y)∈En λ n (x, y) ≤ n -1 ν n < ∞ (3.20)
for all large enough n, hence P n is a well defined the stochastic matrix (namely, its entries obey 0 ≤ p n (x, y) ≤ 1 and y∈Vn p n (x, y) = 1). Denote by Q n = (q n (x, y)) the matrix with entries q n (x, y) : A c × A c → R given by q n (x, y) = p n (x, y). This is the sub-matrix of P n on A c × A c . Thus Q n is sub-stochastic. Similarly, denote by R n = (r n (x, y)) the sub-matrix of P n on A c × A. Let 1 A be the vector of 1's on A and let δ x be the vector on A c taking value 1 at x and zero else. Then, for all x ∈ A c ,

h x n,A (t) = ν n ∞ k=0 (ν n t) k k! e -νnt δ x , Q k n R n 1 A , t ≥ 0, (3.21)
where (•, •) denotes the inner product in R |A c | . Consequently, for all s > 0,

P x (H(A) ≤ s) = s 0 ν n ∞ k=0 (ν n t) k k! e -νnt δ x , Q k n R n 1 A dt. (3.22)
For later reference we also denote by (h x n,y,A (t)) y∈A the vector whose components are, for each y ∈ A, the joint density that A is reached at time t, and that arrival to that set occurs in state y, namely, h x n,y,A (t) is defined as in (3.21) substituting δ y for 1 A therein; as a result h x n,A (t) = y∈A h x n,y,A (t). Returning to (3.18), a first order Tchebychev inequality yields, for all > 0

P x∈T • n P x (H(T • n \ x) ≤ s) ≥ ≤ -1 E x∈T • n P x H T n \ T n,l(x) ≤ s (3.23) where T n ≡ ∪ L l=1
T n,l is defined in (2.10) and l(x) is as in (2.11). Calling W n the expectation in the right-hand side of (3.23) we have, by (3.22) with A = T n \ T n,l(x) ,

W n = x∈Vn s 0 dt ∞ k=1 (ν n t) k k! e -νnt W n,k (x) (3.24)
where

W n,k (x) ≡ E 1 {x∈T • n } ν n δ x , Q k n R n 1 T n \T n,l(x) . (3.25)
Note that the term k = 0 is zero. For k ≥ 1 the matrix term in (3.25) reads,

1 {x∈T • n } ν n δ x , Q k n R n 1 T n \T n,l(x) = 1 {x∈T • n } y∈(T n \T n,l(x) ) c q (k) n (x, y) z∈T n \T n,l (x)
ν n r n (y, z)

(3.
W n,k (x) ≤ r n n E E 1 {x∈T • n } y∈(T n \T n,l(x) ) c q (k) n (x, y) z∈T n \T n,l(x) 1 {z∼y} V n (3.28)
where E[• | V n ] denotes the conditional expectation given a realization of the set V n , namely, setting C n,0 ≡ V n \ V n and using (2.4) to write V n = ∪ 0≤l≤L C n,l , expectation with respect to the measure

P(• | V n ) = P(• ∩ {∀ 1≤l≤L ∀ x∈C n,l τ n (x) ≥ r n } ∩ {∀ x∈C n,0 τ n (x) < r n }) P({∀ 1≤l≤L ∀ x∈C n,l τ n (x) ≥ r n } ∩ {∀ x∈C n,0 τ n (x) < r n }) . ( 3

.29)

Observe now that, conditionally on V n , the entries of the matrix Q n are functions of the variables {τ n (y), y ∈ (T n \ T n,l(x) ) c } only: for off-diagonal entries this is an immediate consequence of (2.6), and for diagonal entries this claim follows from (1.12) and (2.6) if x / ∈ ∂V n and from (2.23) and (2.6) if x ∈ ∂V n . To build on this property let us rewrite the sums in (3.28) in such a way that the variables {τ n (y), y ∈ T n \ T n,l(x) } no longer appear in the summations sets but only in the summands. For this note that the sum over y ∈ (T n \ T n,l(x) ) c in (3.28) can be restricted to the sum over y ∈ ∂V n ⊆ C n,0 and use the definitions (2.10) and (2.11) of T n and T • n to write that for all x ∈ T • n y∈(T n \T n,l(x) ) c q (k) n (x, y)

z∈T n \T n,l(x)

1 {z∼y} = x 1 ∈Vn • • • x k-1 ∈Vn y∈∂V n z∈∂y 0≤l 1 ≤L . . . 0≤l k-1 ≤L 1≤l =l(x)≤L q n (x, x 1 ) . . . q n (x k-1 , y) k-1 i=1 1 {∀ x i ∈C n,l i \{x} τn(x i )<rn(εn)} 1 {∃ z ∈C n,l τn(z )≥rn(εn)} (3.30)
where the starred sums are defined as

0≤l i ≤L • ≡ L l i =0 •1 {C n,l i ∩x i =∅} and 1≤l =l(x)≤L • ≡ L l=1 •1 {l =l(x)} 1 {C n,l ∩z =∅} . (3.31)
Notice that each of the starred sums over l i has only one term given by the index, l i , of the set C n,l i that contains x i . Similarly, the starred sum over l has at most one term.

Since 1 {∀ z ∈C n, \{x} τn(z )<rn(εn)} 1 {∃ z ∈C n, τn(z )≥rn(εn)} = 0 for all = l(x), the starred sum over l in (3.30) can be restricted to 1 ≤ l = l(x), l = l 1 , . . . , l = l k-1 ≤ L . We may now multiply (3.30) by 1 {x∈T • n } and take the conditional expectation. The variables {τ n (z ), z ∈ C n,l } being independent of the variables {τ n (x ), x ∈ ∪ 0≤ =l≤L C n, }, they can be integrated out first, yielding, for all y ∈ ∂V n z∈∂y 1≤l =l(x),l =l 1 ,...,l =l k-1 ≤L

P ∃ z ∈C n,l τ n (z ) ≥ r n (ε n ) V n (3.32) ≤ n max 1≤l≤L |C n,l |2 -(εn-ρ n )n (3.33) ≤ n 2 2 -(εn-ρ n )n , (3.34)
where we used in (3.33) that the starred sum over l contains at most one term while the sum over z contains at most n terms. Eq. (3.34) then follows from (2.13) and so, is valid on Ω for all large enough n. This bound is uniform in y ∈ ∂V n . Therefore, using (3.34) in (3.30) and re-summing, (3.28) becomes

W n,k (x) ≤ r n n n 2 2 -(εn-ρ n )n E E 1 {x∈T • n } y∈∂V n q (k) n (x, y) V n (3.35) ≤ r n n n 2 2 -(εn-ρ n )n P(x ∈ T • n ) (3.36)
where we used in (3.36) that since Q n is sub-stochastic, y∈∂V n q (k) n (x, y) ≤ 1 for all x. Now, by (2.11) and (2.2), P(x ∈ T

• n ) ≤ P(τ n (x) ≥ r n (ε n )) = 2 -εnn . Thus W n,k (x) ≤ r n n2 -2εnn 2 ρ n n = n c +1 r n 2 -2εnn .
(3.37)

The last equality is (2.3). Using this bound in (3.24) finally yields that on Ω , for all large enough n, 1)) for all but a finite number of indices n. Hence

W n = x∈Vn θn 0 dt ∞ k=1 (ν n t) k k! e -νnt S n,k (x) ≤ θ n n c +1 r n 2 n 2 -2εnn . ( 3
(T • n ) = 2 -nεn (1 + o(
P |T • n | -1 x∈T • n P x H(T n \ T n,l(x) ) ≤ s ≥ ≤ -1 sn c +1 r n π n (T • n )(1 + o(1)). Choosing = n 2 n c +1 r n π n (T • n )
, the claim of the proposition follows from Borel-Cantelli Lemma.

Proof of Proposition 3.9. This is a rerun of the proof of Proposition 3.8.

Rough bounds on local times.

