aa r X i v : . [ m a t h . P R ] F e b AGING IN METROPOLIS DYNAMICS OF THE REM: A PROOF
V ´ERONIQUE GAYRARDA
BSTRACT . We study the aging behavior of the Random Energy Model (REM) evolvingunder Metropolis dynamics. We prove that a classical two-time correlation function con-verges almost surely to the arcsine law distribution function that characterizes activatedaging, as predicted in the physics literature, in the optimal domain of the time-scale andtemperature parameters where this result can be expected to hold. In the course of theproof we establish that a certain continuous time clock process, after proper rescaling,converges almost surely to a stable subordinator, improving upon the result of Ref. [15]where a closely related clock is shown to converge in probability only, and in a restrictedregion of the time-scale and temperature parameters. The random rescaling involved inthis convergence is controlled at the fine level of fluctuations. As a byproduct, we refineand prove a conjecture made in Ref. [15].
1. I
NTRODUCTION
While there is as yet no established theory for the description of glasses, a consensusexists that this amorphous state of matter is intrinsically dynamical in nature [19], [29],[26]. Measuring suitable two-time correlation functions indeed reveals that glassy dynam-ics are history dependent and dominated by ever slower transients: they are aging . Therealization in the late 80’s that mean-field spin glass dynamics could provide a mathemati-cal formulation for this phenomenon sparked renewed interest in models, such as Derrida’sREM and p -spin SK models [16], [17], whose statics had, until then, been the main fo-cus of attention [11]. Despite this, Bouchaud’s phenomenological trap models first tookthe center stage as they succeeded in predicting the power-law decay of two-time correla-tion functions observed experimentally, even though they did so at the cost of an ad hocconstruction and drastically simplifying assumptions [10], [12].It was not until 2003 that a trap model dynamics was shown to result for the microscopicGlauber dynamics of a (random) mean-field spin glass Hamiltonian, namely, the REM en-dowed with the so-called Random Hopping dynamics and observed on time-scales nearequilibrium [4, 5, 6]. Quite remarkably, the predicted functional form of two-time correla-tion functions was recovered. Rapid progress followed over the ensuing decade, beginningwith [7]. The optimal domain of temperature and time-scales were this prediction applieswas obtained in Ref. [22] (almost surely in the random environment except for times scalesnear equilibrium where the results hold in probability only) and these results were partiallyextended to the p -spin SK models [3], [13].The choice of the Random Hopping dynamics, however, clearly favored the emergenceof trap models. Just as in trap model constructions, its trajectories are those of a simplerandom walk on the underlying graph, and thus, do not depend on the random Hamil-tonian. This is in sharp contrast with Metropolis [30] dynamics, a choice heralded in the
Date : October 16, 2018.2000
Mathematics Subject Classification.
Key words and phrases. random dynamics, random environments, clock process, L´evy processes, spinglasses, aging, Metropolis dynamics . physic’s literature as the natural microscopic Glauber dynamics [27], whose trajectoriesare biased against increasing the energy. This dependence on the random Hamiltonianmakes the analysis of the two-time correlation functions much harder. This problem wasfirst tackled in [24] were a truncated REM is considered, and a natural two-time correlationfunction is proved to behave as in the Random Hopping dynamics, in the same, optimalrange of time-scales and temperatures for which this result holds almost surely in the ran-dom environment. In the present paper, we free ourselves of the simplifying truncationassumption and prove that the same result holds true almost surely for the full REM. Arecent paper [15], by establishing the convergence of a so-called clock process, suggestedthat this might be the case but failed short of proving aging: the sole clock convergence,indeed, does not suffice to deduce aging, a property of correlation functions.1.1.
Main result.
Let us now specify the model. Denote by V n = {− , } n the n-dimensional discrete cube and by E n its edge set. The Hamiltonian (or energy) of the REMis a collection of independent Gaussian random variables, ( H n ( x ) , x ∈ V n ) , satisfying E H n ( x ) = 0 , E H n ( x ) = n. (1.1)The sequence ( H n ( x ) , x ∈ V n ) , n > , is defined on a common probability space denotedby (Ω , F , P ) . On V n , we consider the Markov jump process ( X n ( t ) , t > with rates λ n ( x, y ) = 1 n e − β [ H n ( y ) −H n ( x )] + , if ( x, y ) ∈ E n , (1.2)and λ n ( x, y ) = 0 else, were a + = max { a, } . This defines the single spin-flip continuoustime Metropolis dynamics of the REM at temperature β − > . Note that the rates arereversible with respect to the measure that assigns to x ∈ V n the mass τ n ( x ) ≡ exp {− β H n ( x ) } . (1.3)When studying aging the choice of the observation time-scale, c n , is all-important. Given < ε < and < β < ∞ , we let c n ≡ c n ( β, ε ) be the two-parameter sequence definedby εn P ( τ n ( x ) ≥ c n ) = 1 . (1.4)Gaussian tails estimates yield the explicit form c n = exp (cid:8) nββ c ( ε ) − (1 / α ( ε )) (cid:0) log( β c ( ε ) n/
2) + log 4 π + o (1) (cid:1)(cid:9) (1.5)where β c ( ε ) = √ ε , (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 theprocess in the same state at the two endpoints of the time interval [ c n t, c n ( t + s )] , C n ( t, s ) ≡ P µ n ( X n ( c n t ) = X n ( c n ( t + s ))) , t, s > . (1.8)Here P µ n denotes the law of X n conditional on F (i.e. for fixed realizations of the randomHamiltonian) when the initial distribution, µ n , is the uniform measure on V n . Theorem 1.1.
For all < ε < and all β > β c ( ε ) , for all t > and s > , P -almostsurely, lim n →∞ P µ n ( X n ( c n t ) = X n ( c n ( t + s ))) = sin α ( ε ) ππ Z t/ ( t + s )0 u α ( ε ) − (1 − u ) − α ( ε ) du. (1.9) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 3
Remark.
We in fact prove the more general statement that (1.9) holds along any n -dependentsequences of the form < ε n ≤ − c ′ β p n − log n + c ′′ n − log n where < c ′ , c ′′ < ∞ areconstants, that satisfy lim n →∞ ε n = ε , < ε ≤ . Relaxation to stationarity is known tooccur, to leading order, on time-scales c n of the form (1.5) with ε n = 1 [21]. At the otherextremity, a behavior known as extremal aging is expected to characterize the process ontimes scales that are sub-exponential in the volume and defined through sequences ε n thatdecay to 0 slowly enough [14], [8]. 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 two-step schemethat 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 linkedto the arcsine law for stable subordinators through the convergence of the suitably rescaledclock process to an α -stable subordinator, < α < . This is provided that the two-timecorrelation function at hand can be brought into a suitable function of the clock. Bothsteps heavily depend on the properties of the exploration process.While this scheme offers the methodological underpinnings of the analysis of aging,two distinct ways of implementing it, through discrete or continuous time objects, respec-tively, have emerged from the literature (we refer to the recent papers [24], [25], and [15]for in-depth bibliographies). The first arose from the study of models whose explorationprocess can be chosen as the simple random walk on the underlying graph. As mentionedearlier, this includes all Random Hopping dynamics and several trap models (e.g. on thecomplete graph or on Z d ). In physically more realistic dynamics the discrete scheme mayquickly become intractable. As shown in Ref. [24] for Metropolis dynamics of a trun-cated REM, the associated exploration process is itself an aging process that presents thesame complexity as the original dynamics. A similar situation arises when consideringasymmetric trap models on Z d . Initiated in that context, the continuous scheme consistsin choosing a (now continuous time) exploration process that mimics the simple randomwalk.Prescribing the exploration process completely determines the clock process. Clearly,having efficient tools available to prove their convergence to stable subordinators is es-sential. Such tools were provided in Ref. [23] and [13] for discrete time clock processesin the general setting of reversible Markov jumps processes in random environment onsequences of finite graphs and, more recently, for both discrete and continuous time clockprocesses of similar Markov jumps processes on infinite graphs [25]. These tools bothallowed one to improve all earlier results on the Random Hopping dynamics of mean-field models [22], [13], [14], turning statements previously obtained in law into almostsure statements in the random environment, and to obtain the first aging results for severaltwo-time correlation functions of asymmetric trap model on Z d [25].In Section 1.2 below we fill the gap left by continuous time clock processes in the caseof sequences of finite graphs and, thus, extent the results of Ref. [13] to that setting. Thisis perhaps no more than an exercise but the results we present (Theorem 1.2 and Theorem1.3) are the cornerstone of our approach and, hopefully, of other papers to come. Weclose this introduction out in Section 1.3 by stating a clock process convergence resultfor Metropolis dynamics of the REM (Theorem 1.4) that is at the heart of the proof ofTheorem 1.1.1.2. Convergence of continuous time clock processes.
We now enlarge our focus to thefollowing abstract setting. Let G n ( V n , E n ) be a sequence of loop-free graphs with set ofvertices V n and set of edges E n . A random environment is a family of possibly dependent GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 4 positive random variables, ( τ n ( x ) , x ∈ V n ) . The sequence ( τ n ( x ) , x ∈ V n ) , n > , is de-fined on a common probability space denoted by (Ω , F , P ) . On V n we consider a Markovjump process, ( X n ( t ) , t > , with initial distribution µ n and jump rates ( λ n ( x, y )) x,y ∈V n 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 > , with state space V n , initial distribution µ n , and jump rates ( e λ n ( x, y )) x,y ∈V n chosen such that X n and Y n have the same trajectories, that is to say, λ n ( x, y ) λ n ( x ) = e λ n ( x, y ) e λ n ( x ) ∀ ( x, y ) ∈ E n , (1.11)where e λ − n ( x ) and λ − n ( x ) are, respectively, the mean holding times at x of Y n and X n : e λ n ( x ) ≡ X y :( x,y ) ∈E n e λ n ( x, y ) , (1.12) λ n ( x ) ≡ X y :( x,y ) ∈E n 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 ( e S ← n ( t )) , t ≥ , (1.14)where e S ← n denotes the generalized right continuous inverse of e S n , and e S n , the so-called continuous time clock process , is given by e S n ( t ) = Z t λ − n ( Y n ( s )) e λ n ( Y n ( s )) ds, t ≥ . (1.15)Note that there is considerable freedom in the choice of the exploration process Y n . Wewill come back to this issue at the end of this subsection and focus, for the time being, onthe 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 fixedrealizations 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 µ n and P µ n . Expectationwith respect to P , P µ n , and P µ n are denoted by E , E µ n , and E µ n , respectively.Our main aim is to obtain simple and robust criteria for the convergence of the (suit-ably rescaled) clock process (1.15) to a stable subordinator. Since the clock is a doublystochastic process, the desired convergence mode must be specified. We will ask whetherthere exist sequences a n and c n that make the rescaled clock process S n ( t ) = c − n e S n ( a n t ) , t ≥ , (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 sincesuch results (also referred to as quenched ) contain the most useful information from thepoint of view of physics.As for discrete time clock processes [23], [13], the driving force behind our approach isa powerful method developed by Durrett and Resnick [20] to prove functional limit theo-rems for sums of dependent variables. Clearly this method does not cover the case of our GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 5 continuous time clock processes. The simple idea (already present in [25] ) is to introducea suitable “blocking” that turns the rescaled clock process (1.16) into a partial sum processto which Durrett and Resnick method can now be applied. For this we introduce a newscale, θ n , and set k n ( t ) ≡ ⌊ a n t/θ n ⌋ . (1.17)The blocked clock process , S bn ( t ) , is defined through S bn ( t ) = k n ( t ) X i =1 Z n,i (1.18)where, for each i ≥ , Z n,i ≡ c − n X x ∈V n (cid:0) λ − n ( x ) e λ n ( x ) (cid:1) [ ℓ xn ( θ n i ) − ℓ xn ( θ n ( i − , (1.19)and where, for each x ∈ V n , ℓ xn ( t ) = Z t { Y n ( s )= x } ds (1.20)is the local time at x . The next theorem gives sufficient conditions for S bn to converge.These conditions are expressed in terms of a small number of key quantities. For each t > , let π Y,tn ( y ) = k − n ( t ) k n ( t ) − X i =1 { Y n ( 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 > , denote by Q un ( y ) ≡ P y ( Z n, > u ) (1.22)the tail distribution of the aggregated jumps when X n (equivalently, Y n ) starts in y . Usingthese quantities, define the functions ν Y,tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π Y,tn ( y ) Q un ( y ) , (1.23) σ Y,tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π Y,tn ( y ) [ Q un ( y )] . (1.24)Observe that the sequence of measures π Y,tn as well as the sequence of functions Q un ( y ) , y ∈V n , are random variables on the probability space (Ω , F , P ) of the random environment.Thus, the functions ν Y,tn and σ Y,tn also are random variables on that space.We now formulate four conditions for the sequence S bn to converge to a subordinator.These conditions refer to a given sequence of initial distributions µ n , given sequences ofnumbers a n , c n , and θ n as well as a given realization of the random environment. Condition (A0).
For all u > , lim n →∞ P µ n ( Z n, > u ) = 0 . (1.25) Condition (A1).
There exists a σ -finite measure ν on (0 , ∞ ) satisfying R ∞ ( x ∧ ν ( dx ) < ∞ and such that for all continuity points x of the distribution function of ν , for all t > and all u > , P µ n (cid:0)(cid:12)(cid:12) ν Y,tn ( u, ∞ ) − tν ( u, ∞ ) (cid:12)(cid:12) < ǫ (cid:1) = 1 − o (1) , ∀ ǫ > . (1.26) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 6
Condition (A2).
For all u > and all t > , P µ n (cid:0) σ Y,tn ( u, ∞ ) < ǫ (cid:1) = 1 − o (1) , ∀ ǫ > . (1.27) Condition (A3).
For all t > , lim ǫ ↓ lim sup n ↑∞ k n ( t ) X y ∈V n E µ n ( π Y,tn ( y )) E y ( Z n, { Z n, ≤ ǫ } ) = 0 . (1.28) Theorem 1.2.
For all sequences of initial distributions µ n and all sequences a n , c n , and ≤ θ n ≪ a n for which Conditions (A0), (A1), (A2), and (A3) are verified, either P -almostsurely or in P -probability, the following holds w.r.t. the same convergence mode: S bn ⇒ J S ν , (1.29) where S ν is the L´evy subordinator with L´evy measure ν and zero drift. Convergence holdsweakly on the space D ([0 , ∞ )) equipped with the Skorokhod J -topology.Remark. Note that the theorem is stated for the blocked process S bn rather than the originalprocess S n (defined in (1.16)). This may falsely appear as an undesirable consequence ofour techniques. We stress that for applications to correlation functions, one needs state-ments that are valid in the strong J topology whereas forming blocks is needed in order tomake sense of writing J convergence statements in the setting of continuous time clocks. Remark.
Also note that convergence of S bn in the strong J topology immediately impliesthe strictly weaker result that S n converges to the same limit in the M topology.As for discrete time clocks of Ref. [13], 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 lawof large numbers for the random variables Q n . Again we formulate three conditions fora given sequence of initial distributions µ n , given sequences a n , c n , and θ n , and a givenrealization 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 ↓ as n ↑ ∞ , such that, forall pairs x, y ∈ V n , and all t ≥ , | 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 > and all u > , ν tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π n ( y ) Q un ( y ) → tν ( u, ∞ ) , (1.31) Condition (B2).
For all t > and all u > , σ tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π n ( y ) [ Q un ( y )] → . (1.32) Condition (B3).
For all t > , lim ǫ ↓ lim sup n ↑∞ k n ( t ) X y ∈V n π n ( y ) E y ( Z n, { Z n, ≤ ǫ } ) = 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 -almostsurely, respectively in P -probability. Then, as in (1.29), S bn ⇒ J S ν , P -almost surely,respectively in P -probability. GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 7
Theorem 1.3 is our key tool for proving convergence of blocked clock processes tosubordinators. It is of course essential for the success of our strategy that the convergencecriteria we obtained be tractable. Going back to (1.11) we thus now ask, in this light, howbest to choose the exploration process Y n .A tentative answer to this question is to mimic the exploration process of the RandomHopping 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 pro-cess X n . The following class of jump rates, inspired from an ingenious choice made inRef. [15], is intended to favor the emergence of these properties. Given a fresh sequence η n ≥ , set e 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 (cid:0) η n , τ n ( x ) (cid:1)P x ∈V n min (cid:0) η n , τ n ( x ) (cid:1) { η n > } + |V n | − { η n =0 } , x ∈ V n , (1.35)and that the clock (1.15) becomes e S n ( t ) = Z t max (cid:0) η n , τ n ( Y n ( s )) (cid:1) ds. (1.36)Let us discuss the role of η n on the example of Metropolis dynamics of REM. When η n = 0 , π n nicely reduces to the uniform measure but the mixing time, κ n , of the resultingexploration process turns out to be of the same order as that of X n , that is to say, of theorder of max ( x,y ) (min ( τ n ( x ) , τ n ( y ))) − = e βn √ log 2(1+ o (1)) . This leaves little hope that theconditions of Theorem 1.3 can be verified. A moment’s thought suffices, however, to seethat such a large mixing time is a side effect of the symmetry of the Hamiltonian (1.1). Bybreaking this symmetry, the term max( η n , τ n ( x )) in (1.34) places an η n -dependent cap on κ n (see Section 3.1). One is then left to choose η n small enough so that π n remains closeto the uniform measure but large enough so that κ n is kept as small as needed. A similarstrategy should hopefully apply to more general mean-field spin glass Hamiltonians. Remark.
