Mixing time for the asymmetric simple exclusion process in a random environment
aa r X i v : . [ m a t h . P R ] F e b MIXING TIME FOR THE ASYMMETRIC SIMPLE EXCLUSION PROCESSIN A RANDOM ENVIRONMENT
HUBERT LACOIN AND SHANGJIE YANG
Abstract.
We consider the simple exclusion process in the integer segment J , N K with k ≤ N/ x ∈ J , N K jumps tosite x − x ≥
2) at rate 1 − ω x and to site x +1 (if x ≤ N −
1) at rate ω x if the target site is notoccupied. The sequence ω = ( ω x ) x ∈ Z is chosen by IID sampling from a probability law whosesupport is bounded away from zero and one (in other words the random environment satisfiesthe uniform ellipticity condition). We further assume E [log ρ ] < ρ := (1 − ω ) /ω ,which implies that our particles have a tendency to move to the right. We prove that themixing time of the exclusion process in this setup grows like a power of N . More precisely, forthe exclusion process with N β + o (1) particles where β ∈ [0 , N asymptotic N max ( , λ ,β + λ ) + o (1) ≤ t N,k mix ≤ N C + o (1) where λ > E [ ρ λ ] = 1 ( λ = ∞ if the equation has no positive root) and C is aconstant which depends on the distribution of ω . We conjecture that our lower bound is sharpup to sub-polynomial correction. introduction From the viewpoint of probability, statistical mechanics and combinatorics, the simple ex-clusion process is an important and intensively studied interacting particle system, which is areasonable toy model to describe the relaxation of a low density gas and we refer to [Lig12,Chapter VIII.6] for a historical introduction.We are interested in the exclusion process on the segment J , N K with k particles and 1 ≤ k ≤ N/ the exclusionrule) . Therefore at all time there are k occupied sites and N − k empty sites.(B) Each of the k particles performs a random walk on the segment, independently of theothers, except that any jump that violates the exclusion rule is cancelled.More precisely, we want to consider the case exclusion process in a random environement wherethe jump rates of the particles are specified by sampling an IID sequence of random variables ω = ( ω x ) x ∈ Z , and the transition rates are given by q ωN ( x, x + 1) = ω x { x ≤ N − } ,q ωN ( x, x −
1) = (1 − ω x ) { x ≥ } ,q ωN ( x, y ) = 0 if y / ∈ { x − , x + 1 } . (1.1)The random walk with transitions q ωN which corresponds to the case k = 1 is an extensivelystudied process, usually referred to as Random Walk in a Random Environment (RWRE). TheRWRE on the full line Z was first studied by Solomon in [Sol75] who established a criterion forrecurrence/transience. The limit law of the random walk in a random environment is studied by Key words and phrases.
Markov chains, interacting particles, mixing time, random environment.
AMS subject classification : 60K37; 60J27 .
Kesten et al. in [KKS75] when the random walk is transient, and by Sinai in [Sin82] when therandom walk is recurrent (we refer to [Szn04, Zei04] for complete introductions to this researchfield). N N + 1 × ω x − ω x × − ω N × y ω y − ω y × − ω z z Figure 1.
A graphical representation of the simple exclusion process in the segment J , N K and environment ω = ( ω x ) x ∈ Z : a bold circle represents a particle, and the number above everyarrow represents the jump rate while a red ” × ” represents an inadmissible jump. We are interested in the following quantitative question: How long does the system needto relax to equilibrium, forgetting the information of its initial configuration in the sense oftotal-variation distance? More precisely we are interested in the asymptotic in the limit when k, N → ∞ of this total-variation mixing time . This question has been extensively studied inthe case where the sequence ω = ( ω x ) x ∈ Z is constant, which we refer to as the homogeneousenvironment case:(1) When ω x ≡ , Wilson in [Wil04] showed that the system takes time of order N log min( k, N − k ) and later Lacoin in [Lac16b] proved that the lower bound in [Wil04] is sharp.(2) When ω x ≡ p = , Benjamini et al. in [BBHM05] told that the system takes time oforder N , and later Labb´e and Lacoin in [LL19] provided the exact constant.