Lemma 3.10. For all 0 ≤ α ≤ 1, all x ∈ V n , and all s > 0,

E x [ x n (s)] α ≥ ( λ -1 n (x)) α Γ(1 + α)[1 -c 1 exp(-c 2 s λ n (x))] + s α exp(-s λ n (x)) (3.
39) where 0 < c 1 , c 2 < ∞ are constants, and if moreover sr n nπ n (x) → 0 as n → ∞,

E x [ x n (s)] α ≤ κ α n + 1 {s>κn} s α (s -κ n )r n nπ n (x)(1 + o(1)).
(3.40)

Proof of Lemma 3.10. The lower bound follows from the trite observation that x n (s) is at least as large as the minimum between the first jump of Y n and s, that is,

x n (s) ≥ λ -1 n (x)e 1 1 s> λ -1 n (x)e 1 + s1 s≤ λ -1 n (x)e 1 , (3.41)
where e 1 is an exponential random variable of mean one. Thus

E x [ x n (s)] α ≥ E x λ -1 n (x)e 1 1 s> λ -1 n (x)e 1 α + s α E x 1 s≤ λ -1 n (x)e 1 α . (3.42)
Eq. (3.39) now readily follows. To get an upper bound write

E x [ x n (s)] α ≤ κ α n if s ≤ κ n . Otherwise write E x [ x n (s)] α ≤ E x κ n +
E πn κ n + s-κn 0 1 {Yn(s)=x} ds α ≤ E πn κ α n 1 {H(x)>s-κn} + s α 1 {H(x)≤s-κn} ≤ κ α n + s α P πn (H(x) ≤ s -κ n ). (3.45)
Eq. (3.40) now follows from (3.9) of Lemma 3.6.

VERIFICATION OF CONDITION (B1)

In this section we prove a strong law of large number for the function ν t n (u, ∞) defined in (1.31). Recall that for r n defined in (1.37), we take η n ≡ (r n ) -1 in (1.34), (1.35), and (1.36). Then by (1.18)-(1.19), (1.22), and (1.34),

ν t n (u, ∞) = k n (t)P πn θn 0 max (c n r n ) -1 , c -1 n τ n (Y n (s)) ds > u (4.1)
where π n is the invariant measure (1.35) of Y n , θ n is the block length of the blocked clock process (1.18), k n (t) = a n t/θ n , and, given 0 < ε < 1, c n and a n are defined in (1.4) and (1.40)-(1.43), respectively. By Theorem 1.3, θ n and a n must obey

n 4 r n (1 + o(1)) ≡ κ n ≤ θ n a n , (4.2)
where the left-most equality is (3.4). Further recall from Section 2 that for ρ n as in (2.3),

ρ n ε n ≡ ε -δ n . (4.3)
(Recall that 0 < x n y n means that x n /y n → 0 as n → ∞.) From now on we take δ n such that 2 nδn = (n 2 θ n ) α(ε) , i.e.

δ n ≡ 1 nβ 2ε log 2 log n 2 θ n . (4.4)
Thus, given 0 < ε < 1 and β > 0, all sequences except θ n are determined.

Proposition 4.1. Given 0 < ε < 1 and β > 0 let the sequences c n and a n be defined as in (1.4) and (1.40)-(1.43), respectively, and let θ n be such that

(r n ) 4 θ 1-α(ε) n , (4.5) n -1 log θ n 1. (4.6)
Then, for all 0 < ε < 1 and β > 0, P-almost surely,

lim n→∞ ν t n (u, ∞) = tu α(ε) , ∀ t > 0, u > 0. (4.7)
Remark. Eq. (4.6) implies that δ n 1 and that θ n c n for all ε > 0. In view of (1.38), (3.3), (4.4) and (3.4), (4.6) also implies that

c 0 n c 1 κc 2 n κ c 3 n (r n ) c 4 θ c 5 n 2 εn and c 0 n c 1 κc 2 n κ c 3 n (r n ) c 4 θ c 5 n 2 εnn (4.8)
for all ε > 0 and any choice of constants 0 ≤ c i < ∞.

Remark. In order to guarantee strict equivalence of the definitions (1.42) and (2.9) of the set T n when δ n is given by (4.4), we should replace the term c n (nθ n ) -1 in (1.42) by

c n exp -log(n 2 θ n ) 1 + (1 + o(1))(2nββ c (ε)) -1 log(n 2 θ n ) (4.9)
(use Lemma 2.2). We didn't state this precise formula to keep the presentation simple.

The rest of the section is organized as follows. In Section 4.1 we show that ν t n (u, ∞) can be reduced to the quantity ν •,t n (u, ∞) defined in (4.32). In Section 4.2 we prove upper et lower bounds on a sequence, b • n , defined as b n with T • n substituted for T n , and show that b n and b • n behave in the same way to leading order. In Section 4.3 we show that ν •,t n (u, ∞) concentrates around its mean value when choosing a n = 2 εn /b • n . The proof of Proposition 4.1 is finally completed in Section 4.2.

Preparations.

To begin with, we bring the function ν t n (u, ∞) given in (4.1) into a form amenable to treatment. Let T n be as in (2.9). For all 0 < ε < 1 and δ n as in (4.4),

0 ≤ θn 0 max (c n r n ) -1 , c -1 n τ n (Y n (s)) 1 {Yn(s) / ∈Tn} ds ≤θ n r n (ε n ) r n (ε) ≤ n -2 (4.10)
as follows from (2.20). Hence visits of Y n outside the set T n only yield a negligible contribution to the event in (4.1), implying that

νt n (u, ∞) ≤ ν t n (u, ∞) ≤ νt n u -n -2 , ∞ (4.11)
where

νt n (u, ∞) ≡ k n (t)P πn θn 0 c -1 n τ n (Y n (s))1 {Yn(s)∈Tn} ds > u . (4.12)
Our next step consists in reducing visits to T n in νt n (u, ∞) to visits to the subset T • n defined in (2.11). Set

νt n (u, ∞) ≡ k n (t)P πn θn 0 c -1 n τ n (Y n (s))1 {Yn(s)∈T • n } ds > u . (4.13)
Lemma 4.2. Assume that (4.6) holds. Then on Ω , for all but a finite number of indices n,

|ν t n (u, ∞) -νt n (u, ∞)| ≤ 2k n (t)θ n r n n 5 2 -2nεn (1 + o(1)). (4.14)
Proof of Lemma 4.2. Decomposing the event appearing in the probability in (4.12) according to whether (4.14) follows from (3.10) of Corollary 3.7 applied with t n = θ n , which is licit by virtue of (4.6) (see also (4.8)).

{H(T n \ T • n ) ≤ θ n } or {H(T n \ T • n ) > θ n },
We next decompose (4.13) according to the hitting time, H(T • n ), and hitting place,

Y n (H(T • n )), of the set T • n . The density of the joint distribution of H(T • n ) and Y n (H(T • n )) is a |T • n |- dimensional vector, (h n,x ) x∈T •
n , whose components are, for each x ∈ T • n , the joint density that T • n is reached at time v, and that arrival to that set occurs in state x,

P πn (H(T • n ) ≤ s, Y n (H(T • n )) = x) = s 0 h n,x (v)dv. (4.15)
For this vector of densities we have

x∈T • n ∞ 0 h n,x (v)dv = 1, (4.16)
and, denoting by h n,T

νt n (u, ∞) = k n (t) x∈T • n θn 0 h n,x (v)P x θn-v 0 c -1 n τ n (Y n (s))1 {Yn(s)∈T • n } ds > u dv. (4.18) Denote by Q u,v
n (x) the probability appearing in (4.18). Notice that Y n starts in x ∈ T • n and further decompose this probability according to whether {H(T

• n \ x) ≤ θ n -v} or {H(T • n \ x) > θ n -v}, that is, write Q u,v n (x) ≡ Q u,v n (x) + Q u,v n (x), Q u,v n (x) = P x θn-v 0 c -1 n τ n (Y n (s))1 {Yn(s)∈T • n } ds > u, H(T • n \ x) ≤ θ n -v , (4.19) Q u,v n (x) = P x θn-v 0 c -1 n τ n (Y n (s))1 {Yn(s)∈T • n } ds > u, H(T • n \ x) > θ n -v , (4.20)
and split (4.18) accordingly. Clearly, for all v > 0

Q u,v n (x) ≤ P x (H(T • n \ x) ≤ θ n ) . (4.21)
This and the bound θn 0 h n,x (v)dv ≤ P πn (H(x) ≤ θ n ) (that follows from (4.15)), yield

k n (t) x∈I • n θn 0 h n,x (v) Q u,v n (x)dv (4.22) ≤ k n (t) x∈T • n P πn (H(T • n ) ≤ θ n , Y n (H(T • n )) = x)P x (H(T • n \ x) ≤ θ n ) (4.23) ≤ ν t n (4.24)
where

ν t n ≡ k n (t) x∈T • n P πn (H(x) ≤ θ n )P x (H(T • n \ x) ≤ θ n ) . (4.25)
Lemma 4.3. Assume that (4.6) holds. Then on Ω , for all but a finite number of indices n, 1)), wich decays to zero as n diverges by (4.6) (see also (4.8)). We may thus use (3.9) of Lemma 3.6 to bound the term P πn (H(x) ≤ θ n ) in (4.25), and by this and (3.2) we get that on Ω , for all large enough n,

ν t n ≤ k n (t)n c +4 (θ n π n (T • n )r n ) 2 (1 + o(1)). ( 4
(T • n )r n = n 1+2α(ε) r n θ 1+α(ε) n 2 -nε (1 + o(
ν t n ≤ k n (t)θ n nπ n (T • n )r n (1 + o(1))|T • n | -1 x∈T • n P x (H(T • n \ x) ≤ θ n ) . (4.27)
The lemma now follows from Proposition 3.8.