We stress that the sole convergence of the clock process does not suffice to deduceaging, namely, the specific power law decay of the two-time correlation function. One stillhas to solve the problem of reducing the behavior of the two-time correlation function, as n → ∞ , to the arcsine law for stable subordinators, and this requires more informationon the exploration process than needed to only prove convergence of the clock. Noticealso that unlike the discrete time clock process, the continuous time clock process is not aphysical time. It thus has no physical meaning on its own.1.3. Application to Metropolis dynamics of the REM.
From that point onwards wefocus on Metropolis dynamics of the REM (see (1.1)-(1.2)) started in the uniform measureon V n . Applying the abstract results of Section 1.2 enables us to prove P -almost sureconvergence of the blocked clock process S bn ( t ) , defined in (1.18), when the continuoustime clock process e 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 e S n ( t ) and S bn ( t ) . We beginby defining a sequence, r ⋆n , that is ubiquitous throughout the rest of the paper: given β > and a constant c ⋆ > , we let r ⋆n ≡ r n ( β, c ⋆ ) be the solution of n c ⋆ P ( τ n ( x ) ≥ r ⋆n ) = 1 . (1.37) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 8
In explicit form r ⋆n = exp n β p c ⋆ n log n (cid:16) − log log n c ⋆ log n (1 + o (1)) (cid:17)o . (1.38)We now take η n ≡ ( r ⋆n ) − in (1.34) which, combined with (1.2), yields e λ n ( x, y ) = 1 nr ⋆n min( τ n ( y ) , τ n ( x ))min (cid:0) r ⋆n , τ n ( x ) (cid:1) , if ( x, y ) ∈ E n , (1.39)and e λ n ( x, y ) = 0 else. The physical observation time-scale, c n , is chosen as in (1.4). It isnaturally the same as in the Random Hopping dynamics. On the contrary, the definitionof the auxiliary time-scale, a n , contrasts sharply with the simple choice a n = 2 εn made inthe 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 α n ( ε ) − log x nβ (cid:0) − log xnββ c ( ε ) (cid:1) − , x > , (1.41)where α n ( ε ) ≡ ( nβ ) − log c n , that is, in view of (1.5), α n ( ε ) = α ( ε ) − log( β c ( ε ) n/ π + o (1)2 nββ c ( ε ) . (1.42)Further introduce the random set T n ≡ (cid:8) x ∈ V n | τ n ( x ) ≥ c n ( n θ n ) − (cid:9) . (1.43)Then, for ℓ xn as in (1.20), we set b n ≡ ( θ n π n ( T n )) − X x ∈ T n E π n [ F β,ε,n, ( ℓ xn ( θ n ))] . (1.44)It now only remains to choose the block length θ n . (The notation x n ≪ y n means thatthe sequences x n > and y n > satisfy x n /y n → as n → ∞ .) Theorem 1.4.
Given < ε < let θ n be any sequence such that − α ( ε ) log r ⋆n < log θ n ≪ n (1.45) and let c n and a n be as in (1.4) and (1.40)-(1.44), respectively. Then, for all < ε < and all β > β c ( ε ) , P -almost surely, S bn ⇒ J V α ( ε ) (1.46) where V α ( ε ) is a stable subordinator with zero drift and L´evy measure ν defined through ν ( u, ∞ ) = u − α ( ε ) , u > , (1.47) and where ⇒ J denotes weak convergence in the space D ([0 , ∞ )) of c`adl`ag functionsequipped with the Skorokhod J -topology. We again emphasize (see the remark below Theorem (1.2)) that the J convergencestatement of Theorem 1.4 is crucial to the control correlation functions. Of course, The-orem 1.4 implies the weaker result that the original (non blocked) clock process (1.16)converges to the same limit in the M topology of Skorokhod. Such a result was proved inRef. [15] (for the clock obtained by taking η n = 1 in (1.36)) albeit only in P -probabilityand in the restricted domain of parameters β > β c ( ε ) and / < ε < . As shown in GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 9 [24] (see lemma 2.1) the graph structure of the set T n when / < ε < reduces to a col-lection of isolated vertices (no element of T n has a neighbor in T n ) and this considerablysimplifies the analysis.Let us now examine the sequence b n introduced in (1.40) and defined in (1.44). Thissequence is a priori random in the random environment and depends on a sequence, θ n , thatcan itself be chosen within the two widely different bounds of (1.45). The next propositionprovides upper and lower bounds on b n that are not affected by the choice of θ n . Proposition 1.5.
Given < ε < , let c n and θ n be as in Theorem 1.4. Then, there existsa subset Ω ′ ⊆ Ω with P (Ω ′ ) = 1 such that on Ω ′ , for all but a finite number of indices n (cid:0) n c − ( r ⋆n ) α n ( ε )+ o (1) (cid:1) − ≤ b n ≤ n c + ( r ⋆n ) α n ( ε ) (1.48) where < c − , c + ≤ ∞ are numerical constants. Thus lim n →∞ n − log a n = ε P -a.s..Remark. The definition (1.40)-(1.44) of a n and that of the sequence R N in (2.10) ofRef. [15] have an obvious family resemblance. Our control of a n through Proposition1.5 implies the behavior conjectured in item 4 page 4 of that paper. Remark.
One may wonder whether the lower bound of (1.45) can be improved. The maintechnical obstacle to doing so is the lower bound on mean hitting times of Lemma 3.5. Inparticular, trying to improve the bound (3.5) on the spectral gap by choosing η n larger, sayas large as 1 as in Ref. [15], can at best improve the constant − α ( ε ) in front of log r ⋆n in(1.45).The rest of the paper is organized as follows. Section 2 is concerned with the propertiesof the REM’s landscape: several level sets that play an important role in our analysisare introduced and their properties collected. Section 3 gathers all needed results on theexploration process Y n . The proof of Theorem 1.4 can then begin. Section 4, 5, and 6 aredevoted, respectively, to the verification of Condition (B1), (B2), and (B3) of Theorem 1.3.The proof of Theorem 1.4 is completed in Section 7. Also in Section 7, the link betweenthe blocked clock process of (1.46) 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 ofthe results of Section 1.2.2. L EVEL SETS OF THE
REM’
S LANDSCAPE : THE T OP 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 ′ ) ≡ P ni =1 | x i − x ′ i | is the graph distance on V n . When dist( x, y ) = 1 wesimply write x ∼ y . We are concerned with the graph properties of level sets of the form V n ( ρ ) = { x ∈ V n | τ n ( x ) ≥ r n ( ρ ) } (2.1)where, given ρ > , the threshold level r n ( ρ ) is the sequence defined through ρn P ( τ n ( x ) ≥ r n ( ρ )) = 1 . (2.2)Observe that V n ( ρ ) can uniquely be decomposed into a collection of subsets V n ( ρ ) = ∪ Ll =1 C n,l ( ρ ) , C n,l ( ρ ) ∩ C n,k ( ρ ) ∀ ≤ l = k ≤ L, L ≡ L n ( ρ ) , (2.3)such that each graph G ( C n,l ( ρ )) is connected but any two distinct graphs G ( C n,l ( ρ )) and G ( C n,k ( ρ )) are disconnected. With a little abuse of terminology we call the sets C n,l ( ρ ) the connected components of the graph G ( V n ( ρ )) . As ρ decreases from ∞ to , the set V n ( ρ ) grows and the graph G ( V n ( ρ )) potentially acquires new edges. It is known [9] that GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 10 the size of the largest connected component C n,l ( ρ ) undergoes a transiton near the criticalvalue ρ c ≈ log nn log 2 , with a unique “giant” component of size O ( n − n ) emerging slightlybelow this value. As ρ decreases the small components merge into the giant one, and totalconnectedness is achieved for ρ slightly smaller than n − . One may naturally think ofthe connected components C n,l ( ρ ) before criticality as containing distinct “valleys” of theREM’s energy landscape, the level of emergence of the totally connected giant componentthen being a “ground level” connecting the local valleys.We now introduce several sets that play key roles in our analysis. • The sets V ⋆n and V ⋆n (of valleys and hills). Let c ⋆ be as in (1.37) and set ρ ⋆n ≡ c ⋆ log nn log 2 . (2.4)Thus, taking ρ = ρ ⋆n in (2.1)-(2.3), r ⋆n ≡ r n ( ρ ⋆n ) and the set V ⋆n ≡ V n ( ρ ⋆n ) decomposes into V ⋆n = ∪ L ⋆ l =1 C ⋆n,l , C ⋆n,l ∩ C ⋆n,k ∀ l = k, L ⋆ ≡ L n ( ρ ⋆n ) , (2.5)where the C ⋆n,l are the connected components of the graph G ( V n ( ρ ⋆n )) . According to ourearlier picture they contain “valleys” of the landscape. Since H n ( x ) is symmetrical the set V ⋆n ≡ V n ( ρ ⋆n ) = (cid:8) x ∈ V n | τ − n ( x ) ≥ r ⋆n (cid:9) (2.6)obtained from V n ( ρ ⋆n ) by substituting −H n ( x ) for H n ( x ) in (1.3) has the same randomgraph properties as V ⋆n but now contains “hills”. As in (2.5) we write V ⋆n ≡ V n ( ρ ⋆n ) = ∪ L ⋆ l =1 C ⋆n,l , C ⋆n,l ∩ C ⋆n,k ∀ l = k, L ⋆ ≡ L n ( ρ ⋆n ) , (2.7)where C ⋆n,l are the connected components of the graph G ( V n ( ρ ⋆n )) . With this definition(1.39) becomes e λ n ( x, y ) = ( n e − β max( H n ( y ) , H n ( x )) , if x / ∈ V ⋆n , nr ⋆n e − β [ H n ( y ) −H n ( x )] + , if x ∈ V ⋆n . (2.8)Furthermore, by (1.12), denoting by ∂A = { x ∈ V n | dist( x, A ) = 1 } the outer boundaryof A ⊂ V n , we have that for all x ∈ ∂V ⋆n , e λ n ( x ) = X y ∈ ( V ⋆n ) c e λ n ( x, y ) + (cid:16) ( nr ⋆n ) − { x ∈ V ⋆n } + τ n ( x ) n − { x ∈ ( V ⋆n ) c } (cid:17) | ∂x ∩ V ⋆n | . (2.9)Hence, conditional on V ⋆n , the mean holding time at x ∈ ( V ⋆n ) c does not depend on thevariables { τ n ( y ) , y ∈ V ⋆n } but only depends on the variables { τ n ( y ) , y ∈ ( V ⋆n ) c } . • Immersions in V ⋆n . Given any subset A ⊂ V ⋆n we call the immersion of A in V ⋆n anddenote by A ⋆ the set A ⋆ ≡ ∪ L ⋆ l =1 A ⋆n,l , A ⋆n,l = ( C ⋆n,l , if C ⋆n,l ∩ A = ∅ , ∅ , else . (2.10)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 (2.8), immersed sets have the property that e λ n ( x, y ) ≤ n − r ⋆n for all x ∼ y such that x ∈ A ⋆ , y / ∈ A ⋆ or y ∈ A ⋆ , x / ∈ A ⋆ . (2.11) • The top, T n , and the associated sets T ⋆n , T ◦ n and I ⋆n . Given a sequence δ n ↓ as n ↑ ∞ ,set ε n ≡ ε − δ n and let the top be the set T n ≡ V n ( ε n ) (2.12) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 11 obtained by taking ρ = ε n in (2.3). ( δ n will later be chosen so that the definitions (2.12)and (1.43) of T n coincide.) T n contains the top of the order statistics or the τ n ( x ) ’s, whenceits name. Since ρ ⋆n ≪ ε n , T n ⊂ V ⋆n , and so T n can be immersed in V ⋆n . According to (2.10)we write T ⋆n ≡ ∪ L ⋆ l =1 T ⋆n,l . (2.13)To each x ∈ T n corresponds a unique index ≤ 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 specialimportance in the sequel is the subset T ◦ n of vertices of T n that are alone in their valley, T ◦ n ≡ (cid:8) x ∈ T n | T ⋆n,l ( x ) ∩ T n = { x } (cid:9) . (2.14)This in particular implies that T ◦ n ⊆ { x ∈ V n | τ n ( x ) ≥ r n ( ε n ) , ∀ y ∼ x τ n ( y ) < r n ( ε n ) } . (2.15)Finally, define I ⋆n ≡ { x ∈ V n | τ n ( x ) ≥ r n ( ε n ) , ∀ y ∼ x ( r ⋆n ) − < τ n ( y ) < r ⋆n } ⊆ T ◦ n . (2.16)This is the largest subset of T ◦ n such that dist (cid:0) ( V ⋆n ∪ V ⋆n ) , I ⋆n (cid:1) ≥ .Most of the content of the next three lemmata is taken from [24]. The first lemma givesestimates on the size of various sets. Lemma 2.1.
There exists Ω ⋆ ⊂ Ω with P (Ω ⋆ ) = 1 such that on Ω ⋆ , for all but a finitenumber of indices n , ≤ | C ⋆n,l | ≤ { ρ ⋆n [1 − c − ⋆ (1 + O (log n/n ))] } − , ≤ l ≤ L ⋆ . (2.17) The same bounds hold replacing C ⋆n,l by C ⋆n,l and L ⋆ by L ⋆ in (2.17). Furthermore, | V ⋆n | = 2 n n − c ⋆ (1 + o ( n − c ⋆ )) and | V ⋆n | = 2 n n − c ⋆ (1 + o ( n − c ⋆ )) , (2.18) | T n | = 2 n (1 − ε n ) (1 + O ( n − nε n / )) , (2.19) | T ◦ n | = 2 n (1 − ε n ) (1 + O ( n − nε n / )) , (2.20) | T n \ T ◦ n | ≤ n n (1 − ε n ) (1 + o (1)) , (2.21) | I ⋆n | = 2 n (1 − ε n ) (1 − n − c ⋆ +1 (1 + o (1)) , (2.22) | T ◦ n \ I ⋆n | = 2 n − c ⋆ +1 n (1 − ε n ) (1 + o (1)) . (2.23) Finally, introducing the set M n ≡ { x ∈ V n | τ n ( x ) > τ n ( y ) for all y ∼ x } (2.24) of local minima of the Hamiltonian, | V ⋆n ∩ M n | = 0 . (2.25) Proof of Lemma 2.1.
Recall that by assumption c ⋆ > > . Eq. (2.17) is (2.9)of Lemma 2.2 of Ref. [24]. That the same bound holds for | C ⋆n,l | follows by symmetry of H n . The estimate (2.18) on | V ⋆n | is (2.11) of Ref. [24] and the estimate on | V ⋆n | followsagain by symmetry of H n . Eq. (2.19) and (2.22) are proved, respectively, as (2.11) of and(2.10) of Ref. [24]. The proof of (2.21) is a simple adaptation of the proof of lemma 7.1 ofRef. [24]. Clearly, (2.20) follows from (2.19) and (2.21), and (2.23) follows from (2.20) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 12 and (2.22). It only remains to prove (2.24). For this note that x ∈ V ⋆n ∩ M n if and only if τ n ( x ) ≤ ( r ⋆n ) − and τ n ( y ) < τ n ( x ) for all y ∼ x . Thus P (cid:0) | V ⋆n ∩ M n | ≥ (cid:1) ≤ X x ∈V n P (cid:0) τ n ( x ) ≤ ( r ⋆n ) − , ∀ y ∼ x τ n ( y ) < ( r ⋆n ) − (cid:1) (2.26) = 2 n n − c ⋆ (cid:0) n − c ⋆ (cid:1) n (2.27)which is summable. Thus, by Borel-Cantelli Lemma, there exists a set of full measuresuch that on that set, for all but a finite number of indices, | V ⋆n ∩ M n | = 0 . (cid:3) The second lemma expresses the function r n ( ρ ) defined through (2.2). Lemma 2.2 (Lemma 2.3 of [24]) . For all ρ > , possibly depending on n , and such that ρn ↑ ∞ as n ↑ ∞ , r n ( ρ ) = exp (cid:8) nββ c ( ρ ) − ( β/ β c ( ρ )) (cid:2) log( β c ( ρ ) n/
2) + log 4 π (cid:3) + o ( β/β c ( ρ )) (cid:9) . (2.28) Corollary 2.3.