(3) The case ω x ≡ p N = + ε N with lim N →∞ ε N = 0 is studied by Levins and Peres in[LP16], Labb´e and Lacoin in [LL20].From the results mentioned above, for homogeneous environment the system takes time atleast of order N and most of order N log N to relax to equilibrium. However, when the sequence ω = ( ω x ) x ∈ Z is chosen by independently sampling a nondegenerate common law, the system canexhibit a very different behavior because the random environment can create wells of potentialwhich trap particles (see Equation (2.11) below for a definition of the potential associated to ω ).Gantert and Kochler has studied the mixing time problem when k = 1 (and transient environ-ment) in [GK13] for random environment and identified the mixing time, which is related withthe depth of the deepest trap and may be much larger than N log N . Schmid [Sch19] studiedthe question in the case of a positive density of particles, when the environment is ballistic tothe right, (that is, when the random walk is transient with positive speed) and provided boundsfor the mixing time, showing in particular that the mixing time is of order larger than N assoon as the local drift (which is equal to 2 ω x −
1) is not uniformly bounded from below by apositive constant and is larger than N δ for some δ > P [ ω x < / > N ?(B) If this is the case, for the exclusion process with k N = N β particles and β ∈ (0 , ν > β and the distribution) which is such thatthe mixing time is of order N ν ? SEP IN RANDOM ENVIRONMENT 3
We provide a positive answer to question ( A ) by proving an upper bound on the mixing timewhich grows like a power of N . This upper bound is achieved by using a censoring procedurewhich allows to transport particles one by one to their equilibrium positions. Concerning ques-tion ( B ), we provide a new lower bound on the mixing time which we believe to be optimaland provide a conjecture concerning the value of ν . The bound is based on an analysis of theeffect of the deepest trap on the particle flow through the system. Significant technical obstaclesprevented us from obtaining a matching upper bound.2. Model and result
An introduction to Random Walk in a Random Environment ω . Let us brieflyrecall the definition for random walk in a random environment. Given ω = ( ω x ) x ∈ Z a sequencewith values in (0 , ω is the continuous time Markov chainon Z whose transition rates are given by q ω ( x, x + 1) = ω x ,q ω ( x, x −
1) = 1 − ω x ,q ω ( x, y ) = 0 if | x − y | 6 = 1 . (2.1)We let ( X t ) t ≥ denote the random walk in environment ω and initial condition 0 (we let Q ω denote the corresponding law). This process has been extensively studied in the case where ω = ( ω x ) x ∈ Z is (the fixed realization of) a sequence of IID random variables (we will use P and E denote the associated law and expectation respectively), and we refer to [Szn04, Zei04] forrecent reviews.Simple criteria have been derived on the distribution of ω as necessary and/or sufficientconditions for recurrence/transience, ballisticity etc... Even though most of the results are validin a more general setup, for the sake of simplicity let us assume in the discussion that thevariables ( ω x ) x ∈ Z are bounded away from 0 and 1, that is, for some α ∈ (0 , /
2) we have P ( ω ∈ [ α, − α ]) = 1 . (2.2)Setting ρ x := (1 − ω x ) /ω x , it has been proved in [Sol75] that ( E [log ρ ] = 0 ⇒ X t is reccurent under Q ω , P -a.s. , E [log ρ ] = 0 ⇒ X t is transient under Q ω , P -a.s. (2.3)More precisely in the second case we have with probability one lim t →∞ X t = ∞ (resp. −∞ ) if E [log ρ ] < E [log ρ ] > X t goes to infinity has also been identified in [KKS75].It can be expressed in terms of a simple parameter of the distribution ω , yielding in particulara necessary and sufficient condition for ballisticity. Let us assume that E [log ρ ] <
0, and set λ = λ P := inf { s > , E [ ρ s ] ≥ } ∈ (0 , ∞ ] . It has been proved in [KKS75] that if λ > ϑ P > t →∞ X t t = ϑ (2.4)and that if λ ∈ (0 ,
1] then lim t →∞ log( X t )log t = λ. (2.5)2.2. The Simple Exclusion process in an environment ω . HUBERT LACOIN AND SHANGJIE YANG
Definition.