Consider now the contribution to (4.18) coming from (4.20). By definition,

Q u,v n (x) = P x c -1 n τ n (x) x n (θ n -v) > u, H(T • n \ x) > θ n -v . (4.28) Thus ν t n (u, ∞) (4.29) ≡ k n (t) x∈T • n θn 0 h n,x (v) Q u,v n (x)dv (4.30) = k n (t) x∈T • n θn 0 h n,x (v)P x c -1 n τ n (x) x n (θ n -v) > u, H(T • n \ x) > θ n -v dv.(4.31) Setting ν •,t n (u, ∞) ≡ k n (t) x∈T • n θn 0 h n,x (v)P x c -1 n τ n (x) x n (θ n -v) > u dv, (4.32) we have ν •,t n (u, ∞) -w t n (u, ∞) ≤ ν t n (u, ∞) ≤ ν •,t n (u, ∞) (4.33)
where

w t n (u, ∞) ≡ k n (t) x∈T • n θn 0 h n,x (v)P x c -1 n τ n (x) x n (θ n -v) > u, H(T • n \ x) ≤ θ n -v dv ≤ k n (t) x∈T • n θn 0 h n,x (v)P x (H(T • n \ x) ≤ θ n -v) dv ≤ ν t n . (4.34)
Inserting our bounds in (4.18), we finally get that for all u > 0

ν •,t n (u, ∞) -νt n (u, ∞) ≤ ν t n . (4.35)
Our aim now is to prove almost sure convergence of ν •,t n (u, ∞). To do so we will need certain properties a sequence, b • n , associated to the sequence b n , that we now define. 4.2. Properties of the sequences b n and b

• n . For F β,ε,n (x) as in (1.41) define b • n ≡ (θ n π n (T • n )) -1 x∈T • n θn 0 h n,x (v)E x [F β,ε,n ( x n (θ n -v))]dv. ( 4
Set I n (a, b) = (θ n π n (T • n )) -1 x∈T • n J x n (a, b), J x n (a, b) = b a h n,x (v)E x [F β,ε,n ( x n (θ n -v))]dv, (4.37)
and given

0 < ζ n < θ n split b • n into b • n = I n (0, κ n ) + I n (κ n , θ n -ζ n ) + I n (θ n -ζ n , θ n ). Lemma 4.4.
Assume that (4.5) and (4.6) hold. Let ζ n > 0 be a sequence satisfying

n -1 | log ζ n | 1, and κn (r n ) 1+αn(ε)+o(1) ζ αn(ε)+o(1) n ↓ 0 as n ↑ ∞. (4.38)
Then, on Ω 1 ∩ Ω • ∩ Ω , for all but a finite number of indices n,

I n (0, κ n ) I n (κ n , θ n -ζ n ) ≤ θ -1 n κn κ 1+αn(ε) n (nr n ) 1+αn(ε)+o(1) , (4.39)
0 ≤ (b n -b • n )/b • n ≤ n(r n ) 1+αn(ε)+o(1) κ 1+αn(ε) n 2 -nεn , (4.40
) and the right-hand sides of (4.39) and (4.40) decay to zero as n diverges. Furthermore

κ -1 n (r n ) -{αn(ε)+o(1)} ≤ b • n ≤ (1 + o(1))nr n κ αn(ε) n . (4.41)
Proof of Lemma 4.4. We first prove a lower bound on I n (κ n , θ n -ζ n ). For this write

J x n (κ n , θ n -ζ n ) ≥ J x n,1 ≡ θn-ζn κn h n,x (v)E x [F β,ε,n ( x n (θ n -v))1 {ζn< x n (θn-v)≤θn} ]dv. Since F β,ε,n (x) = (1 + o(1))x αn(ε)+o(1) for all ζ n < x ≤ θ n , J x n,1 ≥ (1 + o(1)) θn-ζn κn h n,x (v)E x [ x n (θ n -v)] αn(ε)+o(1) (1 -1 { x n (θn-v)<ζn} )dv ≡ J x n,3 -J x n,4 (4.42)
where we used the left-most inequality in (4.72) to relax the constraint x n (θ n -v) ≤ θ n . Let us bound J x n,3 for x ∈ I n . Note that by (2.12) and (2.6)

(r n ) -1 ≤ λ n (x) ≤ r n , ∀x ∈ I n . (4.43)
Thus, setting ζ n ≡ nr n , it follows from (3.39) of Lemma 3.10 that for all x ∈ I n ,

J x n,3 ≥ c 3 ( λ -1 n (x)) αn(ε)+o(1) θn-ζ n κn h n,x (v)dv (4.44)
for some numerical constant 0 < c 3 < ∞. Summing over x, wet get

x∈T • n J x n,3 ≥ x∈I n J x n,3 ≥ c 3 (r n ) -{αn(ε)+o(1)} x∈I n θn-ζ n κn h n,x (v)dv (4.45)
where the last sum in the right-hand side of (4.45) is equal to

P πn (κ n < H(I n ) < θ n -ζ n , H(I n ) < H(T • n \ I n )). (4.46)
Decomposing this probability into

p 1 -p 2 ≡ P πn (κ n < H(I n ) < θ n -ζ n )-P πn (κ n < H(I n ) < θ n -ζ n , H(I n ) > H(T • n \I n ))
we have, by Lemma 3.5 and (3.7), whenever θ n r n nπ n (I n ) → 0,

p 1 ≥ κ-1 n θ n π n (I n )(1 -θ -1 n ζ n )(1 + o(1)) = κ-1 n θ n π n (I n )(1 + o(1)) (4.47)
where the last equality follows from (4.5). To get an upper bound on p 2 , write

p 2 ≤P πn (H(T • n \ I n ) < κ n ) + P πn (H(T • n \ I n ) < H(I n ) < θ n ) ≡ p 3 + p 4 . (4.48)
By (3.9),

p 3 ≤ κ n r n nπ n (T • n \ I n )(1 + o(1)
), whereas proceeding as in (4.22)-(4.25),

p 4 ≤ x∈T • n \I n P πn (H(x) ≤ θ n )P x (H(I n ) ≤ θ n ) (4.49) = n 3 (θ n r n ) 2 π n (T • n \ I n )π n (I n )(1 + o(1)) (4.50)
where the last equality follows from (3.9) and (3.19). By (2.18), (2.19), and (3.2), on Ω and for large enough n, π n (

I n ) = 2 -nεn (1 -n -c (1 + o(1))) and π n (T • n \ I n ) = n -c +1 2 -nεn (1 + o(1)) (thus in particular, π n (I n )/π n (T • n ) = 1 + o(1)
). In view of this, (4.5), and (4.6), one checks that θ n r n nπ n (I n ) → 0 (as requested above (4.47)) and that 1)) and by this, (4.47), and (4.45),

p 2 = o(p 1 ). Thus p 1 -p 2 = p 1 (1 + o(
(θ n π n (T • n )) -1 x∈T • n J x n,3 ≥ κ-1 n (r n ) -{αn(ε)+o(1)} (1 + o(1)). (4.51)
Turning to J x n,4 we have

x∈T • n J x n,4 ≤ (1 + o(1))ζ αn(ε)+o(1) n x∈T • n θn-ζn κn h n,x (v)dv, (4.52)
where the last sum is equal to

P πn (κ n < H(T • n ) < θ n -ζ n ).
Since by Lemma 3.5 and (3.7), P πn (κ n < H(T