Set x n = δ n /ε and assume that x n ↓ as n ↑ ∞ and e nββ c ( ε ) x ≫ r ⋆n . Then r n ( ε n ) /r n ( ε ) = exp (cid:8) − nββ c ( ε ) x (cid:2) x + O ( x ) (cid:3)(cid:9) . (2.29)The third and last lemma states needed bounds, in particular, on the maximal jump rate. Lemma 2.4 (Lemma 2.4 of [24]) . There exists a subset Ω ⊆ Ω with P (cid:0) Ω (cid:1) = 1 such thaton Ω , for all but a finite number of indices n the following holds: e − β min { max( H n ( y ) ,H n ( x )) | ( x,y ) ∈E n } ≤ e βn √ log 2(1+2 log n/n log 2) ≡ nν n , (2.30) e − β min { H n ( x ) | x ∈V n } ≤ e βn √ n/n ) . (2.31) Thus, max ( x,y ) ∈E n e λ n ( x, y ) ≤ ν n .
3. P
ROPERTIES OF THE EXPLORATION PROCESS Y n In this Section we establish the properties of the exploration process needed in the restof the paper. By (1.35) with η n ≡ ( r ⋆n ) − and (2.6), the invariant measure π n of Y n can bewritten as π n ( x ) = { x/ ∈ V ⋆n } + r ⋆n τ n ( x ) { x ∈ V ⋆n } Z β,n , x ∈ V n (3.1)where Z β,n ≡ |V n \ V ⋆n | + P x ∈ V ⋆n r ⋆n τ n ( x ) . Lemma 3.1. On Ω ⋆ , for all but a finite number of indices n , n (1 − n − c ⋆ (1 + o ( n − c ⋆ )) ≤ Z β,n ≤ n . (3.2) Therefore, if A is any of the sets T n , T ◦ n , T n \ T ◦ n , I ⋆n or T ◦ n \ I ⋆n in (2.19)- (2.23), π n ( A ) = | A | − n (1 + o (1)) (3.3) whereas for any x ∈ V n , π n ( x ) ≤ − n (1 + o (1)) . (3.4) Proof.
Since { x ∈ V ⋆n } = { r ⋆n τ n ( x ) ≤ } , |V n \ V ⋆n | ≤ Z β,n ≤ |V n \ V ⋆n | + | V ⋆n | ≤ n .Eq. (3.2) then follows from (2.18) of Lemma 2.1. Eq. (3.4) is then immediate and (3.3)follows from the fact that A ∩ V ⋆n = ∅ for each of the mentioned sets. (cid:3) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 13
Spectral gap and mixing condition.
Denote by e L n the Markov generator matrix of Y n (that is, the matrix with off-diagonal entries e λ n ( x, y ) and diagonal entries − e λ n ( x ) ), andby ϑ n, < ϑ n, ≤ · · · ≤ ϑ n, n − the eigenvalues of − e L n . Proposition 3.2. If c ⋆ > then for all β > , there exists a subset Ω ⊂ Ω with P (Ω ) = 1 such that, on Ω , for all but a finite number of indices n , /ϑ n, ≤ n r ⋆n (1 + o (1)) ≡ ˜ κ n (3.5)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 r ⋆n (1 + o (1)) ⌋ . (3.6) Proposition 3.3.
Under the assumptions of Proposition 3.2, on Ω , for all but a finitenumber of indices n , for all β > , all pairs x, y ∈ V n , and all t ≥ , | P x ( Y n ( t + κ n ) = y ) − π n ( y ) | ≤ ρ n π n ( y ) , (3.7) where κ n is given by (3.6) and ρ n < e − n .Proof of Proposition 3.2. The proof of (3.5) relies on a well known Poincar´e inequality,taken from [18] (see Proposition 1’ p. 38), applied to the stochastic matrix e P n = I + ν − n e L n where I denotes the identity matrix and ν n is defined in Lemma 2.4. By Lemma 2.4, on Ω , for all n large enough, max ( x,y ) ∈E n e λ n ( x, y ) ≤ ν n < ∞ . (3.8)Thus, on Ω , for large enough n , the entries e p n ( x, y ) of e P n obey ≤ e p n ( x, y ) ≤ and P y ∈V n e p n ( x, y ) = 1 . The Poincar´e inequality of interest now reads as follows. For eachpair of distinct vertices x, y ∈ V n , choose a path γ x,y going from x to y in the graph G ( V n ) .Paths may have repeated vertices but a given edge appears at most once in a given path.Let Γ n denote such a collection of paths (one for each pair { x, y } ). Then /ϑ n, ≤ ν − n max e ρ − n ( e ) P γ x,y ∋ e | γ x,y | π n ( x ) π n ( y ) , (3.9)where the max is over all edges e = { x ′ , y ′ } of G ( V n ) , ρ n ( e ) ≡ π n,l ( x ′ ) e p n ( x ′ , y ′ ) , and thesummation is over all paths γ x,y in Γ n that pass through e .The quality of the bound (3.9) now depends on making a judicious choice of the set ofpaths Γ n . We adopt the following clever choice made in Ref. [21]. Given i ∈ { , . . . n } and given two vertices x and x ′ ∈ V n such that x i = x ′ i , let γ ix,x ′ be the path obtained bygoing left to right cyclically from x to x ′ , successively flipping the disagreeing coordinates,starting from the i -th coordinate. Set Γ in = (cid:8) γ ix,x ′ , x, x ′ ∈ V n (cid:9) , ≤ i ≤ n . These pathsare ordered in an obvious way. Given x, x ′ and γ x,x ′ , let γ x,x ′ be the set of vertices visitedby the path γ x,x ′ , and let γ intx,x ′ = γ x,x ′ \ { x, x ′ } be the subset of “interior” vertices. Wenext split the set of vertices V n into good ones and bad ones. Recalling (2.7), we say thata vertex is good if it does not belong to V ⋆n ; otherwise it is bad. We say that a path γ isgood if all its interior points γ int are good, and that a set of paths is good if all its elementsare good.The (random) set of path Γ n is then constructed as follows:(i) Consider pairs x and x ′ such that dist( x, x ′ ) ≥ n/ log n . If { γ ix,x ′ , ≤ i ≤ n } containsa good path, choose the first such for Γ n ; otherwise choose γ x,x ′ . GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 14 (ii) Consider pairs x and x ′ such that dist( x, x ′ ) < n/ log n . If there is a good vertex x ′′ ∈ V n such that dist( x, x ′′ ) ≥ n/ log n and dist( x ′′ , x ′ ) ≥ n/ log n , and if there aregood paths, one in (cid:8) γ ix,x ′′ , ≤ i ≤ n (cid:9) and one in (cid:8) γ ix ′′ ,x ′ , ≤ i ≤ n (cid:9) , such that the unionof these two good paths is a self avoiding path of length less than n , select this union asthe path connecting x to x ′ in Γ n (notice that this is a good path); otherwise choose γ x,x ′ .It turns out that this Γ n is almost surely good. More precisely, set Ω GOOD n = { Γ ′ n is good } , n ≥ , and Ω GOOD = lim inf n →∞ Ω GOOD n . Proposition 3.4 (Proposition 4.1 of [21]) . If c ⋆ > then P (cid:0) Ω GOOD (cid:1) = 1 . From now on we assume that ω ∈ Ω GOOD so that, for all large enough n , Γ n is good.Note that the paths of Γ n have length smaller than n . Hence (3.9) yields /ϑ n, ≤ n max e = { x ′ ,y ′ } (cid:0) π n ( x ′ ) e λ n ( x ′ , y ′ ) (cid:1) − X γ x,y ∋ e π n ( x ) π n ( y )= max e = { x ′ ,y ′ } n min( τ n ( y ′ ) , τ n ( x ′ )) X γ x,y ∋ e min (1 , r ⋆n τ n ( x )) min (1 , r ⋆n τ n ( y )) Z β,n (3.10)where the final equality follows from (1.39), (1.35) (with η n ≡ ( r ⋆n ) − ), and (3.1). Alsonote that since bad vertices (i.e. vertices of V ⋆n ) can appear only at the ends of any path,the paths of Γ n do not contain any edge of the graph G n ( V ⋆n ) . This prompts us to write /ϑ n, ≤ max {K ,n , K ,n , K ,n } where K ,n , K ,n , and K ,n are obtained, respectively, byrestricting the maximum in (3.10) to the maximum over edges e = { x ′ , y ′ } with x ′ ∈ V ⋆n and y ′ / ∈ V ⋆n , x ′ / ∈ V ⋆n and y ′ ∈ V ⋆n , and x ′ / ∈ V ⋆n and y ′ / ∈ V ⋆n . To bound K ,n note thatthe sum over paths that contain e = { x ′ , y ′ } reduces to the sum over all paths starting in x ′ that contain e , so that K ,n = max e = { x ′ ,y ′ } : x ′ ∈ V ⋆n ,y ′ / ∈ V ⋆n n min (cid:0) , r ⋆n τ n ( x ′ ) (cid:1) min( τ n ( y ′ ) , τ n ( x ′ )) X γ x ′ ,y ∋ e π n ( y ) ≤ n r n . (3.11)By symmetry of the bound (3.10), K ,n ≤ n r n . Finally, min( τ n ( y ′ ) , τ n ( x ′ )) ≥ /r ⋆n forall x ′ , y ′ / ∈ V ⋆n and min (1 , r ⋆n τ n ( x )) min (1 , r ⋆n τ n ( y )) ≤ for all x, y ∈ V n . Thus K ,n ≤ n r ⋆n Z − β,n max e ∈ G ( V n ) |{ γ ∈ Γ n | e ∈ γ }| ≤ n r ⋆n Z − β,n (2 n − + 2 n/ log n ) , (3.12)where we used that the number of paths connecting vertices at distance n/ log n or moreapart is at most n − (see e.g. Example 2.2, p. 45 in Ref. [18] for this well known bound)whereas, arguing as in Ref. [21] (see Section 4.2.2, page 934), the number of paths con-necting vertices less than n/ log n apart and containing e is bounded above by the volumeof a hypercube of dimension at most n/ log n around e , and so, is smaller than n/ log n .In view of Lemma 3.1 we have that on Ω ⋆ ∩ Ω GOOD , for all but a finite number of indices n , K ,n ≤ n r ⋆n (1 + o (1)) . (3.13)Collecting our bounds and taking Ω = Ω ∩ Ω ⋆ ∩ Ω GOOD yields (3.5) and ends the proof. (cid:3)
Proof of Proposition 3.3.
It is well know that for reversible irreducible Markov processes,bounds on spectral gaps yield bounds on their total variation distance k · k var to stationarity.For instance, Proposition 3 of Ref. [18] applied to Y n states that for all x ∈ V n and all t > , k P x ( Y n ( t ) = · ) − π n ( · ) k var ≤ − π n ( x ) π n ( x ) e − tϑ n, . (3.14) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 15
By (3.1), Lemma 3.1, and (2.31) of Lemma 2.4, on Ω ∩ Ω ⋆ , for all but a finite numberof indices n , sup z ∈V n π − n ( z ) ≤ (2 n /r ⋆n ) e βn √ n/n ) . The claim of Proposition 3.3now readily follows from this, (3.14), and Proposition 3.2, choosing κ n as in (3.6). (cid:3) Hitting time for the stationary chain.
Drawing heavily on Aldous and Brown’swork [2], this section collects results on hitting times for the process Y n at stationarity. Let H ( A ) = inf { t ≥ | Y n ( t ) ∈ A } (3.15)be the hitting time of A ⊆ V n . We begin with bounds on the mean value of H ( A ) . Lemma 3.5. On Ω , for all but a finite number of indices n , for all A ⊆ V n , (1 − nπ n ( A )) r ⋆n nπ n ( A )(1 − π n ( A )) ≤ E π n H ( A )1 − π n ( A ) ≤ ˜ κ n π n ( A ) . (3.16) (If dist( V ⋆n , A ) > , nπ n ( A ) can be replaced by π n ( A ) in the right-hand side). The next lemma gives bounds on the density function h n,A ( t ) , t > , of H ( A ) when Y n starts in its invariant measure, π n . Lemma 3.6. On Ω , for all but a finite number of indices n , for all A ⊆ V n and all t > , E π n H ( A ) (cid:18) − ˜ κ n E π n H ( A ) (cid:19) (cid:18) − tE π n H ( A ) (cid:19) ≤ h n,A ( t ) ≤ E π n H ( A ) (cid:18) κ n t (cid:19) . The bounds of Lemma 3.6 imply that h n,A ( t ) ≈ E πn H ( A ) when ˜ κ n ≪ t ≪ E π n H ( A ) .Complementing this, Lemma 3.7 is well suited to dealing with “small” values of t . Lemma 3.7. On Ω ⋆ , for all but a finite number of indices n , for all A ⊆ V n and all t > , P π n ( H ( A ) > t ) ≥ (1 − nπ n ( A )) exp (cid:18) − t r ⋆n nπ n ( A )1 − nπ n ( A ) (cid:19) . (3.17) In particular, for any A and any sequence t n such that t n r ⋆n nπ n ( A ) → as n → ∞ , P π n ( H ( A ) ≤ t n ) < t n r ⋆n nπ n ( A )(1 + o (1)) . (3.18) If A ⊂ V n \ V ⋆n the factor n in front of π n ( A ) in (3.17) and (3.18) can be suppressed. The next Corollary is stated for later convenience.
Corollary 3.8.
Under the assumptions of Lemma 3.7 the following holds: For all < ε < , for any sequence t n such that t n r ⋆n n − nε n → as n → ∞ P π n ( H ( T n \ T ◦ n ) ≤ t n ) ≤ t n r ⋆n n − nε n (1 + o (1)) , (3.19) P π n ( H ( T ◦ n ) ≤ t n ) ≤ t n r ⋆n n − nε n (1 + o (1)) . (3.20)We now prove these results, beginning with Lemma 3.7. Proof of Lemma 3.7.
Write A = B ∪ B c where B = A ∩ V ⋆n and B c = A \ B . Let B ⋆ bethe immersion of B in V ⋆n (see (2.10)). Since A ⊆ B ⋆ ∪ B c , H ( A ) ≥ H ( B ⋆ ∪ B c ) , and P π n ( H ( A ) > t ) ≥ P π n ( H ( B ⋆ ∪ B c ) > t ) . (3.21)To bound the right-hand side of (3.21), we use a well know lower bound on hitting timesfor stationary reversible chains taken from Ref. [2] (combine Theorem 3 and Lemma 2therein) that states that for all C ⊆ V n and all t > , P π n ( H ( C ) > t ) ≥ (1 − π n ( C )) exp (cid:18) − t q n ( C, C c )1 − π n ( C ) (cid:19) (3.22) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 16 where, for for any two sets C and e C such that C ∩ e C = ∅ , q n ( C, e C ) ≡ X x ∈ C X y ∈ e C π n ( x ) e λ n ( x, y ) . (3.23)Let us thus evaluate (3.23) 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 (2.8), e λ n ( x, y ) ≤ n − r ⋆n forany x ∈ B c and any y ∼ x . Thus q n ( B c , ( B ⋆ ∪ B c ) c ) ≤ r ⋆n π n ( B c ) . Next, by (2.11), 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 )] . (3.24)Denoting by C ⋆n,l ( x ) the (unique) component of B ⋆ (see (2.10)) that contains x , we have | B ⋆ | ≤ | ∪ x ∈ B C ⋆n,l ( x ) | ≤ | B | max x ∈ B | C ⋆n,l ( x ) | where by (2.17), on Ω ⋆ , | C ⋆n,l ( x ) | ≪ n . Bythis and (3.1) we get π n ( B ⋆ ) = Z − β,n | B ⋆ | ≤ nZ − 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 ) . (3.25)Using (3.25) in the right-and side of (3.24) and plugging the result in (3.22) finally yields(3.17). Clearly, if A ⊂ V n \ V ⋆n then B = ∅ and the right-and side of (3.24) reduces to r ⋆n [ π n ( ∅ ) + π n ( B c )] = r ⋆n π n ( A ) . (cid:3) Proof of Corollary 3.8.
This follows from (3.3) of Lemma 3.1, (2.20), and (2.21). (cid:3)
Proof of Lemma 3.6.
Proceed as in Lemma 13 of Ref. [2] and use Proposition 3.2. (cid:3)
Proof of Lemma 3.5.
The rightmost inequality is that of Lemma 2 of Ref. [2] combinedwith Proposition 3.2. Lemma 2 of Ref. [2] also states that for C ⊆ V n and q n ( C, C c ) defined as in (3.23), E π n H ( C )1 − π n ( C ) ≥ − π n ( C ) q n ( C, C c ) . (3.26)Given A ⊆ V n let B ⋆ and B c be defined as in the first line of the proof of Lemma 3.7. Since H ( A ) ≥ H ( B ⋆ ∪ B c ) , E π n H ( A ) ≥ E π n H ( B ⋆ ∪ B c ) . Using (3.26) with C = B ⋆ ∪ B c ,(3.16) follows from (3.24) and the bound on π n ( B ⋆ ∪ B c ) of (3.25). (cid:3) On hitting the top starting in the top.