Given a sequence ω = ( ω x ) x ∈ Z taking values in (0 , N ≥ ≤ k ≤ N − J , N K (we use thenotation J a, b K := [ a, b ] ∩ Z ) with k particles is a Markov process on the spaceΩ N,k := ( ξ ∈ { , } N : N X x =1 ξ ( x ) = k ) . (2.6)The 1s’ are denoting particles while 0s’ correspond to empty sites. It can be informally describedas follows: each of the k particles performs independently a random walk with transitions givenby q ω in (2.1), with the constraints that particles must remain in the segment and each site canbe occupied by at most one particle. All transitions that would make this constraint violated(that is, a particle tries to jump either on 0, N + 1 or an already occupied site) are cancelled.More formally we let ξ x,y is the configuration obtained by swapping the values of ξ at sites x and y of the configuration ξ , more formally defined by ∀ z ∈ J , N K , ξ x,y ( z ) = ξ ( z ) J ,N K \{ x,y } + ξ ( x ) { y } + ξ ( y ) { x } . (2.7)The simple exclusion process in environment ω is the Markov process with transition rates givenby r ω ( ξ, ξ x,x +1 ) := ( ω x if ξ ( x ) = 1 and ξ ( x + 1) = 0 , − ω x +1 if ξ ( x + 1) = 1 and ξ ( x ) = 0 , for x ∈ J , N − K r ω ( ξ, ξ ′ ) := 0 in all other cases . (2.8)Equivalently the generator of the process is defined for f : Ω N,k → R by L ωN,k ( f )( ξ ) := N − X x =1 r ω ( ξ, ξ x,x +1 ) (cid:2) f ( ξ x,x +1 ) − f ( ξ ) (cid:3) . (2.9)The chain is ergodic and reversible. In order to give a simple compact expression for theequilibrium measure, let us introduce the random potential V ω : N → R defined as follows, V ω (1) := 0 and for x ≥ V ω ( x ) := x X y =2 log (cid:18) − ω y ω y − (cid:19) . (2.10)With a small abuse of notation, we extend V ω to a function of Ω N,k . This extension is obtainedby summing the value of V ω among the positions of the particles in the configuration ξ : V ω ( ξ ) := N X x =1 V ω ( x ) ξ ( x ) . (2.11)We consider the probability measure π ωN,k defined by π ωN,k ( ξ ) := 1 Z ωN,k e − V ω ( ξ ) with Z ωN,k = X ξ ∈ Ω N,k e − V ω ( ξ ) . (2.12)It is immediate to check by inspection that π ωN,k satisfies the detailed balance condition for L ωN,k ,and thus that it is the unique invariant probability measure on Ω N,k .If ξ ∈ Ω N,k , we let ( σ ξt ) t ≥ denote the Markov chain with initial condition ξ . We are goingto provide a construction ( σ ξt ) t ≥ for all ξ ∈ Ω N,k on a common probability space in Section3.2, and we use P and E for the corresponding probability law and expectation respectively.We let ( P t ) t ≥ (the dependence in ω , N , k is omitted in the notation to keep it light) denote SEP IN RANDOM ENVIRONMENT 5 the corresponding Markov semi-group and set P ξt := P ( σ ξt ∈ · ) = P t ( ξ, · ) to be the marginaldistribution of ( σ ξt ) t ≥ at time t . Mixing time and spectral gap.
In a standard fashion, we set the total variation-distance toequilibrium at time t to be d ωN,k ( t ) := max ξ ∈ Ω N,k k P ξt − π ωN,k k TV (2.13)where k ν − ν k TV := sup A ⊂ Ω N,k | ν ( A ) − ν ( A ) | denotes the total variation between two proba-bility measures ν , ν on Ω N,k . Since the Markov chain is irreducible, we know that (cf. [LP17,Theorem 4.9]) lim t →∞ d ωN,k ( t ) = 0 . (2.14)We are interested in having quantitative statements related to the convergence (2.14), and forthis reason we want to evaluate the mixing time and spectral gap of the chain (see [LP17] for amotivated and thorough introduction to these notions). For ε ∈ (0 , ε -mixing time ofthe chain be defined by t N,k,ω mix ( ε ) := inf n t ≥ d ωN,k ( t ) ≤ ε o . (2.15)By convention, we simply write t N,k,ω mix when ε = 1 /
4. The spectral gap of the chain gap ωN,k , inour context, can be defined as the smallest non-zero eigenvalue of −L ωN,k . It can be shown usinga spectral decomposition (see for instance [LP17, Corollary 12.7]) to determine the asymptoticrate of convergence of d ωN,k as lim t →∞ t log d ωN,k ( t ) = − gap ωN,k . (2.16)The mixing time and spectral gap are related to one another by the following relation valid for ε ∈ (0 , /
2) (cf. [LP17, Theorems 12.4 and 12.5])1gap ωN,k log (cid:18) ε (cid:19) ≤ t N,k,ω mix ( ε ) ≤ ωN,k log (cid:18) επ min (cid:19) (2.17)where π min = min ξ ∈ Ω N,k π ωN,k ( ξ ) . Results.
The main object of the paper is the study of the exclusion process in an IIDenvironment. On the way to our main result, we also prove bounds on the mixing time whichare valid for any realization of ω , and which we present first. Universal bounds for the mixing time on the exclusion process.
We assume without loss of gen-erality (by symmetry) that k ≤ N/
2. We prove that the mixing time grows at least linearlywith the size of the system and at most exponentially. Both results are in a sense optimal (seethe discussion in Section 2.4. below).
Proposition 2.1.
For any k ∈ J , N/ K and N ≥ , for any ( ω x ) x ∈ Z we have t N,k,ω mix ≥ N. (2.18) Furthermore, if k N is a sequence such that k N ≤ N/ and lim N →∞ k N = ∞ , (2.19) we have for any ε > , for N ≥ N ( ε ) sufficiently large for any ( ω x ) x ∈ Z t N,k N ,ω mix (1 − ε ) ≥ N. (2.20) HUBERT LACOIN AND SHANGJIE YANG
For the upper bound, we require an assumption similar to (2.2), that is ∀ x ∈ Z , ω x ∈ [ α, − α ] . (2.21) Proposition 2.2.
For any sequence ( ω x ) x ∈ Z satisfying (2.21) all N ≥ and all k ∈ J , N/ K ,we have gap ωN,k ≥ αN − | Ω N,k | − (cid:18) − αα (cid:19) − N/ , (2.22) and as a consequence for all ε ∈ (0 , / t N,k,ω mix ( ε ) ≤ α − N | Ω N,k | (cid:18) − αα (cid:19) N/ (cid:18) log | Ω N,k | + N k log 1 − αα − log ε (cid:19) . (2.23) Mixing time for the exclusion process in a random environment.
Let us now introduce our mainresults concerning the exclusion process in a random environment. We assume that (2.2) holds, E [log ρ ] < ≤ k ≤ N/ . (2.24)Using the various symmetries of the the system (between left and right, particles and emptysites...), assumption (2.24) entails almost no-loss of generality, and the only case being left asideis that of a recurrent environment (that is E [log ρ ] = 0). We are also going to consider that λ P < ∞ , this corresponds to saying that P [ ω < / > P [ ω ≥ /
2] = 1 is discussedin the next section).In order to bet a better intuition on the result, let us provide a description of the equilibriummeasure. We introduce the event A r ⊂ Ω N,k that the leftmost particle and rightmost empty siteare at a distance smaller than 2 r of their respective maximal and minimal possible values: A r := { ξ ∈ Ω N,k : ∀ x ∈ J , N − k − r K , ξ ( x ) = 0 ; ∀ x ≥ N − k + r, ξ ( x ) = 1 } . (2.25)The following result tells us that the mass of π N,k N is essentially concentrated at a finite distanceof the configuration ξ max with all k particles packed to the right (see (3.3)). Lemma 2.3.
Under the assumptions (2.21) and (2.24) , for all N sufficiently large we havew.h.p. lim r →∞ inf N ≥ k ∈ J ,N/ K E (cid:2) π ωN,k N ( A r ) (cid:3) = 1 . (2.26)Our first main result is that if the environment satisfies the assumptions (2.2) and (2.24),the system relaxes to equilibrium in polynomial time, or in other words that t N,k,ω mix grows likea power of N with an explicit upper bound on the growth exponent. In order to describe ourexplicit bound, we need to introduce the function F which is the log-Laplace transform of log ρ that is F ( u ) := log E [ ρ u ] . (2.27)Since V ω is, up to a small modification, a sum of IID variables with the same distributionas log ρ , the function F is used to compute the large deviations of V ω , and in particularto determine the geometry of the deepest potential wells. It is strictly convex and satisfies F (0) = F ( λ ) = 0 (see Figure 2). We let u be defined by F ( u ) = min u ∈ R F ( u ) < . Given a sequence of events ( A N ) N ≥ , we say that A N holds with high probability (which wesometimes abbreviate as w.h.p.) if lim N →∞ P [ A N ] = 1. Given a sequence ( B N,k ) N ≥ ,k ∈ J ,N/ K ,we say that B N,k holds with high probability iflim N →∞ inf k ∈ J ,N/ K P [ B N,k ] = 1 . SEP IN RANDOM ENVIRONMENT 7 uF λ Figure 2.
A graphical description of the function F ( u ) with only two zeros at u = 0 and u = λ . We are now ready to state the result.
Theorem 2.4.
Under the assumptions (2.2) and (2.24) , then with high probability we have t N,k,ω mix ≤ kN α − (cid:18) u + 2 | F ( u ) | log N (cid:19) N u | F ( u | ( − αα +4 log 4 − ) . (2.28)Our second result provides a lower bound for the mixing time which depends both on N and k . Theorem 2.5.
Under the assumptions (2.2) - (2.24) and assuming further that λ P < ∞ , thereexists a positive constant c ( α, P ) such that w.h.p. we have for every N and k ∈ J , N/ K t N,k,ω mix ≥ c max n N, N λ (log N ) − λ , kN λ (log N ) − λ ) o . (2.29)2.4. Related work.
Let us provide now a short review of related results present in the litera-ture.
Mixing time for the exclusion process in a homogeneous environment.