• n ) < θ n -ζ n ) ≤ (1 + o(1))r n nθ n π n (T • n ), we get (θ n π n (T • n )) -1 x∈T • n J x n,4 ≤ (1 + o(1))nr n ζ αn(ε)+o (1) n . (4.53)
At this point we may observe that the right-most condition in (4.38) is tailored to guarantee

that x∈T • n J x n,3 x∈T • n J x n,4
. Hence, collecting our bounds,

I n (κ n , θ n -ζ n ) = 1 + o(1) θ n π n (T • n ) x∈T • n θn-ζn κn h n,x (v)E x [ x n (θ n -v)] αn(ε)+o(1) (4.54) ≥ κ-1 n (r n ) -{αn(ε)+o(1)} . (4.55)
We now prove an upper bound on I n (0, κ n ). Using that F β,ε,n (x) ≤ (1 + o(1))x αn(ε) for all 0 < x ≤ θ n , (3.40) of Lemma 3.10 (which by (4.6) and Lemma 3.1 is licit) gives

J x n (0, κ n ) ≤ (1 + o(1))κ αn(ε) n κn 0 h n,x (v)dv. (4.56)
Summing over x ∈ T • n and using (3.11) and (4.6) to bound the resulting probability,

I n (0, κ n ) ≤ (1 + o(1))nr n θ -1 n κ 1+αn(ε) n . (4.57) variables τ n (x) in (T • n ) c
). The proof now hinges on a simple mean and variance argument. We deal with the variance first. By (4.32) and (4.64),

E • ν •,t n (u n , ∞) = k n (t) x∈T • n E • X n (x), (4.65)
and by independence

E • (ν •,t n (u n , ∞) -E • ν •,t n (u n , ∞)) 2 ≤ k 2 n (t) x∈T • n E • (X n (x)) 2 . (4.66)
Note that since

X n (x) ≤ θn 0 h n,x (v)dv ≤ P πn (H(x) ≤ θ n ) ≤ θ n r n n2 -n (1 + o(1)), (4.67)
(the last inequality is (3.9) combined with (3.2)) then

k 2 n (t) x∈T • n E • (X n (x)) 2 ≤ t(2 εn /b • n )r n n2 -n (1 + o(1))E • ν •,t n (u n , ∞), (4.68)
where we used that for

a n = 2 εn /b • n , θ n k n (t) = θ n t(2 εn /b • n )/θ n = t(2 εn /b • n )(1 + o(1)
). Inserting (4.68) in (4.66), a second order Tchebychev inequality then yields (4.61).

To estimate 4.65) we first use Fubini to write,

E • ν •,t n (u n , ∞) in (
E • X n (x) = θn 0 h n,x (v)E x P • c -1 n τ n (x) x n (θ n -v) > u n dv. (4.69)
Denoting by P x the law of the single variable τ n (x),

P • c -1 n τ n (x) x n (θ n -v) > u n = P x (c -1 n τ n (x) x n (θ n -v) > u n , τ n (x) ≥ r n (ε n )) P x (τ n (x) ≥ r n (ε n )) (4.70) = P x (c -1 n τ n (x) x n (θ n -v) > u n ) P x (τ n (x) ≥ r n (ε n )) (4.71)
where (4.71) follows from the definition of c n (see (1.4)), the a priory bound

x n (θ n -v) ≤ θ n -v c n , 0 ≤ v ≤ θ n , (4.72)
and the fact that δ n in (4.4) in chosen in such a way that θ n r n (ε n )r -1 n (ε) ≤ n -2 ↓ 0 as n ↑ ∞ (see the last inequality in (4.10)). Using classical estimates on the asymptotics of gaussian integrals, Lemma 2.2, and again the definition of c n , simple calculations yield that for all 0 < u < ∞ and 0 ≤ v < θ n , (4.71) is equal to

(1 + o(1))F β,ε,n x n (θn-v) un P (τ n (x) > c n ) P(τ n (x) ≥ r n (ε n )) (4.73)
where F β,ε,n (x) is defined in (1.41). Furthermore, by (1.4), 2 εn P(τ n (x) ≥ c n ) = 1 whereas by (2.2), (2.16), and (3.2), P(

τ n (x) ≥ r n (ε n )) = π n (T • n )(1 + o(1)
). In view of this and (4.36) we get, combining (4.73), (4.69), (4.65), and using the a priori bound (4.72) that

E • ν •,t n (u n , ∞) = (1 + o(1))k n (t)θ n (b • n /2 εn ) I (0,θn) (u n ) I (0,θn) (1) (4.74)
where for w > 0

I (a,b) (w) = x∈T • n θn 0 h n,x (v)E x F β,ε,n x n (θn-v) w 1 {a≤ x n (θn-v)<b} dv. (4.75)
To evaluate the ratio in (4.74) set 0 < ζ n ≡ e -n 9/10 ↓ 0 and split the integral in

I (0,θn) (u n ) into I (0,θn) (u n ) ≡ I (0,ζn) (u n )+I (ζn,θn) (u n ). Note that n -1 | log ζ n | = n -1/10 , n -1 (log ζ n ) 2 = n 4/5
, while for all u > 0, n -1 log u n ↓ 0, n -1 (log u n ) 2 ↓ 0 as n ↑ ∞. Using that F β,ε,n (x) is increasing on the domain (0, ζ n /u n )

I (0,ζn) (u n ) ≤ F β,ε,n ζn un P πn (H(T • n ) < θ n ) (4.76)
where

F β,ε,n ζn un = e o(1) log un F β,ε,n (ζ n )F β,ε,n (u -1 n ) and F β,ε,n (ζ n
) ≤ e -αn(ε)n 9/10 -n 4/5 /2β 2 . By this, (3.9), the lower bound (4.41) on b • n , and our assumptions on u n ,

I (0,ζn) (u n ) I (0,θn) (1) = e o(1) log un F β,ε,n (u -1 n )F β,ε,n (ζ n )nκ n (r n ) 1+αn(ε)+o(1) → 0 (4.77)
as n → ∞. Next, since n -1 log l ↓ 0 as n ↑ ∞ for all ζ n ≤ l ≤ θ n we have, using (4.72),

I (ζn,θn) (u n ) I (0,θn) (1) = e o(1) log un F β,ε,n (u -1 n ) 1 - I (0,ζn) (un) I (0,θn) (1)
→ u -α(ε) (4.78)

as n → ∞ for all u > 0. Inserting (4.77) and (4.78) in (4.74), choosing a n = 2 εn /b • n , and passing to the limit n → ∞ finally gives (4.62). The proof of the lemma is done.

< ε < 1, (2 εn /b • n )r n n 3 2 -n ≤ κ n (r n ) 1+αn(ε)+o(1) n 3 2 nε 2 -n → 0 (4.79)
as n → ∞. Thus, by Proposition 4.5 and Borel-Cantelli Lemma we get that for all u > 0 and all t > 0, lim n→∞ ν •,t n (u, ∞) = tu α(ε) Palmost surely. (4.80)

In the same way we get that for all u > 0 and all t > 0, lim n→∞ ν

νt n (u, ∞) -ν •,t n (u, ∞) (4.82) ≤ t(b • n ) -1 [2r n n 5 θ n 2 -nε+2δnn + n c +4 2 nε (θ n π n (T • n )r n ) 2 ](1 + o(1)) (4.83) ≤ 2tn c +4(1+αn(ε)) (r n ) αn(ε)+2+o(1) κ n θ 2+2α(ε) n 2 -nε (1 + o(1)) (4.84)
where the last inequality follows from (4.41), (2.16), (4.3), and (4.4). Since κ n ≤ θ n , (4.6) (see also (4.8)) implies that (4.84) decays to zero as n → ∞. From this and (4.80) we get that for all u > 0 and all t > 0, lim n→∞ νt n (u, ∞) = tu α(ε) P-almost surely. One proves in the same way that for all u > 0 and all t > 0, lim n→∞ νt n (u -n -2 , ∞) = tu α(ε) P-almost surely. Therefore, by (4.11), for all u > 0 and all t > 0, lim n→∞ ν t n (u, ∞) = tu α(ε) Palmost surely. (4.85) Since ν t n is increasing both in t and u and since its limit continuous in those two variables, (4.85) implies that P-almost surely,

lim n→∞ ν t n (u, ∞) = tu α(ε) , ∀ u > 0, t > 0. (4.86)
The proof of Proposition 4.1 is done.

We continue our parallel with the proof of Proposition 4.1 and decompose (5.4) according to the hitting time and hitting place of the set T • n . We slightly abuse the notation of Section 3 (see the paragraph below (3.22)) and denote by h y n,x (instead of h y n,x,T

≡ x∈T • n θn 0 h y n,x (v) Q u,v n (x) + Q u,v n (x) dv ≡ R u n (y) + R u n (y).
(5.12)

By analogy with (4.30) we also set

σ t n (u, ∞) ≡ k n (t) y∈Vn π n (y) R u n (y) 2 .
(5.13)

The next lemma plays the role of Lemma 4.3.