Let T ◦ n and I ⋆n be as in (2.14) and (2.16). Proposition 3.9.
Given ǫ > 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 > | T ◦ n | − X x ∈ T ◦ n P x ( H ( T ◦ n \ x ) ≤ s ) ≤ sn c ⋆ +3 r ⋆n π n ( T ◦ n ) . (3.27)The next proposition is a variant of Proposition 3.9 that we state for later convenience. Proposition 3.10.
Under the assumptions and with the notation of Proposition 3.9, on Ω ◦ ,for all but a finite number of indices n , for all s > | T ◦ n \ I ⋆n | − X x ∈ T ◦ n \ I ⋆n P x ( H ( I ⋆n ) ≤ s ) ≤ sn r ⋆n π n ( I ⋆n )(1 + o (1)) . (3.28) Proof of Proposition 3.9.
A key ingredient of the proof is an explicit expression of thedensity function h xn,A ( t ) , t ≥ , of the hitting time H ( A ) when Y n starts in x ∈ A c ≡V n \ A . We first state this expression in full generality as given in [28] (see Section 6.2,p. 83). Consider the stochastic matrix e P n = ( e p n ( x, y )) defined above (3.8). 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 ) = GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 17 e p n ( x, y ) . This is the sub-matrix of e 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 e P n on A c × A . Let A be the vector of ’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 xn,A ( t ) = ν n ∞ X k =0 ( ν n t ) k k ! e − ν n t (cid:0) δ x , Q kn R n A (cid:1) , t ≥ , (3.29)where ( · , · ) denotes the inner product in R | A c | . Consequently, for all s > , P x ( H ( A ) ≤ s ) = Z s ν n ∞ X k =0 ( ν n t ) k k ! e − ν n t (cid:0) δ x , Q kn R n A (cid:1) dt. (3.30)For later reference we also denote by ( h xn,y,A ( t )) y ∈ A the vector whose components are, foreach y ∈ A , the joint density that A is reached at time t , and that arrival to that set occursin state y , namely, h xn,y,A ( t ) is defined as in (3.29) substituting δ y for A therein; as a result h xn,A ( t ) = P y ∈ A h xn,y,A ( t ) .Returning to (3.27), a first order Tchebychev inequality yields, for all ǫ > P hP x ∈ T ◦ n P x ( H ( T ◦ n \ x ) ≤ s ) ≥ ǫ i ≤ ǫ − E hP x ∈ T ◦ n P x (cid:16) H (cid:0) T ⋆n \ T ⋆n,l ( x ) (cid:1) ≤ s (cid:17)i ≡ ǫ − W n , (3.31)where T ⋆n ≡ ∪ L ⋆ l =1 T ⋆n,l is defined in (2.13) and ≤ l ( x ) ≤ L ⋆ denotes the (unique) indexsuch that T ⋆n,l ( x ) ∩ T ◦ n = { x } in (2.14). By (3.30) with A = T ⋆n \ T ⋆n,l ( x ) , W n = X x ∈V n Z s dt ∞ X k =1 ( ν n t ) k k ! e − ν n t W n,k ( x ) (3.32)where W n,k ( x ) ≡ E h { x ∈ T ◦ n } ν n (cid:16) δ x , Q kn R n T ⋆n \ T ⋆n,l ( x ) (cid:17)i . (3.33)Note that the term k = 0 is zero. For k ≥ the matrix term in (3.33) reads, { x ∈ T ◦ n } ν n (cid:16) δ x , Q kn R n T ⋆n \ T ⋆n,l ( x ) (cid:17) = { x ∈ T ◦ n } X y ∈ ( T ⋆n \ T ⋆n,l ( x ) ) c q ( k ) n ( x, y ) X z ∈ T ⋆n \ T ⋆n,l ( x ) ν n r n ( y, z ) (3.34)where q ( k ) n ( x, y ) denotes the entries of Q kn . By (2.11), for all y ∈ ( T ⋆n \ T ⋆n,l ( x ) ) c , X z ∈ T ⋆n \ T ⋆n,l ( x ) ν n r n ( y, z ) = X z ∈ T ⋆n \ T ⋆n,l ( x ) e λ n ( y, z ) ≤ n − r ⋆n X z ∈ T ⋆n \ T ⋆n,l ( x ) { z ∼ y } . (3.35)Therefore, inserting (3.35) in (3.34), (3.33) yields W n,k ( x ) ≤ r ⋆n n E " E (cid:20) { x ∈ T ◦ n } X y ∈ ( T ⋆n \ T ⋆n,l ( x ) ) c q ( k ) n ( x, y ) X z ∈ T ⋆n \ T ⋆n,l ( x ) { z ∼ y } (cid:12)(cid:12)(cid:12)(cid:12) V ⋆n (cid:21) (3.36)where E [ · | V ⋆n ] denotes the conditional expectation given a realization of the set V ⋆n = ∪ L ⋆ l =1 C ⋆n,l (see (2.5)), namely, expectation with respect to the measure P ( · | V ⋆n ) = P ( · ∩ {∀ ≤ l ≤ L ⋆ ∀ x ∈ C ⋆n,l τ n ( x ) ≥ r ⋆n } ∩ {∀ x ∈ C ⋆n, τ n ( x ) < r ⋆n } ) P ( {∀ ≤ l ≤ L ⋆ ∀ x ∈ C ⋆n,l τ n ( x ) ≥ r ⋆n } ∩ {∀ x ∈ C ⋆n, τ n ( x ) < r ⋆n } ) , (3.37)where we set C ⋆n, ≡ V n \ V ⋆n for simplicity. Thus V n = ∪ ≤ l ≤ L ⋆ C ⋆n,l and C ⋆n,l ∩ C ⋆n,l ′ = ∅ for all ≤ l = l ′ ≤ L ⋆ , so that if L i ⊂ { , . . . , L ⋆ } , i = 1 , . . . , j , is a collection of disjoint GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 18 sets, functions f i of the variables { τ n ( x ) , x ∈ ∪ l ∈L i C ⋆n,l } , i = 1 , . . . , j , are independentunder the conditional law (3.37). Observe now that conditional on V ⋆n the entries of thematrix Q n are functions of the variables { τ n ( y ) , y ∈ ( T ⋆n \ T ⋆n,l ( x ) ) c } only: for off-diagonalentries, i.e. for q n ( x, y ) with x = y , this is an immediate consequence of (2.8); for diagonalentries, i.e. q n ( x, x ) = 1 − ν − n e λ n ( x ) , this claim follows from (1.12) and (2.8) if x / ∈ ∂V ⋆n and from (2.9) and (2.8) if x ∈ ∂V ⋆n (the boundary set ∂A of A is defined above (2.9)).Next, observe that the sum over y ∈ ( T ⋆n \ T ⋆n,l ( x ) ) c in (3.36) can be restricted to the sumover y ∈ ∂V ⋆n ⊆ C ⋆n, and use the definition of T ⋆n (see (2.13)) to write X y ∈ ( T ⋆n \ T ⋆n,l ( x ) ) c q ( k ) n ( x, y ) X z ∈ T ⋆n \ T ⋆n,l ( x ) { z ∼ y } = X x ∈V n · · · X x k − ∈V n X y ∈ ∂V ⋆n X z ∼ y X ≤ l ≤ L⋆C⋆n,l ∩ x = ∅ · · · X ≤ lk − ≤ L⋆C⋆n,lk − ∩ xk − = ∅ X ≤ l = l ( x ) ≤ L⋆C⋆n,l ∩ z = ∅ q n ( x, x ) . . . q n ( x k − , y ) {∀ x ′ ∈ C⋆n,l \{ x } τ n ( x ′ ) The last equality is (2.4). Using this bound in (3.32) finally yields that on Ω ⋆ , for all largeenough n , W n = X x ∈V n Z θ n dt ∞ X k =1 ( ν n t ) k k ! e − ν n t S n,k ( x ) ≤ θ n n c ⋆ +1 r ⋆n n − ε n n . (3.45)It only remains to observe that by (2.20) and (3.3) of Lemma 3.1, on Ω ⋆ , π n ( T ◦ n ) =2 − nε n (1 + o (1)) for all but a finite number of indices n . Hence P h | T ◦ n | − P x ∈ T ◦ n P x (cid:16) H ( T ⋆n \ T ⋆n,l ( x ) ) ≤ s (cid:17) ≥ ǫ i ≤ ǫ − sn c ⋆ +1 r ⋆n π n ( T ◦ n )(1 + o (1)) . Choosing ǫ = n n c ⋆ +1 r ⋆n π n ( T ◦ n ) , the claim of the proposition follows from Borel-CantelliLemma. (cid:3) Proof of Proposition 3.10. This is a rerun of the proof of Proposition 3.9. (cid:3) Rough bounds on local times.Lemma 3.11. For all ≤ α ≤ , all x ∈ V n , and all s > , E x [ ℓ xn ( s )] α ≥ ( e λ − n ( x )) α Γ(1 + α )[1 − c exp( − c s e λ n ( x ))] + s α exp( − s e λ n ( x )) (3.46) where < c , c < ∞ are constants, and if moreover sr ⋆n nπ n ( x ) → as n → ∞ , E x [ ℓ xn ( s )] α ≤ (1 + o (1)) (cid:2) κ αn + { s>κ n } s α ( s − κ n ) r ⋆n nπ n ( x ) (cid:3) . (3.47) Proof of Lemma 3.6. The lower bound follows from the trite observation that ℓ xn ( s ) is atleast as large as the minimum between the first jump of Y n and s , that is, ℓ xn ( s ) ≥ e λ − n ( x ) e s> e λ − n ( x ) e + s s ≤ e λ − n ( x ) e , (3.48)where e is an exponential random variable of mean one. Thus E x [ ℓ xn ( s )] α ≥ E x he λ − n ( x ) e s> e λ − n ( x ) e i α + s α E x h s ≤ e λ − n ( x ) e i α . (3.49)Explicit calculations yield E x he λ − n ( x ) e s> e λ − n ( x ) e i α ≥ ( e λ − n ( x )) α Γ(1 + α )[1 − c exp( − c s e λ n ( x ))] (3.50)for some constants < c , c < ∞ . Eq. (3.46) now readily follows. To get an upperbound write E x [ ℓ xn ( s )] α ≤ κ αn if s ≤ κ n . Otherwise write E x [ ℓ xn ( s )] α ≤ E x h κ n + R sκ n { Y n ( s )= x } ds i α (3.51) ≤ (1 + ρ n ) E π n h κ n + R s − κ n { Y n ( s )= x } ds i α (3.52)where the last line follows from Proposition 3.3 and the Markov property. Next, E π n h κ n + R s − κ n { Y n ( s )= x } ds i α ≤ E π n (cid:0) κ αn { H ( x ) >s − κ n } + s α { H ( x ) ≤ s − κ n } (cid:1) ≤ κ αn + s α P π n ( H ( x ) ≤ s − κ n ) ≤ κ αn + s α ( s − κ n ) r ⋆n nπ n ( x )(1 + o (1)) , (3.53)the last inequality being (3.18) of Lemma 3.7. Eq. (3.47) is proved. (cid:3) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 20 4. V ERIFICATION OF C ONDITION (B1)In this section we prove a strong law of large number for the function ν tn ( u, ∞ ) definedin (1.31). Recall that for r ⋆n defined in (1.37), we take η n ≡ ( r ⋆n ) − in (1.34), (1.35), and(1.36). Then by (1.18)-(1.19), (1.22), and (1.34), ν tn ( u, ∞ ) = k n ( t ) P π n (cid:18)Z θ n max (cid:0) ( c n r ⋆n ) − , c − n τ n ( Y n ( s )) (cid:1) ds > u (cid:19) (4.1)where π n is the invariant measure (1.35) of Y n , θ n is the block length of the blocked clockprocess (1.18), k n ( t ) = ⌊ a n t/θ n ⌋ , and, given < ε < , c n and a n are defined in (1.4) and(1.40)-(1.44), respectively. By Theorem 1.3, θ n and a n must obey ⌊ n r ⋆n (1 + o (1)) ⌋ ≡ κ n ≤ θ n ≪ a n , (4.2)where the left-most equality is (3.6). Further recall from Section 2 that for ρ ⋆n as in (2.4), ρ ⋆n ≪ ε n ≡ ε − δ n . (4.3)(Recall that < x n ≪ y n means that x n /y n → as n → ∞ .) From now on we take δ n such that nδ n = ( n θ n ) α ( ε ) , i.e. δ n ≡ nβ r ε log 2 log (cid:0) n θ n (cid:1) . (4.4)Thus, given < ε < and β > , all sequences except θ n are determined. Proposition 4.1. Given < ε < and β > let the sequences c n and a n be defined as in(1.4) and (1.40)-(1.44), respectively, and let θ n be such that ( r ⋆n ) ≪ θ − α ( ε ) n , (4.5) n − log θ n ≪ . (4.6) Then, for all < ε < and β > , P -almost surely, lim n →∞ ν tn ( u, ∞ ) = tu α ( ε ) , ∀ t > , u > . (4.7) Remark. Eq. (4.6) implies that δ n ≪ and that θ n ≪ c n for all ε > . In view of (1.38),(3.5), (4.4) and (3.6), (4.6) also implies that c n c ˜ κ c n κ c n ( r ⋆n ) c θ c n ≪ εn and c n c ˜ κ c n κ c n ( r ⋆n ) c θ c n ≪ ε n n (4.8)for all ε > and any choice of constants ≤ c i < ∞ . Remark. In order to guarantee strict equivalence of the definitions (1.43) and (2.12) of theset T n when δ n is given by (4.4), we should replace the term c n ( nθ n ) − in (1.43) by c n exp (cid:8) − log( n θ n ) (cid:2) o (1))(2 nββ c ( ε )) − log( n θ n ) (cid:3)(cid:9) (4.9)(see Corollary 2.3). 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 tn ( u, ∞ ) can be reduced to the quantity ν ◦ ,tn ( u, ∞ ) defined in (4.32). In Section 4.2 we prove upperet 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 ◦ ,tn ( u, ∞ ) concentrates around its mean value when choosing a n = 2 εn /b ◦ n . The proof of Proposition4.1 is finally completed in Section 4.2. GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 21 Preparations. To begin with, we bring the function ν tn ( u, ∞ ) given in (4.1) into aform amenable to treatment. Let T n be as in (2.12). For all < ε < and δ n as in (4.4), ≤ Z θ n max (cid:0) ( c n r ⋆n ) − , c − n τ n ( Y n ( s )) (cid:1) { Y n ( s ) / ∈ T n } ds ≤ θ n r n ( ε n ) r n ( ε ) ≤ n − (4.10)as follows from (2.29). Hence visits of Y n outside the set T n only yield a negligible con-tribution to the event in (4.1), implying that ˇ ν tn ( u, ∞ ) ≤ ν tn ( u, ∞ ) ≤ ˇ ν tn (cid:0) u − n − , ∞ (cid:1) (4.11)where ˇ ν tn ( u, ∞ ) ≡ k n ( t ) P π n (cid:18)Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T n } ds > u (cid:19) . (4.12)Our next step consists in reducing visits to T n in ˇ ν tn ( u, ∞ ) to visits to the subset T ◦ n definedin (2.14). Set ¯ ν tn ( u, ∞ ) ≡ k n ( t ) P π n (cid:18)Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T ◦ n } ds > u (cid:19) . (4.13) Lemma 4.2. Assume that (4.6) holds. Then on Ω ⋆ , for all but a finite number of indices n , | ˇ ν tn ( u, ∞ ) − ¯ ν tn ( u, ∞ ) | ≤ k n ( t ) θ n r ⋆n n − nε n (1 + o (1)) . (4.14) Proof of Lemma 4.2. Decomposing the event appearing in the probability in (4.12) accord-ing to whether { H ( T n \ T ◦ n ) ≤ θ n } or { H ( T n \ T ◦ n ) > θ n } , (4.14) follows from (3.19) ofCorollary 3.8 applied with t n = θ n , which is licit by virtue of (4.6) (see also (4.8)). (cid:3) 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 densitythat T ◦ n is reached at time v , and that arrival to that set occurs in state x , P π n ( H ( T ◦ n ) ≤ s, Y n ( H ( T ◦ n )) = x ) = Z s h n,x ( v ) dv. (4.15)For this vector of densities we have X x ∈ T ◦ n Z ∞ h n,x ( v ) dv = 1 , (4.16)and, denoting by h n,T ◦ n the density of H ( T ◦ n ) , h n,T ◦ n = X x ∈ T ◦ n h n,x . (4.17)In the notation of Section 3.3 (see the paragraph below (3.30)) h n,x = P y ∈V n π n ( y ) h yn,x,T ◦ n where, for y ∈ T ◦ n , h yn,x,T ◦ n = δ y . From this and the strong Markov property it follows that ¯ ν tn ( u, ∞ ) = k n ( t ) X x ∈ T ◦ n Z θ n h n,x ( v ) P x (cid:18)Z θ n − v c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T ◦ n } ds > u (cid:19) dv. (4.18) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 22 Denote by Q u,vn ( 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,vn ( x ) ≡ e Q u,vn ( x ) + b Q u,vn ( x ) , e Q u,vn ( x ) = P x (cid:18)Z θ n − v c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T ◦ n } ds > u, H ( T ◦ n \ x ) ≤ θ n − v (cid:19) , (4.19) b Q u,vn ( x ) = P x (cid:18)Z θ n − v c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T ◦ n } ds > u, H ( T ◦ n \ x ) > θ n − v (cid:19) , (4.20)and split (4.18) accordingly. Clearly, for all v > e Q u,vn ( x ) ≤ P x ( H ( T ◦ n \ x ) ≤ θ n ) . (4.21)This and the bound R θ n h n,x ( v ) dv ≤ P π n ( H ( x ) ≤ θ n ) (that follows from (4.15)), yield k n ( t ) X x ∈ I ◦ n Z θ n h n,x ( v ) e Q u,vn ( x ) dv (4.22) ≤ k n ( t ) X x ∈ T ◦ n P π n ( H ( T ◦ n ) ≤ θ n , Y n ( H ( T ◦ n )) = x ) P x ( H ( T ◦ n \ x ) ≤ θ n ) (4.23) ≤ e ν tn (4.24)where e ν tn ≡ k n ( t ) X x ∈ T ◦ n P π n ( 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 , e ν tn ≤ k n ( t ) n c ⋆ +4 ( θ n π n ( T ◦ n ) r ⋆n ) (1 + o (1)) . (4.26) Proof of Lemma 4.3. By (3.3), (2.20), (4.3) and (4.4), on Ω ⋆ , for all large enough n , θ n nπ n ( T ◦ n ) r ⋆n = n α ( ε ) r ⋆n θ α ( ε ) n − nε (1 + o (1)) , wich decays to zero as n divergesby (4.6) (see also (4.8)). We may thus use (3.18) of Lemma 3.7 to bound the term P π n ( H ( x ) ≤ θ n ) in (4.25), and by this and (3.3) we get that on Ω ⋆ , for all large enough n , e ν tn ≤ k n ( t ) θ n nπ n ( T ◦ n ) r ⋆n (1 + o (1)) | T ◦ n | − P x ∈ T ◦ n P x ( H ( T ◦ n \ x ) ≤ θ n ) . (4.27)The lemma now follows from Proposition 3.9. (cid:3) Consider now the contribution to (4.18) coming from (4.20). By definition, b Q u,vn ( x ) = P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u, H ( T ◦ n \ x ) > θ n − v (cid:1) . (4.28)Thus b ν tn ( u, ∞ ) (4.29) ≡ k n ( t ) X x ∈ T ◦ n Z θ n h n,x ( v ) b Q u,vn ( x ) dv (4.30) = k n ( t ) X x ∈ T ◦ n Z θ n h n,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u, H ( T ◦ n \ x ) > θ n − v (cid:1) dv. (4.31)Setting ν ◦ ,tn ( u, ∞ ) ≡ k n ( t ) X x ∈ T ◦ n Z θ n h n,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u (cid:1) dv, (4.32) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 23 we have ν ◦ ,tn ( u, ∞ ) − w tn ( u, ∞ ) ≤ b ν tn ( u, ∞ ) ≤ ν ◦ ,tn ( u, ∞ ) (4.33)where w tn ( u, ∞ ) ≡ k n ( t ) X x ∈ T ◦ n Z θ n h n,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u, H ( T ◦ n \ x ) ≤ θ n − v (cid:1) dv ≤ k n ( t ) X x ∈ T ◦ n Z θ n h n,x ( v ) P x ( H ( T ◦ n \ x ) ≤ θ n − v ) dv ≤ e ν tn . (4.34)Inserting our bounds in (4.18), we finally get that for all u > (cid:12)(cid:12) ν ◦ ,tn ( u, ∞ ) − ¯ ν tn ( u, ∞ ) (cid:12)(cid:12) ≤ e ν tn . (4.35)Our aim now is to prove almost sure convergence of ν ◦ ,tn ( u, ∞ ) . To do so we will needcertain 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 )) − X x ∈ T ◦ n Z θ n h n,x ( v ) E x [ F β,ε,n ( ℓ xn ( θ n − v ))] dv. (4.36)Thus b ◦ n is nothing but b n (see (1.44)) with T ◦ n substituted for T n . The next lemma collectsproperties of the sequences b n and b ◦ n needed in the verification of both Condition (B1)and (B2). Set I n ( a, b ) = ( θ n π n ( T ◦ n )) − P x ∈ T ◦ n J xn ( a, b ) , J xn ( a, b ) = Z ba h n,x ( v ) E x [ F β,ε,n ( ℓ xn ( θ n − v ))] dv, (4.37)and given < 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 > be a sequence satisfying n − | log ζ n | ≪ , and ˜ κ n ( r ⋆n ) α n ( ε )+ o (1) ζ α n ( ε )+ o (1) n ↓ as n ↑ ∞ . (4.38) Then, on Ω ∩ 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 − n ˜ κ n κ α n ( ε ) n ( nr ⋆n ) α n ( ε )+ o (1) , (4.39) ≤ ( b n − b ◦ n ) /b ◦ n ≤ n ( r ⋆n ) α n ( ε )+ o (1) κ α n ( ε ) n − nε n , (4.40) and the right-hand sides of (4.39) and (4.40) decay to zero as n diverges. Furthermore κ − n ( r ⋆n ) −{ α n ( ε )+ o (1) } ≤ b ◦ n ≤ (1 + o (1)) nr ⋆n κ α n ( 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 xn ( κ n , θ n − ζ n ) ≥ J xn, ≡ Z θ n − ζ n κ n h n,x ( v ) E x [ F β,ε,n ( ℓ xn ( θ n − v )) { ζ n <ℓ xn ( θ n − v ) ≤ θ n } ] dv. Since F β,ε,n ( x ) = (1 + o (1)) x α n ( ε )+ o (1) for all ζ n < x ≤ θ n , J xn, ≥ (1 + o (1)) Z θ n − ζ n κ n h n,x ( v ) E x [ ℓ xn ( θ n − v )] α n ( ε )+ o (1) (1 − { ℓ xn ( θ n − v ) <ζ n } ) dv ≡ J xn, − J xn, (4.42) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 24 where we used the left-most inequality in (4.74) to relax the constraint ℓ xn ( θ n − v ) ≤ θ n .Let us bound J xn, for x ∈ I ⋆n . Note that by (2.16) and (2.8) ( r ⋆n ) − ≤ e λ n ( x ) ≤ r ⋆n , ∀ x ∈ I ⋆n . (4.43)Thus, setting ζ ′ n ≡ nr ⋆n , it follows from (3.46) of Lemma 3.11 that for all x ∈ I ⋆n , J xn, ≥ c ( e λ − n ( x )) α n ( ε )+ o (1) Z θ n − ζ ′ n κ n h n,x ( v ) dv (4.44)for some numerical constant < c < ∞ . Summing over x , wet get X x ∈ T ◦ n J xn, ≥ X x ∈ I ⋆n J xn, ≥ c ( r ⋆n ) −{ α n ( ε )+ o (1) } X x ∈ I ⋆n Z θ n − ζ ′ n κ n h n,x ( v ) dv (4.45)where the last sum in the right-hand side of (4.45) is equal to P π n ( κ n < H ( I ⋆n ) < θ n − ζ ′ n , H ( I ⋆n ) < H ( T ◦ n \ I ⋆n )) . (4.46)Decomposing this probability into p − p ≡ P π n ( κ n < H ( I ⋆n ) < θ n − ζ ′ n ) − P π n ( κ n < H ( I ⋆n ) < θ n − ζ ′ n , H ( I ⋆n ) > H ( T ◦ n \ I ⋆n )) we have, by Lemma 3.6 and (3.16), whenever θ n r ⋆n nπ n ( I ⋆n ) → , p ≥ ˜ κ − n θ n π n ( I ⋆n )(1 − θ − n ζ ′ n )(1 + o (1)) = ˜ κ − 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 , write p ≤ P π n ( H ( T ◦ n \ I ⋆n ) < κ n ) + P π n ( H ( T ◦ n \ I ⋆n ) < H ( I ⋆n ) < θ n ) ≡ p + p . (4.48)By (3.18), p ≤ κ n r ⋆n nπ n ( T ◦ n \ I ⋆n )(1 + o (1)) , whereas proceeding as in (4.22)-(4.25), p ≤ X x ∈ T ◦ n \ I ⋆n P π n ( H ( x ) ≤ θ n ) P x ( H ( I ⋆n ) ≤ θ n ) (4.49) = n ( θ n r ⋆n ) π n ( T ◦ n \ I ⋆n ) π n ( I ⋆n )(1 + o (1)) (4.50)where the last equality follows from (3.18) and (3.28). By (2.22), (2.23), and (3.3), on Ω ⋆ and for large enough n , π n ( I ⋆n ) = 2 − nε n (1 − n − c ⋆ (1 + o (1))) and π n ( T ◦ n \ I ⋆n ) = n − c ⋆ +1 − nε n (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 ) → (as requested above (4.47)) and that p = o ( p ) . Thus p − p = p (1 + o (1)) and by this, (4.47), and (4.45), ( θ n π n ( T ◦ n )) − X x ∈ T ◦ n J xn, ≥ ˜ κ − n ( r ⋆n ) −{ α n ( ε )+ o (1) } (1 + o (1)) . (4.51)Turning to J xn, we have X x ∈ T ◦ n J xn, ≤ (1 + o (1)) ζ α n ( ε )+ o (1) n X x ∈ T ◦ n Z θ n − ζ n κ n h n,x ( v ) dv, (4.52)where the last sum is equal to P π n ( κ n < H ( T ◦ n ) < θ n − ζ n ) . Since by Lemma 3.6 and(3.16), P π n ( 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 )) − X x ∈ T ◦ n J xn, ≤ (1 + o (1)) nr ⋆n ζ α n ( ε )+ o (1) n . (4.53) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 25 At this point we may observe that the right-most condition in (4.38) is tailored to guaranteethat P x ∈ T ◦ n J xn, ≫ P x ∈ T ◦ n J xn, . Hence, collecting our bounds, I n ( κ n , θ n − ζ n ) = 1 + o (1) θ n π n ( T ◦ n ) X x ∈ T ◦ n Z θ n − ζ n κ n h n,x ( v ) E x [ ℓ xn ( θ n − v )] α n ( ε )+ o (1) (4.54) ≥ ˜ κ − n ( r ⋆n ) −{ α n ( 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 α n ( ε ) for all < x ≤ θ n together with (3.47) of Lemma 3.11 (which by (4.6) and (3.4) is licit), J xn (0 , κ n ) ≤ (1 + o (1)) κ α n ( ε ) n Z κ n h n,x ( v ) dv. (4.56)Summing over x ∈ T ◦ n and using (3.20) and (4.6) to bound the resulting probability, I xn (0 , κ n ) ≤ (1 + o (1)) nr ⋆n θ − n κ α n ( ε ) n . (4.57)One proves in the same way that I xn (0 , θ n ) ≤ (1 + o (1)) nr ⋆n κ α n ( ε ) n (cid:2) θ α n ( ε ) n r ⋆n nκ − α n ( ε ) n − n (cid:3) , (4.58)where by (4.6) the term in square brackets (that comes from (3.47)) is equal to o (1) .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 toprove (4.40). By definition (see (1.44), (4.36), and the second remark below (4.7) on thedefinition of T n ) | T n | b n − | T ◦ n | b ◦ n = 2 n θ − n X x ∈ T n \ T ◦ n E π n [ F β,ε,n ( ℓ xn ( θ n ))] . (4.59)Conditioning on the time of the first visit to x , and proceeding as in (4.57)-(4.58) to boundthe expectation starting in x , E π n [ F β,ε,n ( ℓ xn ( θ n ))] ≤ (1 + o (1)) P π n ( H ( x ) ≤ θ n ) κ α n ( ε ) n .From this and (3.18), | T n | b n − | T ◦ n | b ◦ n ≤ (1 + o (1)) r ⋆n n n π n ( T n \ T ◦ n ) κ α n ( ε ) n . Now by(2.19)-(2.21), | T n | = | T ◦ n | (1 + o (1)) and | T n \ T ◦ n | = | T ◦ n | n − nε n (1 + o (1)) . Hence b n − b ◦ n ≤ (1 + o (1)) n r ⋆n κ α n ( ε ) n − nε n . Combining this and (4.55) yields (4.40). The proofof Lemma 4.4 is now complete. (cid:3) Proof of Proposition 1.5. This is a straightforward consequence of (4.40), (4.41), the as-sumptions of (1.45), and (1.38). (cid:3) Concentration of ν ◦ ,tn ( u, ∞ ) . Let us now focus on the term ν ◦ ,tn ( u, ∞ ) of (4.32).Recall the definitions of k n ( t ) and b ◦ n from (1.17) and (4.36), respectively. Proposition 4.5. Choose a n = 2 εn /b ◦ n in k n ( t ) and assume that (4.6) holds. Let P ◦ denotethe law of the collection { τ n ( x ) , x ∈ T ◦ n } conditional on T ◦ n , P ◦ ( ∩ x ∈ T ◦ n { τ n ( x ) ∈ ·} ) = P ( ∩ x ∈ T ◦ n { τ n ( x ) ∈ ·} | T ◦ n ) . (4.60) Then, for any sequence u n > such that < u − u n < n − and all u > and t > , P ◦ (cid:18)(cid:12)(cid:12) ν ◦ ,tn ( u n , ∞ ) − E ◦ ν ◦ ,tn ( u n , ∞ ) (cid:12)(cid:12) > n q t Ξ n E ◦ ν ◦ ,tn ( u n , ∞ ) (cid:19) ≤ n − (1 + o (1)) (4.61) where Ξ n ≡ (2 εn /b ◦ n ) nr ⋆n − n and lim n →∞ E ◦ ν ◦ ,tn ( u n , ∞ ) = tu α ( ε ) . (4.62) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 26 Proof of Proposition 4.5. We assume throughout that ω ∈ Ω ⋆ . A key ingredient of theproof is the observation that the generator e L n of Y n is independent of the values of theHamiltonian at its local minima. More precisely, recalling the definition of the set, M n ,of local minima from (2.24), it follows from (2.8) and (2.25) that on Ω ⋆ , for all n largeenough, for all x ∈ M n , and y ∼ x , e λ n ( x, y ) = n − τ n ( y ) and e λ n ( y, x ) = ( n − τ n ( y ) , if y / ∈ V ⋆n ,n − , if y ∈ V ⋆n , (4.63)(note that if x ∈ M n and y ∼ x then y / ∈ M n ). Hence the law of Y n does not depend onthe τ n ( x ) ’s in M n (but it does depend on M n ). Now by (2.15), T ◦ n ⊆ M n ∩ T n ⊆ M n . (4.64)Furthermore, one easily checks that P ◦ in (4.60) is the product measure P ◦ ( ∩ x ∈ T ◦ n { τ n ( x ) ∈ ·} ) = Y x ∈ T ◦ n P ( τ n ( x ) ∈ · , τ n ( x ) ≥ r n ( ε n )) P ( τ n ( x ) ≥ r n ( ε n )) . (4.65)Consequently, for fixed T ◦ n , the collection { X n ( x ) , x ∈ T ◦ n } , X n ( x ) ≡ Z θ n h n,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u n (cid:1) dv, (4.66)viewed as a collection of r.v.’s on the sub-sigma field F ◦ n = σ ( { τ n ( x ) , x ∈ T ◦ n } ) , formsa collection of independent random variables under P ◦ (that of course still depend on thevariables 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 (4.66), E ◦ ν ◦ ,tn ( u n , ∞ ) = k n ( t ) X x ∈ T ◦ n E ◦ X n ( x ) , (4.67)and by independence E ◦ ( ν ◦ ,tn ( u n , ∞ ) − E ◦ ν ◦ ,tn ( u n , ∞ )) ≤ k n ( t ) X x ∈ T ◦ n E ◦ ( X n ( x )) . (4.68)Note that since X n ( x ) ≤ Z θ n h n,x ( v ) dv ≤ P π n ( H ( x ) ≤ θ n ) ≤ θ n r ⋆n n − n (1 + o (1)) , (4.69)(the last inequality is (3.18) combined with (3.3)) then k n ( t ) X x ∈ T ◦ n E ◦ ( X n ( x )) ≤ t (2 εn /b ◦ n ) r ⋆n n − n (1 + o (1)) E ◦ ν ◦ ,tn ( u n , ∞ ) , (4.70)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 (4.70) in (4.68), a second order Tchebychev inequality then yields (4.61).To estimate E ◦ ν ◦ ,tn ( u n , ∞ ) in (4.67) we first use Fubini to write, E ◦ X n ( x ) = Z θ n h n,x ( v ) E x P ◦ (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u n (cid:1) dv. (4.71) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 27 Denoting by P x the law of the single variable τ n ( x ) , P ◦ (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u (cid:1) = P x ( c − n τ n ( x ) ℓ xn ( θ n − v ) > u n , τ n ( x ) ≥ r n ( ε n )) P x ( τ n ( x ) ≥ r n ( ε n )) (4.72) = P x ( c − n τ n ( x ) ℓ xn ( θ n − v ) > u n ) P x ( τ n ( x ) ≥ r n ( ε n )) (4.73)where (4.73) follows from the definition of c n (see (1.4)), the a priory bound ℓ xn ( θ n − v ) ≤ θ n − v ≪ c n , ≤ v ≤ θ n , (4.74)and the fact that δ n in (4.4) in chosen in such a way that θ n r n ( ε n ) r − n ( ε ) ≤ n − ↓ as n ↑ ∞ (see the last inequality in (4.10)). Using classical estimates on the asymptotics ofgaussian integrals (see e.g. [1] p. 932), Lemma 2.2, and again the definition of c n , simplecalculations yield that for all < u < ∞ and ≤ v < θ n , (4.73) is equal to (1 + o (1)) F β,ε,n (cid:16) ℓ xn ( θ n − v ) u n (cid:17) P ( τ n ( x ) > c n ) P ( τ n ( x ) ≥ r n ( ε n )) (4.75)where F β,ε,n ( x ) is defined in (1.41). Furthermore, by (1.4), εn P ( τ n ( x ) ≥ c n ) = 1 whereasby (2.2), (2.20), and (3.3), 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 (4.75), (4.71), (4.67), and using the a priori bound (4.74) that E ◦ ν ◦ ,tn ( u n , ∞ ) = (1 + o (1)) k n ( t ) θ n ( b ◦ n / εn ) I (0 ,θ n ) ( u n ) I (0 ,θ n ) (1) (4.76)where for w > I ( a,b ) ( w ) = X x ∈ T ◦ n Z θ n h n,x ( v ) E x (cid:2) F β,ε,n (cid:0) ℓ xn ( θ n − v ) w (cid:1)(cid:3) { a ≤ ℓ xn ( θ n − v ) , n − log u n ↓ , n − (log u n ) ↓ as n ↑ ∞ . Using that F β,ε,n ( x ) is increasing on the domain (0 , ζ n /u n ) I (0 ,ζ n ) ( u n ) ≤ F β,ε,n (cid:0) ζ n u n (cid:1) P π n ( H ( T ◦ n ) < θ n ) (4.78)where F β,ε,n (cid:0) ζ n u n (cid:1) = e o (1) log u n F β,ε,n ( ζ n ) F β,ε,n ( u − n ) and F β,ε,n ( ζ n ) ≤ e − α n ( ε ) n / − n / / β .By this, (3.18), the lower bound (4.41) on b ◦ n , and our assumptions on u n , I (0 ,ζ n ) ( u n ) I (0 ,θ n ) (1) = e o (1) log u n F β,ε,n ( u − n ) F β,ε,n ( ζ n ) nκ n ( r ⋆n ) α n ( ε )+ o (1) → (4.79)as n → ∞ . Next, since n − log l ↓ as n ↑ ∞ for all ζ n ≤ l ≤ θ n we have, using (4.74), I ( ζ n ,θ n ) ( u n ) I (0 ,θ n ) (1) = e o (1) log u n F β,ε,n ( u − n ) h − I (0 ,ζn ) ( u n ) I (0 ,θn ) (1) i → u − α ( ε ) (4.80)as n → ∞ for all u > . Inserting (4.79) and (4.80) in (4.76), choosing a n = 2 εn /b ◦ n , andpassing to the limit n → ∞ finally gives (4.62). The proof of the lemma is done. (cid:3) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 28 Proof of Proposition 4.1. By (4.6), (4.3)-(4.4), and the bound κ n ≤ θ n , (4.40) im-plies that on Ω ∩ 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 notethat by (4.41), (3.6), (1.38), and (4.6) (see also (4.8)), for all < ε < , (2 εn /b ◦ n ) r ⋆n n − n ≤ κ n ( r ⋆n ) α n ( ε )+ o (1) n nε − n → (4.81)as n → ∞ . Thus, by Proposition 4.5 and Borel-Cantelli Lemma we get that for all u > and all t > , lim n →∞ ν ◦ ,tn ( u, ∞ ) = tu α ( ε ) P − almost surely . (4.82)In the same way we get that for all u > and all t > , lim n →∞ ν ◦ ,tn ( u, ∞ ) = tu α ( ε ) P − almost surely . (4.83)Next, by Lemma 4.2, Lemma 4.3, and (4.35) we have that on Ω ⋆ , for all but a finitenumber of indices n , (cid:12)(cid:12) ˇ ν tn ( u, ∞ ) − ν ◦ ,tn ( u, ∞ ) (cid:12)(cid:12) (4.84) ≤ t ( b ◦ n ) − [2 r ⋆n n θ n − nε +2 δ n n + n c ⋆ +4 nε ( θ n π n ( T ◦ n ) r ⋆n ) ](1 + o (1)) (4.85) ≤ tn c ⋆ +4(1+ α n ( ε )) ( r ⋆n ) α n ( ε )+2+ o (1) κ n θ α ( ε ) n − nε (1 + o (1)) (4.86)where the last inequality follows from (4.41), (2.20), (4.3), and (4.4). Since κ n ≤ θ n , (4.6)(see also (4.8)) implies that (4.86) decays to zero as n → ∞ . From this and (4.82) we getthat for all u > and all t > , lim n →∞ ˇ ν tn ( u, ∞ ) = tu α ( ε ) P -almost surely. One proves inthe same way that for all u > and all t > , lim n →∞ ˇ ν tn ( u − n − , ∞ ) = tu α ( ε ) P -almostsurely. Therefore, by (4.11), for all u > and all t > , lim n →∞ ν tn ( u, ∞ ) = tu α ( ε ) P − almost surely . (4.87)Since ν tn is increasing both in t and u and since its limit continuous in those two variables,(4.87) implies that P -almost surely, lim n →∞ ν tn ( u, ∞ ) = tu α ( ε ) , ∀ u > , t > . (4.88)The proof of Proposition 4.1 is done.5. V ERIFICATION OF C ONDITION (B2)By (1.18)-(1.19), (1.22), and (1.34), Condition (B2) in (1.32) states that σ tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π n ( y ) (cid:20) P y (cid:18)Z θ n max (cid:0) ( c n r ⋆n ) − , c − n τ n ( Y n ( s )) (cid:1) ds > u (cid:19)(cid:21) (5.1)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 < ε < and β > , P -almost surely, lim n →∞ σ tn ( u, ∞ ) = 0 , ∀ t > , u > . (5.2)As in the proof of Proposition 4.1 we first bring σ tn ( u, ∞ ) into a suitable form. Proceed-ing as in (4.11)-(4.12), we first write ˇ σ tn ( u, ∞ ) ≤ σ tn ( u, ∞ ) ≤ ˇ σ tn ( u − n − , ∞ ) (5.3) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 29 where ˇ σ tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π n ( y ) (cid:20) P y (cid:18)Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T n } ds > u (cid:19)(cid:21) , (5.4)and next reduce visits to T n in (5.4) to visits to visits to T ◦ n , just as in Lemma 4.2. Set ¯ σ tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π n ( y ) (cid:20) P y (cid:18)Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T ◦ n } ds > u (cid:19)(cid:21) . (5.5) Lemma 5.2. Assume that (4.6) holds. Then on Ω ⋆ , for all but a finite number of indices n , | ˇ σ tn ( u, ∞ ) − ¯ σ tn ( u, ∞ ) | ≤ k n ( t ) θ n n r ⋆n − nε n (1 + o (1)) . (5.6) Proof of lemma 5.2. As in the Proof of Lemma 4.2 we decompose the event appearing inthe probability in (5.4) according to whether { H ( T n \ T ◦ n ) ≤ θ n } or not, that is, setting q ( y ) = P y (cid:0)R θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T n } ds > u, H ( T n \ T ◦ n ) ≤ θ n (cid:1) , (5.7) q ( y ) = P y (cid:0)R θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T n } ds > u, H ( T n \ T ◦ n ) > θ n (cid:1) , (5.8)we write ˇ σ tn ( u, ∞ ) = k n ( t ) P y ∈V n π n ( y )[ q ( y )+ q ( y )] . In the same way write ¯ σ tn ( u, ∞ ) = k n ( t ) P y ∈V n π n ( y )[¯ q ( y ) + ¯ q ( y )] where ¯ q ( y ) and ¯ q ( y ) are defined as in (5.7) and (5.8),respectively, substituting T ◦ n for T n . Note that [ x + x ] ≤ x + x , ≤ x , x ≤ . (5.9)Applying (5.9) to the terms [ q ( y )+ q ( y )] and [¯ q ( y )+¯ q ( y )] , and observing that q = ¯ q ,we get | ˇ σ tn ( u, ∞ ) − ¯ σ tn ( u, ∞ ) | ≤ k n ( t ) X y ∈V n π n ( y )( q ( y ) + ¯ q ( y )) (5.10) ≤ k n ( t ) P π n ( H ( T n \ T ◦ n ) ≤ θ n ) . (5.11)The Lemma now follows from (3.19) of Corollary 3.8. (cid:3) We continue our parallel with the proof of Proposition 4.1 and decompose (5.5) ac-cording to the hitting time and hitting place of the set T ◦ n . We slightly abuse the notationof Section 3 (see the paragraph below (3.30)) and denote by h yn,x (instead of h yn,x,T ◦ n ) thejoint 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 = P y ∈V n π n ( y ) h yn,x . Proceeding as in (4.18)-(4.20) we then get ¯ σ tn ( u, ∞ ) = k n ( t ) X y ∈V n π n ( y ) (cid:2) R un ( y ) (cid:3) (5.12)where, using (4.19) and (4.20), R un ( y ) ≡ X x ∈ T ◦ n Z θ n h yn,x ( v ) (cid:16) e Q u,vn ( x ) + b Q u,vn ( x ) (cid:17) dv ≡ e R un ( y ) + b R un ( y ) . (5.13)By analogy with (4.30) we also set b σ tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π n ( y ) (cid:2) b R un ( y ) (cid:3) . (5.14)The next lemma plays the role of Lemma 4.3. GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 30 Lemma 5.3. Assume that (4.6) holds. Then Ω ⋆ , for all but a finite number of indices n , ≤ ¯ σ tn ( u, ∞ ) − b σ tn ( u, ∞ ) ≤ k n ( t ) n c ⋆ +4 ( θ n π n ( T ◦ n ) r ⋆n ) (1 + o (1)) . (5.15) Proof of Lemma 5.3. As in the proof of Lemma 5.2, the proof of Lemma 5.3 relies on theobservation that since ≤ e R un ( y ) , b R un ( y ) ≤ in (5.13) for all y ∈ V n , then by (5.9), < ¯ σ tn ( u, ∞ ) − b σ tn ( u, ∞ ) ≤ k n ( t ) X y ∈V n π n ( y ) e R un ( y ) (5.16) = 3 k n ( t ) X x ∈ T ◦ n Z θ n h n,x ( v ) e Q u,vn ( x ) dv ≤ e ν tn . (5.17)The equality in (5.17) follows from the identity h n,x ( v ) = P y ∈V n π n ( y ) h yn,x ( v ) , and thefinal inequality is (4.24). The claim of the lemma now follows from Lemma 4.3. (cid:3) We now need an upper bound on b σ tn ( u, ∞ ) . For this we proceed as in (4.31)-(4.33) andwrite that ≤ b σ tn ( u, ∞ ) ≤ σ ◦ ,tn ( u, ∞ ) where, by analogy with (4.33), σ ◦ ,tn ( u, ∞ ) = k n ( t ) X y ∈V n π n ( y ) " X x ∈ T ◦ n Z θ n h yn,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u (cid:1) dv (5.18)Again, the quantity in between the square brackets is in [0 , . Thus, splitting the integralinto the sum of the integrals over [0 , κ n ] and [ κ n , θ n ] , we get, using (5.9) and reasoning asin (5.16)-(5.17), σ ◦ ,tn ( u, ∞ ) ≤ η ◦ ,tn ( u, ∞ ) + η ◦ ,tn ( u, ∞ ) (5.19)where ¯ η ◦ ,tn ( u, ∞ ) ≡ k n ( t ) X x ∈ T ◦ n Z κ n h n,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u (cid:1) dv, (5.20) η ◦ ,tn ( u, ∞ ) ≡ k n ( t ) X y ∈V n π n ( y ) " X x ∈ T ◦ n Z θ n κ n h yn,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u (cid:1) dv . (5.21)The next two propositions bound (5.20) and (5.21) in terms of the quantities ν ◦ ,tn ( u n , ∞ ) and E ◦ ν ◦ ,tn ( u n , ∞ ) defined in (4.32) and (4.67), respectively. Proposition 5.4. Choose a n = 2 εn /b ◦ n in (1.17). Then, for any sequence u n > such that < u − u n < n − and all u > , P (cid:0) ¯ η ◦ ,tn ( u n , ∞ ) ≥ t E ◦ ν ◦ ,tn ( u n , ∞ ) n θ − n ˜ κ n κ α n ( ε ) n ( nr ⋆n ) α n ( ε )+ o (1) (cid:1) ≤ n − . (5.22) Proposition 5.5. On Ω ⋆ ∩ Ω , for all but a finite number of indices n and all u > , η ◦ ,tn ( u, ∞ ) ≤ ν ◦ ,tn ( u, ∞ ) θ n r ⋆n n − nε n (1 + o (1)) . (5.23) Proof of Proposition 5.4. As in the proof of Proposition 4.5 denote by P ◦ the law of thecollection { τ n ( x ) , x ∈ T ◦ n } conditional on T ◦ n . By a first order Tchebychev inequality, P (cid:0) ¯ η ◦ ,tn ( u n , ∞ ) ≥ ǫ (cid:1) ≤ ǫ − E (cid:2) E ◦ ¯ η ◦ ,tn ( u n , ∞ ) (cid:3) . (5.24)Note that E ◦ ¯ η ◦ ,tn ( u, ∞ ) only differs from the term E ◦ ν ◦ ,tn ( u n , ∞ ) of (4.67) in that the inte-gral in (5.20) is over [0 , κ n ] instead of [0 , θ n ] . Taking a n = 2 εn /b ◦ n , a simple adaptation ofthe proof of (4.62) (see (4.71)-(4.80)) yields E ◦ ¯ η ◦ ,tn ( u n , ∞ ) = t (1 + o (1)) E ◦ ν ◦ ,tn ( u n , ∞ ) I n (0 , κ n ) I n (0 , θ n ) (5.25) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 31 where I n ( a, b ) is defined above (4.37). Eq. (4.39) of Lemma 4.4 was designed preciselyto control the ratio in (5.25). 