The mixing time of theexclusion process on the line segment has been extensively studied in the case where the sequence ω is constant, i.e. ω ≡ p . In that case, not only the right order of magnitude has been identifiedfor the mixing time, but also the sharp asymptotic equivalent. The case of the exclusion withno bias that is p = 1 / N and at most N (log N ) . It was later established (see [Wil04] for the lower bound and [Lac16b] for the upperbound) that if k N satisfies (2.19), we have t N,k N mix ( ε ) = (1 + o (1)) π N log k N . (2.30)In the case where the walk presents a bias, that is p = 1 /
2, it was shown in [BBHM05] that themixing time is of order N . This result was refined in [LL19] by identifying the proportionalityconstant, showing that if k N satisfies lim N →∞ k N /N = θ , then t N,k N mix ( ε ) = [1 + o (1)] ( √ θ + √ − θ ) | p − | N. (2.31)The case where p is allowed to depend on N was investigated in [LP16, LL20] where the orderof magnitude and the sharp asymptotic of the mixing time were respectively determined. Notethat in (2.30) and (2.31) the asymptotic behavior of t N,k N mix ( ε ) does not display any dependenceon ε at first order. This implies that d N,k N ( t ) abruptly drops from 1 to 0 on the time scale N log k N and N respectively. This phenomenon, called cutoff, is expected to hold for a largeclass of Markov chains, we refer to [LP17, Chapter 18] for an introduction. HUBERT LACOIN AND SHANGJIE YANG
Let us also mention that the mixing time for the one-dimensional exclusion process has alsobeen investigated for a variety of different boundary conditions. We refer to [Lac16a] for a sharpestimate of the convergence profile to equilibrium for the periodic boundary condition in thesymmetric case and to [GNS20] (and references therein) for the study of a variety of boundaryconditions, with or without bias. The case of higher dimension has also been considered, see e.g.[Mor06] where the order of magnitude of the mixing time is determined up to a constant.
Mixing time for the random walk in a random environment.
In [GK13], the case of the mixingtime for a random walk in the segment with a transient random environment (which correspondsto the case k = 1 in the present paper) was investigated. It is shown that whenever λ P > t N, ,ω mix ( ε ) = [1 + o (1)] N E [ Q ω [ T ω ]] , (2.32)where T ω is the first hitting time of 1 for the random walk in a random environment ω startingfrom 0 (the result in [GK13] is slightly more precise and the assumption is more general than(2.2)). When λ P <
1, it is shown that the mixing time is of a much larger magnitude but thatcutoff does not hold. More precisely, for λ P ≤ N →∞ log t N, ,ω mix ( ε )log N = 1 λ P . (2.33)The asymptotic N /λ P + o (1) corresponds to the time that is required to overcome the largestpotential barrier present in the system, whose height is of order (1 /λ ) log N . Mixing time for the exclusion in a ballistic environment.
In [Sch19], the mixing time t N,k N ,ω mix were investigated under the assumption that lim N →∞ k N /N = θ ∈ (0 , /
2] and λ P >
1. Threedifferent cases are considered. • When ess inf ω > /
2, it is shown that the mixing t N,k N ,ω mix is of order N , by a simplecomparison with the case of homogeneous asymmetric environment. • When ess inf ω < /
2, it is shown that there exists a positive δ such that the mixingtime satisfies t N,k N ,ω mix ≥ N δ . • When ess inf ω = 1 /
2, it is shown thatlim inf N →∞ t N,k N ,ω mix ( ε ) /N = ∞ and t N,k N ,ω mix ( ε ) ≤ CN (log N ) , (2.34)together with a quantitative lower bound if P [ ω = 1 / > Other perspectives concerning the exclusion process and random environments.
The exclusionprocess with other types of random environment has also been considered in the literature.One possibility is to consider a random environment on bonds instead of sites. A particularchoice which makes the uniform measure on Z reversible for the random walk is the model ofrandom conductance. In that case the mixing property of the system strongly differs from modelconsidered here: the equilibrium measure is uniform on Ω N,k so that there is no trapping bypotential. It is expected that for a large class of environment in that case the mixing propertiesare very similar to that of the homogeneous system. The hydrodynamic limit of exclusionprocesses with bond–dependent random transition rates have been studied in [Fag08, Jar11](see also [Fag20] for a recent work going slightly beyond the random conductance model).Another corpus of work has been considering the (homogeneous) exclusion process itself as adynamical random environment, which determines the transition probability of the random walk.The asymptotic behavior of a random walker in this setup is studied in [HKT20, HS15], and thehydrodynamic limit for the exclusion process as seen by this walker is studied in [AFJV15]. In
SEP IN RANDOM ENVIRONMENT 9 a more general setup for the jump rates of the walker, an invariance principle about the randomwalk when the exclusion process starts from equilibrium is studied in [JM20].2.5.