Lemma 5.3. Assume that (4.6) holds. Then Ω , for all but a finite number of indices n,

0 ≤ σt n (u, ∞) -σ t n (u, ∞) ≤ 3k n (t)n c +4 (θ n π n (T • n )r n ) 2 (1 + o(1)). (5.14)
Proof of Lemma 5.3. As in the proof of Lemma 5.2, the proof of Lemma 5.3 relies on the observation that since 0 ≤ R u n (y), R u n (y) ≤ 1 in (5.12) for all y ∈ V n , then by (5.8),

0 < σt n (u, ∞) -σ t n (u, ∞) ≤ 3k n (t) y∈Vn π n (y) R u n (y) (5.15) = 3k n (t) x∈T • n θn 0 h n,x (v) Q u,v n (x)dv ≤ 3 ν t n .
(5.16)

The equality in (5.16) follows from the identity h n,x (v) = y∈Vn π n (y)h y n,x (v), and the final inequality is (4.24). The claim of the lemma now follows from Lemma 4.3.

We now need an upper bound on σ t n (u, ∞). For this we proceed as in (4.31)-(4.33) and write that 0 ≤ σ t n (u, ∞) ≤ σ •,t n (u, ∞) where, by analogy with (4.33),

σ •,t n (u, ∞) = k n (t) y∈Vn π n (y) x∈T • n θn 0 h y n,x (v)P x c -1 n τ n (x) x n (θ n -v) > u dv 2
Again, the quantity in between the square brackets is in [0, 1]. Thus, splitting the integral into the sum of the integrals over [0, κ n ] and [κ n , θ n ], we get, using (5.8) and reasoning as in (5.15)-(5.16),

σ •,t n (u, ∞) ≤ 3η •,t n (u, ∞) + η •,t n (u, ∞) (5.17) where η•,t n (u, ∞) ≡ k n (t) x∈T • n κn 0 h n,x (v)P x c -1 n τ n (x) x n (θ n -v) > u dv, (5.18)
η •,t n (u, ∞) ≡ k n (t) y∈Vn π n (y) x∈T • n θn κn h y n,x (v)P x c -1 n τ n (x) x n (θ n -v) > u dv 2 .
(5.19)

The next two propositions bound (5.18) and (5.19) in terms of the quantities ν •,t n (u n , ∞) and E • ν •,t n (u n , ∞) defined in (4.32) and (4.65), respectively. Proposition 5.4. Choose a n = 2 εn /b • n in (1.17). Then, for any sequence u n > 0 such that 0 < u -u n < n -1 and all u > 0,

P η•,t n (u n , ∞) ≥ tE • ν •,t n (u n , ∞)n 2 θ -1 n κn κ 1+αn(ε) n (nr n ) 1+αn(ε)+o(1) ≤ n -2 .
(5.20)

Proposition 5.5. On Ω ∩ Ω 1 , for all but a finite number of indices n and all u > 0,

η •,t n (u, ∞) ≤ ν •,t n (u, ∞)θ n r n n2 -nεn (1 + o(1)). ( 5
E • η•,t n (u n , ∞) = t(1 + o(1))E • ν •,t n (u n , ∞) I n (0, κ n ) I n (0, θ n ) (5.23)
where I n (a, b) is defined above (4.37). Eq. (4.39) of Lemma 4.4 was designed precisely to control the ratio in (5.23). Namely, on Ω • ∩ Ω , for all but a finite number of indices n,

I n (0, κ n ) I n (0, θ n ) ≤ I n (0, κ n ) I n (κ n , θ n -ζ n ) ≤ θ -1 n κn κ 1+αn(ε) n (nr n ) 1+αn(ε)+o(1) . (5.24)
The combination of (5.22), (5.23), and (5.24) gives (5.20). The proof is complete.

Proof of Proposition 5.5. To prove (5.21) first observe that

x∈T • n θn κn h y n,x (v)P x c -1 n τ n (x) x n (θ n -v) > u dv ≤ P y (κ n < H(T • n ) ≤ θ n ) ≤ (1 + o(1))P πn (H(T • n ) ≤ θ n )
where the last line follows from Proposition 3.3 and the Markov property, and is valid on Ω 1 , for all but a finite number of indices n. Applying this bound to one of the two square brackets in (5.19) and using (4.32) to bound the remaining term, we get, under the same assumptions as above, that (nr n ) 1+αn(ε)+o(1) ↓ 0 as n ↑ ∞ and by (4.62), for all u > 0 and t > 0 ε) . Thus, by Proposition 5.4 and Borel-Cantelli Lemma we get that for all u > 0 and t > 0, lim n→∞ η•,t n (u, ∞) = 0 Palmost surely.

η •,t n (u, ∞) ≤ (1 + o(1))ν •,t n (u, ∞)P πn (H(T • n ) ≤ θ n ). ( 5
lim n→∞ E • ν •,t n (u n , ∞) = tu α(
(5.26)

Turning to (5.21) and invoking (4.6) (see also (4.8)), it follows from Proposition 5.4 that for all 0 < ε < 1 and for all u > 0 and t > 0, lim n→∞ η

M n (Y n (s))1 { θn 0 Mn(Yn(s))ds≤ } = 0, ∀t > 0. (6.1)
where

M n (Y n (s)) = max ((c n r n ) -1 , c -1 n τ n (Y n (s))). The Lemma below is central to the proof. Recall that α n (ε) ≡ (nβ 2 ) -1 log c n , that is, in view of (1.5), α n (ε) = α(ε) -(2nββ c (ε)) -1 [log(β 2 c (ε)n/2) + log 4π + o(1)]. (6.2)
Lemma 6.2. There are constants K, K < ∞ such that for α n (ε) as in (6.2) and any sequence n > 0 such that iα -1 c (ε) -1 -log n nββc(ε) > 0 where i = 1 in (6.3) and i = 2 in (6.4), we have, for all large enough n,

E2 εn c -1 n τ n (x)1 {c -1 n τn(x)≤ n} ≤ K 1-αn(ε)-log n 2nβ 2 n α -1 c (ε) -1 -log n nββc(ε) , (6.3)
E 2 εn c -1 n τ n (x)1 {c -1 n τn(x)≤ n} 2 ≤ K 2-αn(ε)-log n 2nβ 2 n 2α -1 c (ε) -1 -log n nββc(ε) . (6.4)
Proof of Lemma 6.2. Using standard estimates on the asymptotics of Gaussian integrals the claimed result follows from straightforward computations.

Proof of Proposition 6.1. We assume throughout that ω ∈ Ω 1 ∩ Ω • ∩ Ω and that n is as large as desired. Note that

M n (Y n (s)) ≤ (c n r n ) -1 + c -1 n τ n (Y n (s)
) and that the contribution to (6.1) coming from the term (c n r n ) -1 if or order o(1). Indeed by (1.17), (1.40), the lower bound on b n obtained by combining (4.41) and (4.40), the expression (1.5) of c n , the expression (3.4) of κ n , and the fact, that follows from (1.6), that

2 n = e nβ 2 c (ε)/2 , k n (t)θ n (c n r n ) -1 ≤ 2tn 4 (r n ) αn(ε)+o(1) e nβ 2 c (ε)/2 e -nββc(ε)(1+o(1)) (6.5)
and so, for all 0 < ε < 1 and β > β c (ε), by virtue of (4.6) (see also (4.8))

k n (t)θ n (c n r n ) -1 ≤ 2tn 4 (r n ) αn(ε)+o(1) e -nβ 2 c (ε)(1+o(1))/2 → 0
as n → ∞. To prove Proposition 6.1 it thus suffices to establish that P-almost surely,

lim ↓0 lim sup n↑∞ k n (t)E πn θn 0 c -1 n τ n (Y n (s))1 { θn 0 c -1
n τn(Yn(s))ds≤ } = 0, ∀t > 0. (6.6)

For T n as in (2.9) with δ n given by (4.4), set

S (1) n, (t) ≡ k n (t)E πn θn 0 c -1 n τ n (Y n (s))1 {Yn(s)∈Tn} 1 { θn 0 c -1 n τn(Yn(s))ds≤ } ds, S (2) n, (t) ≡ k n (t)E πn θn 0 c -1 n τ n (Y n (s))1 {Yn(s) / ∈Tn} 1 { θn 0 c -1
n τn(Yn(s))ds≤ } ds.