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 − n ˜ κ n κ α n ( ε ) n ( nr ⋆n ) α n ( ε )+ o (1) . (5.26)The combination of (5.24), (5.25), and (5.26) gives (5.22). The proof is complete. (cid:3) Proof of Proposition 5.5. To prove (5.23) first observe that X x ∈ T ◦ n Z θ n κ n h yn,x ( v ) P x (cid:0) c − n τ n ( x ) ℓ xn ( θ n − v ) > u (cid:1) dv ≤ P y ( κ n < H ( T ◦ n ) ≤ θ n ) (5.27) ≤ (1 + o (1)) P π n ( H ( T ◦ n ) ≤ θ n ) (5.28)where the last line follows from Proposition 3.3 and the Markov property, and is valid on Ω , for all but a finite number of indices n . Applying this bound to one of the two squarebrackets in (5.21) and using (4.32) to bound the remaining term, we get, under the sameassumptions as above, that η ◦ ,tn ( u, ∞ ) ≤ (1 + o (1)) ν ◦ ,tn ( u, ∞ ) P π n ( H ( T ◦ n ) ≤ θ n ) . (5.29)Using Corollary (3.20) to bound the last probability yields the claim of the proposition. (cid:3) We are now ready to complete the Proof of Proposition 5.1. Recall from the proof of Proposition 4.1 that on Ω ∩ Ω ◦ ∩ Ω ⋆ a n = 2 εn /b n = 2 εn /b ◦ n (1 + o (1)) for large enough n and consider (5.22). By (4.5), n θ − n ˜ κ n κ α n ( ε ) n ( nr ⋆n ) α n ( ε )+ o (1) ↓ as n ↑ ∞ and by (4.62), for all u > and t > n →∞ E ◦ ν ◦ ,tn ( u n , ∞ ) = tu α ( ε ) . Thus, by Proposition 5.4 and Borel-Cantelli Lemma weget that for all u > and t > , lim n →∞ ¯ η ◦ ,tn ( u, ∞ ) = 0 P − almost surely . (5.30)Turning to (5.23) and invoking (4.6) (see also (4.8)), it follows from Proposition 5.4 thatfor all < ε < and for all u > and t > , lim n →∞ η ◦ ,tn ( u, ∞ ) = 0 P − almost surely . (5.31)Hence by (5.19), for all u > and t > , lim n →∞ σ ◦ ,tn ( u, ∞ ) = 0 P − almost surely . (5.32)From there on the proof is a rerun of the proof of Proposition 4.1 with Lemma 5.2 andLemma 5.3 playing the role of Lemma 4.2 and Lemma 4.3, respectively. We omit thedetails. (cid:3) 6. V ERIFICATION OF C ONDITION (B3)By (1.18)-(1.20), (1.22), and (1.34), Condition (B3) in (1.33) will be verified if we canestablish that: Proposition 6.1. Under the assumptions of Proposition 4.1, for all < ε < and all β > β c ( ε ) , P -almost surely, lim ǫ ↓ lim sup n ↑∞ k n ( t ) E π n Z θ n M n ( Y n ( s )) { R θn M n ( Y n ( s )) ds ≤ ǫ } = 0 , ∀ t > . (6.1) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 32 where M n ( Y n ( s )) = max (( c n r ⋆n ) − , c − n τ n ( Y n ( s ))) . The Lemma below is central to the proof. Lemma 6.2. There are constants K, K ′ < ∞ such that for α n ( ε ) as in (1.42) and anysequence ǫ n > such that iα − c ( ε ) − − log ǫ n nββ c ( ε ) > where i = 1 in (6.2) and i = 2 in(6.3), we have, for all large enough n , E εn c − n τ n ( x ) { c − n τ n ( x ) ≤ ǫ n } ≤ K ǫ − α n ( ε ) − log ǫn nβ n α − c ( ε ) − − log ǫ n nββ c ( ε ) , (6.2) E (cid:16) εn c − n τ n ( x ) { c − n τ n ( x ) ≤ ǫ n } (cid:17) ≤ K ′ ǫ − α n ( ε ) − log ǫn nβ n α − c ( ε ) − − log ǫ n nββ c ( ε ) . (6.3) Proof of Lemma 6.2. Using standard estimates on the asymptotics of Gaussian integrals(see e.g. [1] p. 932) the claimed result follows from straightforward computations. (cid:3) Proof of Proposition 6.1. We assume throughout that ω ∈ Ω ∩ Ω ◦ ∩ Ω ⋆ and that n is aslarge as desired. Note that M n ( Y n ( s )) ≤ ( c n r ⋆n ) − + c − n τ n ( Y n ( s )) and that the contribu-tion to (6.1) coming from the term ( c n r ⋆n ) − if or order o (1) . Indeed by (1.17), (1.40), thelower bound on b n obtained by combining (4.41) and (4.40), the expression (1.5) of c n ,the expression (3.6) of κ n , and the fact, that follows from (1.6), that n = e nβ c ( ε ) / , k n ( t ) θ n ( c n r ⋆n ) − ≤ tn ( r ⋆n ) α n ( ε )+ o (1) e nβ c ( ε ) / e − nββ c ( ε )(1+ o (1)) (6.4)and so, for all < ε < and β > β c ( ε ) , by virtue of (4.6) (see also (4.8)) k n ( t ) θ n ( c n r ⋆n ) − ≤ tn ( r ⋆n ) α n ( ε )+ o (1) e − nβ c ( ε )(1+ o (1)) / → (6.5)as n → ∞ . To prove Proposition 6.1 it thus suffices to establish that P -almost surely, lim ǫ ↓ lim sup n ↑∞ k n ( t ) E π n Z θ n c − n τ n ( Y n ( s )) { R θn c − n τ n ( Y n ( s )) ds ≤ ǫ } = 0 , ∀ t > . (6.6)For T n as in (2.12) with δ n given by (4.4), set S (1) n,ǫ ( t ) ≡ k n ( t ) E π n Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T n } { R θn c − n τ n ( Y n ( s )) ds ≤ ǫ } ds, (6.7) S (2) n,ǫ ( t ) ≡ k n ( t ) E π n Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) / ∈ T n } { R θn c − n τ n ( Y n ( s )) ds ≤ ǫ } ds. (6.8)To bound S (2) n,ǫ ( t ) simply note that, using (3.4), S (2) n,ǫ ( t ) ≤ k n ( t ) E π n Z θ n c − n τ n ( Y n ( s )) { τ n ( Y n ( s )) ≤ r n ( ε n ) } ds (6.9) ≤ k n ( t ) θ n − n (1 + o (1)) X x ∈V n c − n τ n ( x ) { τ n ( x ) ≤ r n ( ε n ) } . (6.10)Take ǫ n = c − n r n ( ε n ) and note that by (2.29), the definition of c n , and (4.6), − ( nββ c ( ε )) − log ǫ n = o (1) and (cid:16) n c ⋆ /α ( ε )) θ n (cid:17) − ≤ ǫ n ≤ ( n θ n ) − . (6.11)Thus, by Lemma 6.2 and a first order Tchebychev inequality, for all large enough n , P (cid:0) S (2) n,ǫ ( t ) ≥ n tb − n ( c − n r n ( ε n )) − α ( ε )+ o (1) (cid:1) ≤ n − K ′′ (6.12) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 33 for some constant K ′′ > . Using the upper bound on ǫ n of (6.11) and the lower bound on b n of Lemma 4.4 obtained by combining (4.41) and (4.40), n b − n ( c − n r n ( ε n )) − α ( ε )+ o (1) ≤ n κ n ( r ⋆n ) α n ( ε )+ o (1) (cid:0) n θ n (cid:1) − α ( ε )+ o (1) → (6.13)as n → ∞ by (4.5). Hence by (6.12), (6.13), and Borel-Cantelli Lemma, for all ǫ > , lim n →∞ S (2) n,ǫ ( t ) = 0 , P − almost surely. (6.14)To deal with S (1) n,ǫ ( t ) we further decompose it into S (1) n,ǫ ( t ) = S (3) n,ǫ ( t ) + S (4) n,ǫ ( t ) , where S (3) n,ǫ ( t ) ≡ k n ( t ) E π n Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T ◦ n } { R θn c − n τ n ( Y n ( s )) ds ≤ ǫ } ds, (6.15) S (4) n,ǫ ( t ) ≡ k n ( t ) E π n Z θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ T n \ T ◦ n } { R θn c − n τ n ( Y n ( s )) ds ≤ ǫ } ds. (6.16)Since S (4) n,ǫ ( t ) is non zero only if the event { H ( T n \ T ◦ n ) ≤ θ n } occurs, S (4) n,ǫ ( t ) ≤ ǫk n ( t ) E π n { H ( T n \ T ◦ n ) ≤ θ n } . (6.17)Using assertion (ii) of Corollary 3.8 with t n = θ n as in the proof of Lemma 4.2, we get,assuming (4.6), that on Ω ⋆ , for all but a finite number of indices n , S (4) n,ǫ ( t ) ≤ ǫk n ( t ) θ n r ⋆n n − nε n (1 + o (1)) , (6.18)Proceeding as in (6.13) to bound b n , (4.6) (see also (4.8)) guarantees that for all ǫ > n →∞ S (4) n,ǫ ( t ) = 0 , P − almost surely. (6.19)Using next that R θ n c − n τ n ( Y n ( s )) { Y n ( s ) ∈ A } = P x ∈ A c − n τ n ( x ) ℓ xn ( θ n ) for any A ⊆ V n , S (3) n,ǫ ( t ) ≤ S (5) n,ǫ ( t ) ≡ k n ( t ) E π n X x ∈ T ◦ n c − n τ n ( x ) ℓ xn ( θ n ) { P x ∈ T ◦ n c − n τ n ( x ) ℓ xn ( θ n ) ≤ ǫ } . (6.20)With the notation of (4.15)-(4.17), S (5) n,ǫ ( t ) = k n ( t ) X y ∈ T ◦ n Z θ n dvh n,y ( v ) E y X x ∈ T ◦ n c − n τ n ( x ) ℓ xn ( θ n − v ) { P x ∈ T ◦ n c − n τ n ( x ) ℓ xn ( θ n − v ) ≤ ǫ } . We further split the sum over x above into x = y and x = y . The latter contribution is S (6) n,ǫ ( t ) ≡ k n ( t ) X y ∈ T ◦ n Z θ n dvh n,y ( v ) E y X x ∈ T ◦ n \ y c − n τ n ( x ) ℓ xn ( θ n − v ) { P x ∈ T ◦ n c − n τ n ( x ) ℓ xn ( θ n − v ) ≤ ǫ } . Observing that E y X x ∈ T ◦ n \ y c − n τ n ( x ) ℓ xn ( θ n − v ) { P x ∈ T ◦ n c − n τ n ( x ) ℓ xn ( θ n − v ) ≤ ǫ } ≤ ǫP y ( H ( T ◦ n \ y ) ≤ θ n ) , (6.21)yields the bound S (6) n,ǫ ( t ) ≤ ǫ e ν tn where e ν tn is defined in (4.25). Thus by Lemma 4.3, reason-ing as in the paragraph below (4.86), we get that for all ǫ > n →∞ S (6) n,ǫ ( t ) = 0 , P − almost surely. (6.22)It remains to bound S (5) n,ǫ ( t ) − S (6) n,ǫ ( t ) . For this we write S (5) n,ǫ ( t ) − S (6) n,ǫ ( t ) ≤ S (7) n,ǫ ( t ) where S (7) n,ǫ ( t ) ≡ k n ( t ) X y ∈ T ◦ n Z θ n dvh n,y ( v ) E y c − n τ n ( y ) ℓ yn ( θ n − v ) { c − n τ n ( y ) ℓ xn ( θ n − v ) ≤ ǫ } . (6.23) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 34 Let us now establish that for b ◦ n as in (4.36), S (7) n,ǫ ( t ) obeys the following Lemma 6.3. Let the sequences a n , c n , θ n be as in Proposition 6.1. Then, under theassumptions and with the notation of Proposition 4.5, P ◦ (cid:0)(cid:12)(cid:12) S (7) n,ǫ ( t ) − E ◦ S (7) n,ǫ ( t ) (cid:12)(cid:12) > tǫ / n − n (1 − ε ) / (cid:1) ≤ n − (1 + o (1)) (6.24) for all ǫ > , and lim ǫ → lim n →∞ E ◦ S (7) n,ǫ ( t ) = 0 . (6.25) Proof of lemma 6.3. The proof closely follows that of Proposition 4.5. We only point outthe main differences. The random variables (4.66) are now replaced by X n ( y ) ≡ Z θ n dvh n,y ( v ) E y c − n τ n ( y ) ℓ yn ( θ n − v ) { c − n τ n ( y ) ℓ yn ( θ n − v ) ≤ ǫ } (6.26)and E ◦ S (7) n,ǫ ( t ) = k n ( t ) P y ∈ T ◦ n E ◦ X n ( y ) . (6.27)Proceeding as in (4.72)-(4.74) to deal with the conditional expectation and using that P ( τ n ( x ) ≥ r n ( ε n )) = π n ( T ◦ n )(1 + o (1)) (see the paragraph below (4.75)), we get E ◦ S (7) n,ǫ ( t ) = k n ( t )(1 + o (1)) π n ( T ◦ n ) X y ∈ T ◦ n Z θ n dvh n,y ( v ) E y ℓ yn ( θ n − v ) E y c − n τ n ( y ) { c − n τ n ( y ) ≤ ǫ n } where P y denotes the law of τ n ( y ) and where ǫ n ≡ ǫ n ( y ) = ǫ/ℓ yn ( θ n − v ) . Using (6.2) if ℓ yn ( θ n − v ) > ǫe − nββ c ( ε )( α − c ( ε ) − and using that if ℓ yn ( θ n − v ) ≤ ǫe − nββ c ( ε )( α − c ( ε ) − then E y ℓ yn ( θ n − v ) E y c − n τ n ( y ) { c − n τ n ( y ) ≤ ǫ n } ≤ ǫe − nββ c ( ε )( α − c ( ε ) − c − n e nβ / , (6.28)we readily see that E ◦ S (7) n,ǫ ( t ) ≤ C t ǫ − α n ( ε ) − log ǫ nβ b ◦ n θ n π n ( T ◦ n ) X y ∈ T ◦ n Z θ n dvh n,y ( v ) E y e F β,ε,ǫ,n ( ℓ yn ( θ n − v ))+ C ǫn α n ( ε ) / e − nβ / k n ( t )( π n ( T ◦ n )) − P π n ( H ( T ◦ n ) ≤ θ n ) (6.29)where here and below C i > , i = 1 , , . . . are constants, and for F β,ε,n as in (1.41), e F β,ε,ǫ,n ( z ) = F β,ε,n ( z ) z log ǫnβ (cid:16) − log znββ c ( ε ) (cid:17) α − c ( ε ) − − log ǫnββ c ( ε ) + log znββ c ( ε ) n z>ǫe − nββc ( ε )( α − c ( ε ) − o . (6.30)By the leftmost inequality of (4.74) and (4.6), e F β,ε,ǫ,n ( z ) ≤ C F β,ε,n ( z ) . Thus, by (4.36),the first summand in (6.29) is bounded above by C tǫ − α n ( ε ) − log ǫ nβ . (6.31)Using (3.20) and proceeding as in (6.4) to bound k n ( t ) , the second summand is boundedabove by C te − n ( β − β c ( ε )) / κ n n α n ( ε ) / ( r ⋆n ) α n ( ε )+ o (1) → (6.32)as n → ∞ by virtue of (3.6), (1.38), and the assumption that β > β c ( ε ) where < ε < .Note in particular that lim n →∞ α n ( ε ) = α ( ε ) < . Hence, inserting (6.31) and (6.32) in(6.29) and passing to the limit lim ǫ → lim sup n →∞ E ◦ S (7) n,ǫ ( t ) = 0 , ∀ t > . (6.33) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 35 This proves (6.25). Turning to the variance we have, as in (4.68), by independence, that V ◦ ( S (7) n,ǫ ( t )) ≡ E ◦ ( S (7) n,ǫ ( t ) − E ◦ S (7) n,ǫ ( t )) ≤ k n ( t ) X y ∈ T ◦ n E ◦ ( X n ( y )) . (6.34)Proceeding as in the proof of (6.29) but using (6.3) and the line below (6.30), we get that V ◦ ( S (7) n,ǫ ( t )) ≤ C t ǫ − α n ( ε ) − log ǫ nβ ( b ◦ n θ n ) π n ( T ◦ n ) X y ∈ T ◦ n (cid:18)Z θ n dvh n,y ( v ) E y F β,ε,ǫ,n ( ℓ yn ( θ n − v )) (cid:19) + C ǫn α n ( ε ) / e − nββ c ( ε ) k n ( t ) θπ n ( T ◦ n ) X y ∈ T ◦ n (cid:18)Z θ n dvh n,y ( v ) (cid:19) . From the bound R θ n dvh n,y ( v ) E y F β,ε,ǫ,n ( ℓ yn ( θ n − v )) ≤ (1 + o (1)) R θ n dvh n,y ( v ) θ α n ( ε ) n ≤ (1 + o (1)) θ α n ( ε ) n P π n ( H ( y ) ≤ θ n ) and (3.18), (4.41), we get that on Ω ⋆ , for all but a finitenumber of indices n , the first summand is bounded above by C t ǫ − α n ( ε ) − log ǫ nβ (cid:0) nκ n θ α n ( ε ) n ( r ⋆n ) α n ( ε )+ o (1) (cid:1) − n . (6.35)Using the bound P y ∈ T ◦ n (cid:0)R θ n dvh n,y ( v ) (cid:1) ≤ sup y ∈ T ◦ n P π n ( H ( y ) ≤ θ n ) P π n ( H ( T ◦ n ) ≤ θ n ) ,and proceeding as in (6.32), the second summand is bounded above by C t ǫn α n ( ε ) / (cid:0) n κ n ( r ⋆n ) α n ( ε )+ o (1) (cid:1) θ n e − nβ c ( ε )( β − β c ( ε )) − n . (6.36)Since by assumption β > β c ( ε ) and < ε < , (4.6) (see also (4.8)) enables us toconclude that on Ω ⋆ , for all large enough n , V ◦ ( S (7) n,ǫ ( t )) ≤ C t ǫ − n (1 − ε ) . (6.37)This yields (6.24) and concludes the proof of the Lemma. (cid:3) Arguing as in the proof of Proposition 4.1 that b n = b ◦ n (1 + o (1)) on Ω ∩ Ω ◦ ∩ Ω ⋆ forall large enough n , it follows from Lemma 6.3 and Borel-Cantelli Lemma that lim ǫ → lim n →∞ (cid:0) S (5) n,ǫ ( t ) − S (6) n,ǫ ( t ) (cid:1) = 0 , P − almost surely. (6.38)Collecting (6.14), (6.19), (6.22) and (6.38) yields (6.6). The proof of Proposition 6.1 iscomplete. (cid:3) 7. P ROOF OF T HEOREM AND T HEOREM Proof of Theorem 1.4. By Proposition (3.3), Proposition (4.1), Proposition (5.1) and Propo-sition (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 Condi-tion (A0), i.e. to prove that P -a.s., for all u > , lim n →∞ P µ n ( Z n, > u ) = 0 (7.1)where Z n, = R θ n max (( c n r ⋆n ) − , c − n τ n ( Y n ( s ))) ds and µ n is the uniform measure on V n .By (3.3) and (3.4) P µ n ( Z n, > u ) ≤ (1 + o (1)) P π n ( Z n, > u ) + P x ∈ V ⋆n µ n ( x ) P x ( Z n, > u ) (7.2) ≤ (1 + o (1)) P π n ( Z n, > u ) + n − c ⋆ (1 + o (1)) (7.3)where the last line is (2.18). 