Interpretation of our results and conjectures.
Comments on Propositions 2.1 and 2.2.
The asymptotic for the mixing time for ASEP in ho-mogeneous environment (2.31) shows that the lower bound of Proposition 2.1 is sharp up to aconstant factor. An important observation is also that (2.20) is not true without the assumptionthat k N goes to infinity, even if 1 /
30 is replaced by an arbitrarily small constant provided thatit is not allowed to depend on ε .However the constant in our bounds (2.18) and (2.20) are clearly not optimal. Let us statenow a natural conjecture. We believe that if lim N →∞ k N /N = θ ∈ (0 , / ω x ∈ [ α, − α ]for all x ∈ Z (with the possibility of having α = 0) then we should havelim inf N →∞ N t
N,k N mix (1 − ε ) ≥ ( √ θ + √ − θ ) − α . (2.35)One can obtain counter examples to (2.35) in the zero density case by considering the case ω x = 1 − α in the first half of the segment J , N K and ω = α in the second half of the segment,and k N diverging to infinity such that lim N →∞ k N / (log N ) = 0. In that case, one can with someminor efforts, show that the mixing time is asymptotically equivalent N − α (which is half of thelower bound in (2.35)).Proposition 2.2 can also be shown to be sharp within constant in the sense that there existsa constant C α , and for given N and k it is always possible to construct an environment ω suchthat gap ωN,k ≥ e − C α N . (2.36)We conjecture that the best possible lower bound on the spectral gap when lim N →∞ k N / (log N ) = θ ∈ (0 , /
2] is the followinglim inf N →∞ log gap ωN,k N N = − (1 − θ )2 log (cid:18) − αα (cid:19) . (2.37)The lim inf is reached asymptotically by the environment ( ω x = α if 1 ≤ x ≤ (1 − θ ) N ,ω x = 1 − α if (1 − θ ) N < x ≤ N. (2.38) Comments on Theorems 2.4 and 2.4.
Our paper brings a complement to the results in [Sch19],in the case when ess inf ω < /
2. Firstly it provides a complementary upper bound result,which shows that the mixing time in transient environment always scales like a power of N ,even in the non-ballistic case λ P ≤ • Mass transport cannot be faster than ballistic:
Which is explored in Proposition 2.1 isthat particle cannot move faster than ballistically (and this is independent of the choiceof ω ), so that the time required to transport the mass of particles to equilibrium has tobe at least of order N . This idea is already present in [BBHM05]. • Individual particles may be blocked by traps in the potential profile:
As soon as ess inf ω < /
2, the potential profile V is non-monotone and will create energy barriers. It is knownsince [KKS75] that these energy barriers can slow down particles to subballistic speedin λ P ≤ • Potential barrier may also create bottleneck for the flow of particles:
The third mecha-nism which was partially identified in [Sch19] is that potential barrier may also limit theflow of particles throughout the system. The limitation on the flow does not correspondto the inverse of the time that a particle needs to cross the trap, but rather to the squareroot of this inverse. The reason for this is that when particles are flowing through thesystem, the particle are “filling” half of the potential well, so that the remaining potentialbarrier to be crossed is halved. This reasoning yields the third term in (2.29).We believe that the three mechanism described above are the only possible limiting factor tomixing, and thus that the lower bound give in Theorem 2.5 is sharp as far as the exponent isconcerned. Let us formulate this as a conjecture. Let us assume that k N satisfieslim N →∞ log k N log N = β, and then we should have the following convergence w.h.p.lim N →∞ t N,k N mix log N = max (cid:18) , λ , λ + β (cid:19) . (2.39)We refer to Figure 3 for the phase diagram concerning the conjectured exponent of the mixingtime. β − axis λ − axis max (cid:0) , λ , λ + β (cid:1) = 1 Ballistic begime max (cid:0) , λ , λ + β (cid:1) = λ One particle limitation max (cid:0) , λ , λ + β (cid:1) = λ + β Flow limitation
Figure 3.
The phase diagram for the exponent of the mixing time (the lower bound is provedrigorously and the upper bound is only conjectured). The transition between the blue and red(hatched) regions of the diagramm corresponds to the transition of the RWRE from the balisticphase to the transient-with-zero-speed phase. A third phase represented by the white regionappears when one considers a large number of particles, in this phase the main limitation tomixing is the flow of particle through the deepest trap.
In particular this means that when β ≤ / (2 λ ) then the mixing time of the exclusion processon the segment coincides (as far as the exponent is concerned) with that of the random walk inthe segment. SEP IN RANDOM ENVIRONMENT 11
Organization.