To bound S

n, (t) simply note that, using Lemma 3.1,

S (2) n, (t) ≤ k n (t)E πn θn 0 c -1 n τ n (Y n (s))1 {τn(Yn(s))≤rn(εn)} ds ≤ k n (t)θ n 2 -n (1 + o(1)) x∈Vn c -1 n τ n (x)1 {τn(x)≤rn(εn)} .
Take n = c -1 n r n (ε n ) and note that by (2.20), the definition of c n , and (4.6),

-(nββ c (ε)) -1 log n = o(1) and n 2(1+c 6 2 /α(ε)) θ n -1 ≤ n ≤ (n 2 θ n ) -1 . (6.7)
Thus, by Lemma 6.2 and a first order Tchebychev inequality, for all large enough n,

P S (2) n, (t) ≥ n 2 tb -1 n (c -1 n r n (ε n )) 1-α(ε)+o(1) ≤ n -2 K (6.8)
for some constant K > 0. Using the upper bound on n of (6.7) and the lower bound on b n of Lemma 4.4 obtained by combining (4.41) and (4.40),

n 2 b -1 n (c -1 n r n (ε n )) 1-α(ε)+o(1) ≤ n 2 κ n (r n ) αn(ε)+o(1) n 2 θ n -1+α(ε)+o(1) → 0 (6.9)
as n → ∞ by (4.5). Hence by (6.8), (6.9), and Borel-Cantelli Lemma, for all > 0, lim n→∞ S (2) n, (t) = 0, Palmost surely. (6.10)

To deal with S

n, (t) we further decompose it into S

n, (t) = S

n, (t) + S

n, (t), where

S (3) n, (t) ≡ k n (t)E πn θn 0 c -1 n τ n (Y n (s))1 {Yn(s)∈T • n } 1 { θn 0 c -1 n τn(Yn(s))ds≤ } ds, S (4) n, (t) ≡ k n (t)E πn θn 0 c -1 n τ n (Y n (s))1 {Yn(s)∈Tn\T • n } 1 { θn 0 c -1 n τn(Yn(s))ds≤ } ds. Since S (4) n, (t) is non zero only if the event {H(T n \ T • n ) ≤ θ n } occurs, S (4) n, (t) ≤ k n (t)E πn 1 {H(Tn\T • n )≤θn} . ( 6
n, (t) ≤ k n (t)θ n r n n2 -2nεn (1 + o(1)), Proceeding as in (6.9) to bound b n , (4.6) (see also (4.8)) guarantees that for all > 0 lim n→∞ S (4) n, (t) = 0, Palmost surely. (6.12)

Using next that

θn 0 c -1 n τ n (Y n (s))1 {Yn(s)∈A} = x∈A c -1 n τ n (x) x n (θ n ) for any A ⊆ V n , S (3)
n, (t) ≤ S (5) n, (t

) ≡ k n (t)E πn x∈T • n c -1 n τ n (x) x n (θ n )1 { x∈T • n c -1 n τn(x) x n (θn)≤ } .
With the notation of (4.15)-(4.17),

S (5) n, (t) = k n (t) y∈T • n θn 0 dvh n,y (v)E y x∈T • n c -1 n τ n (x) x n (θ n -v)1 { x∈T • n c -1 n τn(x) x n (θn-v)≤ } .
We further split the sum over x above into x = y and x = y. The latter contribution is

S (6) n, (t) ≡ k n (t) y∈T • n θn 0 dvh n,y (v)E y x∈T • n \y c -1 n τ n (x) x n (θ n -v)1 { x∈T • n c -1 n τn(x) x n (θn-v)≤ } .
Observing that

E y x∈T • n \y c -1 n τ n (x) x n (θ n -v)1 { x∈T • n c -1 n τn(x) x n (θn-v)≤ } ≤ P y (H(T • n \ y) ≤ θ n )
yields the bound S

n, (t) ≤ ν t n , where ν t n is defined in (4.25). Thus by Lemma 4.3, reasoning as in the paragraph below (4.84), we get that for all > 0 lim n→∞ S (6) n, (t) = 0, Palmost surely. (

It remains to bound S

n, (t) -S

n, (t). For this we write S

n, (t) -S

n, (t) ≤ S

n, (t) where S (7) n, (t) ≡ k n (t) Let us now establish that for b • n as in (4.36), S

n, (t) obeys the following Lemma 6.3. Let the sequences a n , c n , θ n be as in Proposition 6.1. Then, under the assumptions and with the notation of Proposition 4.5, P • S (7) n, (t) -E • S (7) n, (t) > t 1/2 n2 -n(1-ε)/2 ≤ n -2 (1 + o(1)) (6.14) for all > 0, and lim →0 lim n→∞ E • S (7) n, (t) = 0. (6.15)

Proof of lemma 6. where P y denotes the law of τ n (y) and where n ≡ n (y) = / y n (θ n -v). Using (6.3) if y n (θ n -v) > e -nββc(ε)(α -1 c (ε)-1) and using that if y n (θ n -v) ≤ e -nββc(ε)(α -1 c (ε)-1) then E y y n (θ n -v)E y c -1 n τ n (y)1 {c -1 n τn(y)≤ n} ≤ e -nββc(ε)(α -1 c (ε)-1) c -1 n e nβ 2 /2 , we readily see that E • S (7) n, (t) ≤ C 1 t

1-αn(ε)-log 1 z> e -nββc(ε)(α -1 c (ε)-1) . (6.18)

By the leftmost inequality of (4.72) and (4.6), F β,ε, ,n (z) ≤ C 3 F β,ε,n (z). Thus, by (4.36), the first summand in (6.17 Using (3.11) and proceeding as in (6.5) to bound k n (t), the second summand is bounded above by C 5 te -n(β 2 -β 2 c (ε))/2 κ n n αn(ε)/2+1 (r n ) 1+αn(ε)+o(1) → 0 (6.20) as n → ∞ by virtue of (3.4), (1.38), and the assumption that β > β c (ε) where 0 < ε < 1. Note in particular that lim n→∞ α n (ε) = α(ε) < 1. Hence, inserting (6. [START_REF] Durrett | Functional limit theorems for dependent variables[END_REF]) and (6.20) in (6.17) and passing to the limit lim →0 lim sup n→∞ E • S (7) n, (t) = 0, ∀t > 0.

This proves (6.15). Turning to the variance we have, as in (4.66), by independence, that V • (S (7) n, (t)) ≡ E • (S (7) n, (t) -E • S (7) n, (t)) 2 ≤ k 2 n (t)

y∈T • n E • (X n (y)) 2 .
Proceeding as in the proof of (6.17) but using (6.4) and the line below (6.18), we get that V • (S (7) n, (t)) ≤ C 6 t 2 ≤ sup y∈T • n P πn (H(y) ≤ θ n )P πn (H(T • n ) ≤ θ n ), and proceeding as in (6.20), the second summand is bounded above by C 9 t 2 n αn(ε)/2 n 2 κ n (r n ) 1+αn(ε)+o(1) 2 θ n e -nβc(ε)(β-βc(ε)) 2 -n .

Since by assumption β > β c (ε) and 0 < ε < 1, (4.6) (see also (4.8)) enables us to conclude that on Ω , for all large enough n, V • (S (7) n, (t)) ≤ C 10 t 2 2 -n(1-ε) . This yields (6.14) and concludes the proof of the Lemma.

Arguing as in the proof of Proposition 4.1 that b n = b • n (1 + o(1)) on Ω 1 ∩ Ω • ∩ Ω for all large enough n, it follows from Lemma 6.3 and Borel-Cantelli Lemma that lim →0 lim n→∞ S (5) n, (t) -S (6) n, (t) = 0, Palmost surely. (6.21)

Collecting (6.10), (6.12), (6.13) and (6.21) yields (6.6). The proof of Proposition 6.1 is complete. where the last line is (2.14). Thus (7.1) is an immediate consequence of Proposition (4.1). One readily checks that the assumptions on a n , c n , and θ n of the theorem imply that the conditions (4.5) and (4.6) of Proposition (4.1) are verified. The proof of 1.4 is done.

Proof of Theorem 1.1. Reasoning as in the proof of Theorem 1.4, we may assume that the process starts in its invariant measure π n . The main idea behind the proof is now classical. Suppose that P πn (A n (t, s)) = P πn ({R n ∩ (t, t + s) = ∅}) + o(1) (7.2) where A n (t, s) ≡ {X(c n t) = X(c n (t + s))} and where R n denotes the range of the rescaled blocked clock process S b n (t). Then Theorem 1.1 is a direct consequence of Theorem 1.4 and the arcsine law for stable subordinators. We refer to Ref. [START_REF] Gayrard | Convergence of clock process in random environments and aging in Bouchaud's asymmetric trap model on the complete graph[END_REF] for a detailed proof (see the proof of Theorem 1.6 therein) and again stress that the J 1 topology in (1.45) is necessary for this statement to hold.