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 GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 36 conditions (4.5) and (4.6) of Proposition (4.1) are verified. The proof of 1.4 is complete. (cid:3) Proof of Theorem 1.1. Reasoning as in the proof of Theorem 1.4, we may assume that theprocess starts in its invariant measure π n . The main idea behind the proof is now classical.Suppose that P π n ( A n ( t, s )) = P π n ( {R n ∩ ( t, t + s ) = ∅} ) + o (1) (7.4)where A n ( t, s ) ≡ { X ( c n t ) = X ( c n ( t + s )) } and where R n denotes the range of therescaled blocked clock process S bn ( t ) . Then Theorem 1.1 is a direct consequence of The-orem 1.4 and the arcsine law for stable subordinators. We refer to Ref. [23] for a detailedproof of this statement (see the proof of Theorem 1.6 therein) and focus on establishing(7.4). For k ≥ and Z n,i as in (1.19) set B k = nP ki =1 Z n,i < t, P k +1 i =1 Z n,i > t + s o . (7.5)Then by (1.18), {R n ∩ ( t, t + s ) = ∅} = {∪ k ≥ B k } . Furthermore, for any T > , P π n (cid:0) ∪ k>k n ( T ) B k (cid:1) ≤ P π n (cid:0) S bn ( T ) < t (cid:1) −→ n →∞ P (cid:0) V α ( ε ) ( T ) < t (cid:1) ≤ δ (7.6)where convergence is almost sure in the random environment as follows from Theorem1.4, and where δ can be made as small as desired by taking T large enough. Therefore ≤ P π n ( {R n ∩ ( t, t + s ) = ∅} ) − P π n (cid:0) ∪ ≤ k ≤ k n ( T ) B k (cid:1) ≤ δ. (7.7)Note that the event B k is non empty if and only if the increment Z n,k +1 straddles overthe interval ( t, t + s ) . To show that (7.4) holds it now suffices to establish the followingtwo facts: Fact 1. Almost surely in the random environment, with overwhelming probability, non-empty events B k , k ≤ k n ( T ) , are produced by visits of the process Y n to the set T ◦ n and,more precisely, by (many) visits of the process to one and the same element of T ◦ n , noother element of T ◦ n being visited in the time interval ( t, t + s ) . This implies that P -a.s. P π n (cid:0) A n ( t, s ) ∩ {∪ ≤ k ≤ k n ( T ) B k } (cid:1) ≥ P π n (cid:0) ∪ ≤ k ≤ k n ( T ) B k (cid:1) + o (1) (7.8) Fact 2. If B k and B ′ k , ≤ k = k ′ ≤ k n ( T ) , are two non-empty events then, almost surelyin the random environment they are produced by visits to two distinct elements of T ◦ n withoverwhelming probability. This implies that P -a.s. P π n (cid:0) A n ( t, s ) ∩ ( ∩ ≤ k ≤ k n ( T ) B ck ) (cid:1) → , n → ∞ (7.9)Combining (7.7), (7.8), and (7.9) then establishes that | P π n ( A n ( t, s )) − P π n ( {R n ∩ ( t, t + s ) = ∅} ) | ≤ δ + o (1) (7.10)which is tantamount to (7.4).The proofs of Facts 1 and 2 mostly use information already obtained in the course of theverification of Conditions (B1)-(B3). We present them succinctly below, beginning withthe proof of Fact 1. Fix < T < ∞ and assume that the assumption of Proposition (4.1)are satisfied. Let H k ( A ) = inf { t ≥ θ n k | Y n ( t ) ∈ A } be the first hitting time of A ⊆ V n after time θ n k . Note first that B k = B k ∩ { Z n,k +1 > s } and thus, by (4.10), P π n (cid:0) ∪ ≤ k ≤ k n ( T ) ( B k ∩ { H k ( T n ) > θ n } ) (cid:1) = 0 (7.11)for all large enough n . Note next that reasoning as in (6.17)-(6.19), on Ω ◦ ∩ Ω ⋆ , P π n (cid:0) ∪ ≤ k ≤ k n ( T ) ( B k ∩ { H k ( T n \ T ◦ n ) ≤ θ n } ) (cid:1) ≤ k n ( T ) P π n ( H k ( T n \ T ◦ n ) ≤ θ n ) → GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 37 as n → ∞ by virtue of (4.6). Hence on Ω ◦ ∩ Ω ⋆ , for all large enough n , P π n (cid:0) ∪ ≤ k ≤ k n ( T ) B k (cid:1) = P π n (cid:0) ∪ ≤ k ≤ k n ( T ) ( B k ∩ { H k ( T ◦ n ) ≤ θ n } ∩ { H k ( T n \ T ◦ n ) > θ n ) } (cid:1) + o (1) . (7.12)This means that for B k to be non-empty, the increment Z n,k +1 must be produced by visitsof Y n to T ◦ n , and T ◦ n only. To prove that all these visits, if there are several of them, mustbe to a single vertex it suffices to show that as n → ∞ , p n ≡ P π n (cid:0) ∪ ≤ k ≤ k n ( T ) ( B k ∩ { H k ( T ◦ n ) ≤ θ n } ∩ C n ( Y n ( H k ( T ◦ n ))) (cid:1) → , (7.13)where C n ( Y n ( H k ( T ◦ n ))) ≡ (cid:8) inf { t > H k ( T ◦ n ) | Y n ( t ) ∈ T ◦ n \ Y n ( H k ( T ◦ n )) } ≤ θ n (cid:9) . (7.14)Now, p n = P π n (cid:0) ∪ ≤ k ≤ k n ( T ) ∪ x ∈ T ◦ n ( B k ∩ { H k ( T ◦ n ) ≤ θ n , Y n ( H k ( T ◦ n )) = x } ∩ C n ( x ) (cid:1) ≤ e ν Tn (7.15)where e ν Tn is defined in (4.25) and bounded in Lemma 4.3. Reasoning as in the para-graph below (4.86) then yields that under the assumptions (4.5) and (4.6), on Ω ◦ ∩ Ω ⋆ , lim n →∞ e ν Tn = 0 . Fact 1 is now proved.Fact 2 will be established if we can prove that as n → ∞ , ¯ p n ≡ P π n (cid:0) ∪ ≤ k ≤ k n ( T ) ( { H k ( T ◦ n ) ≤ θ n } ∩ D n,k ( Y n ( H k ( T ◦ n )))) (cid:1) → , (7.16)where D n ( Y n ( H k ( T ◦ n ))) ≡ (cid:8) inf { t > ( k + 1) θ n | Y n ( t ) = Y n ( H k ( T ◦ n )) } ≤ θ n k n ( T ) (cid:9) . (7.17)To prove this observe that the event in (7.16) can be written as ∪ x ∈ T ◦ n ∪ y ∈ T ◦ n ( { H k ( T ◦ n ) ≤ θ n , Y n ( H k ( T ◦ n )) = x } ∩ { Y n ( θ n ( k + 1)) = y } ∩ D n,k ( x )) Thus, by the Markov property we have, using the notation of (4.15)-(4.17) and the bound P y ( H ( x ) ≤ θ n ( k n ( T ) − ( k + 1))) ≤ P y ( H ( x ) ≤ θ n k n ( T )) , ¯ p n ≤ X ≤ k ≤ k n ( T ) X x ∈ T ◦ n X y ∈ T ◦ n Z θ n dvh n,x ( v ) P x ( Y n ( θ n − v ) = y ) P y ( H ( x ) ≤ θ n k n ( T )) . To proceed, we split the domain of integration into [0 , θ n − κ n ) ∪ [ θ n − κ n , θ n ] . Using thatby Proposition 3.3, on Ω , for all n large enough, P x ( Y n ( θ n − v ) = y ) = π n ( y )(1 + o (1)) for all v ∈ [0 , θ n − κ n ) , the contribution coming from this domain is at most (1 + o (1)) X ≤ k ≤ k n ( T ) X x ∈ T ◦ n Z θ n dvh n,x ( v ) X y ∈ T ◦ n π n ( y ) P y ( H ( x ) ≤ θ n k n ( T )) (7.18) ≤ (1 + o (1)) k n ( T ) P π n ( H ( T ◦ n ) ≤ θ n ) sup y ∈ T ◦ n P π n ( H ( y ) ≤ θ n k n ( T )) (7.19) ≤ (1 + o (1)) (cid:0) θ n k n ( T ) r ⋆n n − n (cid:1) n π n ( T ◦ n ) (7.20)where we used (3.20) with t n = θ n (which is licit as we many times saw) and (3.18) with t n = θ n k n ( T ) , which is licit provided that θ n k n ( T ) r ⋆n n − n → as n → ∞ , and this isguaranteed by our assumptions on a n . Indeed, proceeding as in the proof of Proposition4.1 (see (4.81) and the paragraph above) we get that on Ω ◦ ∩ Ω ⋆ ∩ Ω , for large enough n , θ n k n ( T ) r ⋆n n − n ≤ κ n ( r ⋆n ) α n ( ε )+ o (1) n − (1 − ε ) n → (7.21) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 38 as n → ∞ for all < ε < . Since furthermore n π n ( T ◦ n ) = (1 + o (1))2 (1 − ε ) n ( n θ n ) α ( ε ) by (4.4), (3.3), and (2.20), and we get that on Ω ◦ ∩ Ω ⋆ ∩ Ω , (7.20) is bounded above by (1 + o (1)) (cid:0) κ n ( r ⋆n ) α n ( ε )+ o (1) n (cid:1) (cid:0) n θ n (cid:1) α ( ε ) − (1 − ε ) n , (7.22)and by (4.6) this decays to zero as n → ∞ for all < ε < .Consider next the domain [ θ n − κ n , θ n ] and note that since X y ∈ T ◦ n P x ( Y n ( θ n − v ) = y ) P y ( H ( x ) ≤ θ n k n ( T )) ≤ (7.23)the corresponding contribution is bounded above by k n ( T ) P π n ( θ n − κ n ≤ H ( T ◦ n ) ≤ θ n ) .By the upper bound of (3.6) and the lower bound of (3.5), on Ω ⋆ , for all but a finite numberof indices n , this is in turn bounded above by n α n ( ε ) θ − (1 − α ( ε )) n κ n ( r ⋆n ) α n ( ε )+ o (1) → (7.24)as n → ∞ , where we again used that nδ n = ( n θ n ) α ( ε ) by (4.4) whereas < α ( ε ) < by assumption; the final convergence then follows from (4.5). Combining the conclusionsof (7.21) and (7.24) we get that on Ω ◦ ∩ Ω ⋆ ∩ Ω , lim n →∞ ¯ p n = 0 . (7.25)This concludes the proof of Fact 2. The proof of Theorem 1.1 is now complete. (cid:3) 8. A PPENDIX : P ROOF OF T HEOREM AND T HEOREM Proof of Theorem 1.2. The proof closely follows that of Theorem 1.2 of Ref. [13]. Through-out we fix a realization ω ∈ Ω of the random environment but do not make this explicit inthe notation. We set b S bn ( t ) ≡ S bn ( t ) − Z n, . (8.1)Condition (A0) ensures that S bn − b S bn converges to zero, uniformly. Thus we must showthat under Conditions (A1), (A2), and (A3), b S bn ⇒ J S ν . (8.2)For this we rely on Theorem 1.1 of Ref. [13]. (This result is itself a specialized form ofTheorem 4.1 of Ref. [20] suited to the present setting.) Namely, we want to show thatConditions (A1), (A2), and (A3) imply the conditions of Theorem 1.1 of Ref. [13].To this end let {F n,i , n ≥ , i ≥ } 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 ≥ . Note that for each n and i ≥ , Z n,i is F n,i measurable and F n,i − ⊂ F n,i . Next observe that by the Markovproperty and the fact that, for all i ≥ and y ∈ V n , P y ( Z n,i > u ) = P y ( Z n, > u ) , P µ n (cid:0) Z n,i > u (cid:12)(cid:12) F n,i − (cid:1) = X y ∈V n { Y n (( i − θ )= y } P y ( Z n, > u ) . (8.3)In view of this, (1.21), (1.22), and (1.23) k n ( t ) X i =2 P µ n ( Z n,i > u | F n,i − ) = ν Y,tn ( u, ∞ ) , (8.4)and in view of (1.24) k n ( t ) X i =2 [ P µ n ( Z n,i > u | F n,i − )] = σ Y,tn ( u, ∞ ) . (8.5) GING IN METROPOLIS DYNAMICS OF THE REM: A PROOF 39 From (8.4) and (8.5) it follows that Conditions (A1) and (A2) of Theorem 1.2 are exactlythe conditions of Theorem 1.1 of Ref. [13]. Similarly Condition (A3) is condition (1.9).Therefore the conditions of Theorem 1.1 of Ref. [13] are verified, and so b S bn ⇒ J S ν in D ([0 , ∞ )) where S ν is a subordinator with L´evy measure ν and zero drift. (cid:3) The proof of Theorem 1.3 centers of the Proposition 8.1. Assume that Condition (B1) is satisfied. Then, choosing θ n ≥ κ n , thefollowing holds for all initial distributions µ n : for all t > , all u > , and all ǫ > , P µ n (cid:0)(cid:12)(cid:12) ν Y,tn ( u, ∞ ) − ν tn ( u, ∞ ) (cid:12)(cid:12) ≥ ǫ (cid:1) ≤ ǫ − h ρ n (cid:0) ν tn ( u, ∞ ) (cid:1) + σ tn ( u, ∞ ) i , (8.6) and P µ n (cid:0) σ Y,tn ( u, ∞ ) ≥ ǫ (cid:1) ≤ ǫ − (1 + ρ n ) σ tn ( u, ∞ ) . (8.7) Proof of Proposition 8.1. We assume throughout that θ n ≥ κ n . To prove (8.7), simplynote that by a first order Tchebychev inequality P µ n (cid:0) σ Y,tn ( u, ∞ ) ≥ ǫ (cid:1) ≤ ǫ − k n ( t ) P y ∈V n E µ n ( π Y,tn ( y )) [ Q un ( y )] (8.8) ≤ ǫ − (1 + ρ n ) σ tn ( u, ∞ ) , (8.9)where we used in the last line that by (1.30), | E µ n ( π Y,tn ( y )) − π n ( y ) | ≤ ρ n π n ( y ) . (8.10)Turning to (8.6), a second order Chebychev inequality yields P µ n (cid:0)(cid:12)(cid:12) ν Y,tn ( u, ∞ ) − ν tn ( u, ∞ ) (cid:12)(cid:12) ≥ ǫ (cid:1) ≤ ǫ − E µ n h k n ( t ) P y ∈V n (cid:0) π Y,tn ( y ) − π n ( y ) (cid:1) Q un ( y ) i = ǫ − P x ∈V n P y ∈V n Q un ( x ) Q un ( y ) P k n ( t ) − i =1 P k n ( t ) − j =1 ∆ ij ( x, y ) (8.11)where ∆ ij ( x, y ) ≡ P µ n ( Y n ( iθ n ) = x, Y n ( jθ n ) = y ) + π n ( x ) π n ( y ) − π n ( y ) P µ n ( Y n ( iθ n ) = x ) − π n ( x ) P µ n ( Y n ( jθ n ) = y ) . (8.12)Using again (1.31) yields | ∆ ij ( x, y ) | ≤ ρ n (4 + ρ n ) π n ( x ) π n ( y ) , if i = j, (1 + ρ n ) π n ( x ) + (1 + 2 ρ n ) π n ( x ) , if i = j and x = y, . (8.13)Thus (8.11) is bounded above by ǫ − ρ n (4 + ρ n ) h k n ( t ) X y ∈V n π n ( y ) Q un ( y ) i + ǫ − (2 + 3 ρ n ) k n ( t ) X y ∈V n π n ( y ) [ Q un ( y )] (8.14)Since by assumption ρ n ↓ as n ↑ ∞ , (8.14) is tantamount to the right-hand side of (8.6).Proposition 8.1 is proven. 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