Section 3 is devoted to some technical preliminaries including the particle de-scription, equilibrium estimates, partial order, a graphical construction and a composed censor-ing inequality.Section 4 is devoted to universal lower and upper bounds on the mixing time for all randomenvironments, that is, the proof of Propositions 2.1 and Proposition 2.2.Section 5 is devoted to lower bounds on the mixing time, that is Theorem 2.5. There are threebounds to prove, one of them is a consequence of Proposition 2.1, the other two are presentedas two distinct results (Proposition 5.1 and Proposition 5.2) and proved in separate subsections.The first bound rely on controlling the displacement of the leftmost particle while the other isbased on a control of the particle flow.Section 6 is concerned with the upper bound on the mixing time (Theorem 2.4). The proofis based on application of the censoring inequality and of our upper bound from Proposition2.2: blocking the transition along carefully chosen edges (in a way that varies throught time) weguide all particlesto the right of the segment (where they are typically located at equilibrium)in polynomial time.
Notation.
We use c ( α, P ) and C ( α, P ) to stress that the constants c and C depend on α andthe law of the random environment ω . Moreover, we use “ := ” (or “ =: ”) to define a newquantity on the left-hand (right-hand, resp.) side, and J a, b K := [ a, b ] ∩ Z . Furthermore, we letΩ [ a,b ] ,k := ξ ∈ { , } J a,b K : X x ∈ J a,b K ξ ( x ) = k (2.40)denote the state space of k particles performing exclusion process restricted in the interval J a, b K and environment ω , and let π ω [ a,b ] ,k denote the corresponding equilibrium probability measure. Acknowledgment.
The authors thank Milton Jara, Roberto Imbuzeiro Oliveira and AugustoTeixeira for enlightening discussions. This work was realized in part during H.L. extended stay inAix-Marseille University funded by the European Union’s Horizon 2020 research and innovationprogramme under the Marie Sk lodowska-Curie grant agreement No 837793.3.
Technical preliminaries
Partial order on Ω N,k . Given ξ ∈ Ω N,k we define ¯ ξ : J , k K → J , N K as an increasingfunction which provides the positions of the particles of ξ from left to right: { ¯ ξ ( i ) = x } ⇐⇒ ξ ( x ) = 1 and x X y =1 ξ ( x ) = i . (3.1)We introduce a natural partial order relation “ ≤ ” on Ω N,k × Ω N,k as follows( ξ ≤ η ) ⇔ (cid:0) ∀ i ∈ J , k K , ¯ ξ ( i ) ≤ ¯ η ( i ) (cid:1) . (3.2)Informally ξ ≤ η means that the particles in the configuration η are located “more to theright” than those of ξ . Let ξ max and ξ min denote the maximal and minimal configurations of(Ω N,k , “ ≤ ”) respectively, given by ξ max := { N − k +1 ≤ x ≤ N } and ξ min := { ≤ x ≤ k } . (3.3)This order plays a special role for our dynamic ( σ ξt ) t ≥ , and the next two subsections providetools to exploit this link. Canonical coupling via graphical construction.
Let us present a construction of agrand coupling for the exclusion process on the segment J , N K which has the property of con-serving the order defined above.To each site x ∈ J , N K we associate an independent rate 1 Poisson clock process ( T ( x ) i ) i ≥ (the increments of the sequence ( T ( x ) i ) i ≥ are independent exponential variables of parameter1) and an independent sequence of IID variables ( U ( x ) i ) i ≥ with uniform distribution on [0 , ω = ( ω x ) x ∈ Z , and the trajectory ( σ ξt ) t ≥ for each ξ is a deterministic function of ( T ( x ) i , U ( x ) i ) i ≥ ,x ∈ J ,N K . In the remainder of the paper, P denote the joint law of ( T ( x ) i , U ( x ) i ) i ≥ ,x ∈ J ,N K , and E denotes the corresponding expectation.Let us also introduce a natural filtration ( F t ) t ≥ in this probability space setting i ( x, t ) := max { i ≥ T ( x ) i ≤ t } (3.4)with the convention that max ∅ = 0 and set F t := σ (cid:16) T ( x ) i , U ( x ) i , x ∈ Z , i ≤ i ( x, t ) (cid:17) . (3.5)Now, given 1 ≤ k ≤ N − ξ ∈ Ω N,k , we construct the trajectory( σ ξt ) t ≥ as follows:(1) ( σ ξt ) t ≥ is c`adl`ag and may change its value only at times T ( x ) i , x ∈ J , N K and i ≥ σ ξ = ξ and modifying it sequentially at theupdate times ( T ( x ) i ) i ≥ ,x ∈ J ,N K . For instance if t = T ( x ) i we obtain σ ξt − from σ ξt as follows:(A) If U ( x ) i ≤ ω x , x ≤ N − σ ξt − ( x ) = 1 and σ ξt − ( x + 1) = 0, then σ ξt ( x + 1) = 1 and σ ξt ( x ) = 0 (and σ ξt ( y ) = σ ξt − ( y ) for y / ∈ { x, x + 1 } ).(B) If U ( x ) i > ω x , x ≥ σ ξt − ( x ) = 1 and σ ξt − ( x −
1) = 0, then σ ξt ( x −
1) = 1 and σ ξt ( x ) = 0 (and σ ξt ( y ) = σ ξt − ( y ) for y / ∈ { x − , x } ).(C) In all other cases σ ξt = σ ξt − .It is immediate by inspection to check that the above construction corresponds indeed to theMarkov chain with generator L ωN,k . Note also that our process is adapted and Markov withrespect to the filtration ( F t ) t ≥ . In the same manner, the reader can check that it preserves theorder in the following sense. Proposition 3.1.