We now focus on establishing (7.2). For k ≥ 1 and Z n,i as in (1.19) set

B k = k i=1
Z n,i < t, k+1 i=1 Z n,i > t + s . where convergence is almost sure in the random environment, as follows from Theorem 1.4, and where δ can be made as small as desired by taking T large enough. Therefore 0 ≤ P πn ({R n ∩ (t, t + s) = ∅}) -P πn ∪ 1≤k≤kn(T ) B k ≤ δ. (7.5)

  ) e -β min{Hn(x) | x∈Vn} ≤ e βn √ 2 log 2(1+2 log n/n) .

  .7) follows from (3.15) and the bound on π n (B ∪ B c ) of (3.16).

n

  (x, y) denotes the entries of Q k n . By(2.8), for all y ∈ (T n \ T n,l(x) ) c ,z∈T n \T n,l(x) ν n r n (y, z) = z∈T n \T n,l(x) λ n (y, z) ≤ n -1 r n z∈T n \T n,l(x)1 {z∼y} . (3.27) Therefore, inserting (3.27) in (3.26), (3.25) yields

≤ ( 1 +

 1 ρ n )E πn κ n + line follows from Proposition 3.3 and the Markov property. Next,

  .36) Thus b • n is nothing but b n (see (1.43)) with T • n substituted for T n . The next lemma collects properties of the sequences b n and b • n needed in the verification of both Condition (B1) and (B2).

4. 4 .

 4 Proof of Proposition 4.1. By (4.6), (4.3)-(4.4), and the bound κ n ≤ θ n , (4.40) implies that on Ω 1 ∩ Ω • ∩ Ω , for large enough n, b n = b • n (1 + o(1)). The assumption that a n = 2 εn /b n in (4.1) can thus be replaced by a n = 2 εn /b • n . Consider now (4.61) and note that by (4.41), (3.4), (1.38), and (4.6) (see also (4.8)), for all 0

. 25 )

 25 Using Corollary(3.11) to bound the last probability yields the claim of the proposition.We are now ready to complete the Proof of Proposition 5.1. Recall from the proof of Proposition 4.1 that onΩ 1 ∩ Ω • ∩ Ω a n = 2 εn /b n = 2 εn /b • n (1 + o(1)) for large enough n and consider(5.20). By (4.5), n 2 θ -1 n κn κ 1+αn(ε) n

1 n

 1 y (v)E y c -1 n τ n (y) y n (θ n -v)1 {c -τn(y) x n (θn-v)≤ } .

3 . 1 n

 31 The proof closely follows that of Proposition 4.5. We only point out the main differences. The random variables (4.64) are now replaced byX n (y) ≡ θn 0 dvh n,y (v)E y c -1 n τ n (y) y n (θ n -v)1 {c -τn(y) y n (θn-v)≤ } (6.16)andE • S (7) n, (t) = k n (t) y∈T • n E • X n (y).Proceeding as in (4.70)-(4.72) to deal with the conditional expectation and using thatP(τ n (x) ≥ r n (ε n )) = π n (T • n )(1 + o(1)) (see the paragraph below (4.73)), we getE • S (7) n, (t) = k n (t)(1 + o(1)) π n (T • n ) y∈T • n θn 0 dvh n,y (v)E y y n (θ n -v)E y c -1 n τ n (y)1 {c -1n τn(y)≤ n}

  y (v)E y F β,ε, ,n ( y n (θ n -v)) + C 2 n αn(ε)/2 e -nβ 2 /2 k n (t)(π n (T • n )) -1 P πn (H(T • n ) ≤ θ n ) (6.17)where here and below C i > 0, i = 1, 2, . . . are constants, and for F β,ε,n as in (1.41),F β,ε, ,n (z) = F β,ε,n (z) ) -1 -log nββc(ε) + log z nββc(ε)

  ) is bounded above byC 4 t 1-αn(ε)-log 2nβ 2 . (6.19)

2 + C 7 n 2 .P

 272 y (v)E y F β,ε, ,n ( y n (θ n -v)) αn(ε)/2 e -nββc(ε) k 2 n (t)θ π n (T • n )From the boundθn 0 dvh n,y (v)E y F β,ε, ,n ( y n (θ n -v)) ≤ (1 + o(1)) πn (H(y) ≤ θ n ) and (3.9), (4.41), we get that on Ω , for all but a finite number of indices n, the first summand is bounded above byC 8 t 2 2-αn(ε)-log 2nβ 2 nκ n θ αn(ε) n (r n ) 1+αn(ε)+o(1) 2 2 -n .Using the bound y∈T • n θn 0 dvh n,y (v)

7 . 4 Proof of Theorem 1 . 4 . 1 P

 74141 PROOF OF THEOREM 1.1 AND THEOREM 1.By Proposition(3.3), Proposition (4.1), Proposition (5.1) and Proposition (6.1), under the assumptions of Proposition (4.1) and Proposition (6.1), Conditions (B0), (B1), (B2), and (B3) of Theorem 1.3 are satisfied P-a.s.. It remains to check Condition (A0), i.e. to prove that P-a.s., for all u > 0,lim n→∞ P µn (Z n,1 > u) = 0 (7.1)whereZ n,1 = θn 0 max ((c n r n ) -1 , c -1 n τ n (Y n (s)))ds and µ n is the uniform measure on V n . By (3.2) and Lemma 3.µn (Z n,1 > u) ≤ (1 + o(1))P πn (Z n,1 > u) + x∈V n µ n (x)P x (Z n,1 > u)≤ (1 + o(1))P πn (Z n,1 > u) + n -c (1 + o(1))

(7. 3 )

 3 Then by(1.18), {R n ∩ (t, t + s) = ∅} = {∪ k≥1 B k }. Furthermore, for any T > 0,P πn ∪ k>kn(T ) B k ≤ P πn S b n (T ) < t -→ n→∞ P V α(ε) (T ) < t ≤ δ (7.4)

  .19) Proof of Lemma 2.1. Recall that c > 2. Eq. (2.13) is (2.9) of Lemma 2.2 of Ref. [24]. The estimate (2.14) on |V n | is (2.11) of Ref. [24]. and the estimate on |V n | follows by symmetry of H n . Eq. (2.15) and (2.18) are proved, respectively, as (2.11) of and (2.10) of Ref.[START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF]. The proof of (2.17) is a simple adaptation of the proof of lemma 7.1 of Ref.[START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF]. Clearly,(2.16) follows from (2.15) and (2.17), and (2.19) follows from (2.16) and (2.18).

	Lemma 2.2 (Lemma 2.3 of [24]

  .15) Denoting by C n,l(x) the (unique) component of B (see (2.7)) that contains x, we have |B | ≤ | ∪ x∈B C n,l(x) | ≤ |B| max x∈B |C n,l(x) | where by (2.13), on Ω , |C n,l(x) | n. By this and (3.1) we get π n

  .38) It only remains to observe that by (2.16) and (3.2) of Lemma 3.1, on Ω , π n

  • n ) the joint density that T • n is reached at time t, and that arrival to that set occurs in state x, given that the process starts in y. As already observed (see the paragraph below (4.17)),

	h n,x = y∈Vn π n (y)h y n,x . Proceeding as in (4.18)-(4.20) we then get
		σt n (u, ∞) = k n (t)	π n (y) R	u n (y)	2	(5.11)
		y∈Vn
	where, using (4.19) and (4.20),
	R	u n (y)

  .21) Proof of Proposition 5.4. As in the proof of Proposition 4.5 denote by P • the law of the collection {τ n(x), x ∈ T • n } conditional on T • n . By a first order Tchebychev inequality, P η•,t n (u n , ∞) ≥ ≤ -1 E E • η•,t n (u n , ∞) . (5.22) Note that E • η•,t n (u, ∞) only differs from the term E • ν •,t n (u n , ∞) of (4.65) in that the integral in (5.18) is over [0, κ n ] instead of [0, θ n ]. Taking a n = 2 εn /b • n ,a simple adaptation of the proof of(4.62) (see (4.69)-(4.78)) yields

  Proposition 6.1. Under the assumptions of Proposition 4.1, for all 0 < ε < 1 and all β > β c (ε), P-almost surely,

					•,t n (u, ∞) = 0 P -almost surely.	(5.27)
	Hence by (5.17), for all u > 0 and t > 0,
			lim n→∞	σ •,t n (u, ∞) = 0 P -almost surely.	(5.28)
	From there on the proof is a rerun of the proof of Proposition 4.1 with Lemma 5.2 and
	Lemma 5.3 playing the role of Lemma 4.2 and Lemma 4.3, respectively. We omit the
	details.
			6. VERIFICATION OF CONDITION (B3)
	By (1.18)-(1.20), (1.22), and (1.34), Condition (B3) in (1.33) will be verified if we can
	establish that:
					θn
	lim ↓0	lim sup n↑∞	k n (t)E πn	0

One proves in the same way that

where by (4.6) the term in square brackets (that comes from (3.40)) is equal to 1 + o [START_REF] Aldous | Inequalities for rare events in time-reversible Markov chains. I. In Stochastic inequalities[END_REF]. Combining (4.57) and (4.55) proves (4.39). Since I n (κ n , θ n -ζ n ) ≤ b • n = I n (0, θ n ), (4.55) and (4.58) yield, respectively, the lower and upper bounds of (4.41). It remains to prove (4.40). By definition (see (1.43), (4.36), and the second remark below (4.7) on the definition of T n ) Proof of Proposition 1.5. This is a straightforward consequence of (4.40), (4.41), the assumptions of (1.44), and (1.38).

Concentration of ν

Then, for any sequence u n > 0 such that 0 < u -u n < n -1 and all u > 0 and t > 0,

where

Consequently, for fixed

viewed as a collection of r.v.'s on the sub-sigma field

decays to zero as n diverges. We prove in this section that this holds true P-almost surely.

Proposition 5.1. Under the assumptions of Proposition 4.1, for all 0 < ε < 1 and β > 0, P-almost surely, lim

As in the proof of Proposition 4.1 we first bring σ t n (u, ∞) into a suitable form. Proceeding as in (4.11)-(4.12), we first write

where

and next reduce visits to T n in (5.3) to visits to visits to T • n , just as in Lemma 4.2. Set

(5.4) Lemma 5.2. Assume that (4.6) holds. Then on Ω , for all but a finite number of indices n,

).

(5.5)

Proof of lemma 5.2. As in the Proof of Lemma 4.2 we decompose the event appearing in the probability in (5.3) according to whether

we write σt 2 where q1 (y) and q2 (y) are defined as in (5.6) and (5.7), respectively, substituting

(5.8)

Applying (5.8) to the terms [q 1 (y)+q 2 (y)] 2 and [q 1 (y)+q 2 (y)] 2 , and observing that q 2 2 = q2 2 , we get

(5.10)

The Lemma now follows from (3.10) of Corollary 3.7.

Note that the event B k is non empty if and only if the increment Z n,k+1 straddles over the interval (t, t + s). To show that (7.2) holds it now suffices to prove the following two facts: Fact 1. P-a.s,

Fact 2. P-a.s,

(7.7) Combining (7.5), (7.6) and (7.7) then establishes that

which is tantamount to (7.2). The proofs of Facts 1 and 2 follow a now classical pattern (see e.g. Ref. [START_REF] Gayrard | Convergence of clock processes and aging in Metropolis dynamics of a truncated REM[END_REF], [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF]) which mostly uses information already obtained in the course of the verification of Conditions (B1)-(B3).

Proof of Fact 1 . Fix 0 < T < ∞ and assume that the assumptions of Proposition (4.1)

> s} so that, by (4.10),

for all large enough n. Note next that reasoning as in (6.11)-(6.12), on Ω • ∩ Ω , (4.6). Hence on Ω • ∩ Ω , for all large enough n,

. This means that for B k to be non-empty asymptotically, the increment Z n,k+1 must be produced by visits of Y n to T • n , and T • n only. Let us now prove that all these visits, if there are several of them, must be to a single vertex. For this it suffices to show that as n → ∞,

where ν T n is defined in (4.25) and bounded in Lemma 4.3. Reasoning as in the paragraph below (4.84) then yields that under the assumptions (4.5) and (4.6), on Ω • ∩ Ω , lim n→∞ ν T n = 0. Thus, the increment Z n,k+1 in B k cannot be produced by visits of Y n to two or more distinct vertices of T

and combining our results, we get that for all large enough n, A n (t, s) ⊇ W n so that

on Ω • ∩ Ω . Eq. (7.6) of Fact 1 is now proved.

Proof of Fact 2. In view of the information gathered in the proof of Fact 1, Fact 2 will be established if we can prove that no two distinct clock increments Z n,k+1 and Z n,k +1 can be produced by visits to the same vertex T • n , asymptotically. More precisely, as n → ∞,

where

To prove this, observe that the event in (7.10) can be written as

) Thus, by the Markov property we have, using the notation of (4.15)-(4.17) and the bound

To proceed, we split the domain of integration into [0,

Using that by Proposition 3.3, on Ω 1 , for all n large enough,

, the contribution coming from this domain is at most

where we used (3.11) with t n = θ n (which is licit as we many times saw) and (3.9) with t n = θ n k n (T ), which is licit provided that θ n k n (T )r n n2 -n → 0 as n → ∞, and this is guaranteed by our assumptions on a n . Indeed, proceeding as in the proof of Proposition 4.1 (see (4.79) and the paragraph above) we get that on

by (4.4), (3.2), and (2.16), and we get that on Ω • ∩ Ω ∩ Ω 1 , (7.11) is bounded above by

and by (4.6) this decays to zero as n → ∞ for all 0 < ε < 1.

Consider next the domain [θ n -κ n , θ n ] and note that since

By the upper bound of (3.5) and the lower bound of (3.4), on Ω , for all but a finite number of indices n, this is in turn bounded above by

as n → ∞, where we again used that 2 nδn = (n 2 θ n ) α(ε) by (4.4) whereas 0 < α(ε) < 1 by assumption; the final convergence then follows from (4.5). Combining the conclusions of (7.12) and (7.15) we get that on

, are two non-empty events then, on Ω • ∩ Ω ∩ Ω 1 , the increments Z n,k+1 and Z n,k +1 are produced by visits to two distinct elements of T • n with probability 1 -o(1). This readily implies (7.7) and concludes the proof of Fact 2.

The proof of Theorem 1.1 is now complete. 8. APPENDIX: PROOF OF THEOREM 1.2 AND THEOREM 1.3

Proof of Theorem 1.2. The proof closely follows that of Theorem 1.2 of Ref. [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF]. Throughout we fix a realization ω ∈ Ω of the random environment but do not make this explicit in the notation. We set

Condition (A0) ensures that S b n -S b n converges to zero, uniformly. Thus we must show that under Conditions (A1), (A2), and (A3), S b n ⇒ J 1 S ν . For this we rely on Theorem 1.1 of Ref. [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF]. Namely, we want to show that Conditions (A1), (A2), and (A3) imply the conditions of Theorem 1.1 of Ref. [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF]. To this end let {F n,i , n ≥ 1, i ≥ 0} be the array of sub-sigma fields of F Y defined (with obvious notation) through F n,i = σ (Y n (s), s ≤ θ n i), for i ≥ 0. Note that for each n and i ≥ 1, Z n,i is F n,i measurable and F n,i-1 ⊂ F n,i . Next observe that by the Markov property and the fact that, for all i ≥ 1 and y ∈ V n , P y (Z n,i > u) = P y (Z n,1 > u),

In view of this, (1.21), (1.22), and (1.23)

and in view of (1.24)

From (8.3) and (8.4) it follows that Conditions (A1) and (A2) of Theorem 1.2 are exactly the conditions of Theorem 1.1 of Ref. [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF]. Similarly Condition (A3) is condition (1.9). Therefore the conditions of Theorem 1.1 of Ref. [START_REF] Bovier | Convergence of clock processes in random environments and ageing in the p-spin SK model[END_REF] are verified, and so

where S ν is a subordinator with Lévy measure ν and zero drift.

The proof of Theorem 1.3 centers of the Proposition 8.1. Assume that Condition (B1) is satisfied. Then, choosing θ n ≥ κ n , the following holds for all initial distributions µ n : for all t > 0, all u > 0, and all > 0,

and Proof of Theorem 1.3. Condition (B2) combined with the conclusions of Proposition 8.1 implies both conditions (A1) and (A2), and Condition (B3) combined with (8.9) implies Condition (A3).