For the coupling constructed above, we have for all ξ, ξ ′ ∈ Ω N,k ξ ≤ ξ ′ ⇒ P h ∀ t ≥ , σ ξt ≤ σ ξ ′ t i = 1 . (3.6)3.3. Composed censoring inequality.
We are going to use a variant of the censoring inequal-ity introduced by Peres and Winckler [PW13]. Let E N = {{ n, n + 1 } : n ∈ J , N − K } be theset of edges in J , N K , and a censoring scheme C : [0 , ∞ ) → P ( E N ) is a deterministic c`adl`agfunction where P ( E N ) is the set of all subsets of E N .The censored chain ( σ ξ, C t ) t ≥ is a time inhomegenous Markov chain, with a generator obtainedby cancelling the transition using edges in C ( t ) L C ,tN,k ( f )( ξ ) := N − X x =1 r ωN,k ( ξ, ξ x,x +1 ) {{ x,x +1 } / ∈C ( t ) } (cid:2) f ( ξ x,x +1 ) − f ( ξ ) (cid:3) , (3.7) SEP IN RANDOM ENVIRONMENT 13 where r ωN,k ( ξ, ξ x,x +1 ) is defined in (2.8). We let P C t be the associated semigroup (the solution of ∂ t P t = P t L C ,tN,k with initial condition given by the identity). We will use the following corollaryof the censoring inequality [PW13, Theorem 1] (recall (3.3)). Proposition 3.2.
For any ξ ∈ Ω N,k and any censoring scheme C , we have P t ( ξ, ξ max ) ≥ P C t ( ξ min , ξ max ) (3.8) Sketch of proof.
Proposition 3.1 implies that P t ( ξ, ξ max ) ≥ P t ( ξ min , ξ max ). To compare P t ( ξ min , ξ max )with P C t ( ξ min , ξ max ), we rely on the censoring inequality [PW13, Theorem 1] (to see that theexclusion process fits the setup in [PW13], one uses the height function representation see e.g.[Lac16b, Section A.2]) which implies that P t ( ξ min , · ) stochastically dominates P C t ( ξ min , · ). (cid:3) We consider a modified censored dynamics, where on top of censoring, at fixed time, wereplace the current configuration by one which is lower for the order ≥ by moving some particlesto the left. For the application we have in mind, we can consider that these replacements areperformed deterministically (although the result would hold also for random replacements).Let ( s i ) Ii =1 be an increasing time sequence tending to infinity and let ( Q i ) Ii =1 be a sequenceof stochastic matrices on Ω N,k such that for all ξ in Ω N,k there exists ξ ′ (depending on ξ and i )such that ξ ′ ≤ ξ,Q i ( ξ, ξ ′ ) = 1 ,Q i ( ξ, ξ ′′ ) = 0 , when ξ ′′ = ξ ′ . (3.9)We consider e P t the semigroup defined by e P = Id ,∂ t e P t = e P t L C ,t if t / ∈ { s i } Ii =1 , e P s i = e P ( s i ) − Q i . (3.10) Proposition 3.3.
For any choice of ( s i ) Ii =1 , ( Q i ) Ii =1 and C , we have for all t ≥ P C t ( ξ min , ξ max ) ≥ e P t ( ξ min , ξ max ) . (3.11) Proof.
We construct both ( e σ min t ) t ≥ with transition probability e P t with initial condition ξ min and ( σ min , C t ) t ≥ the censored dynamics with the same initial condition on the same probabilityspace, using the variables ( T ( x ) i , U ( x ) i ) i ≥ ,x ∈ J ,N K .For ( σ min , C t ) t ≥ we use the same procedure as for ( σ ξt ) t ≥ (for ξ = ξ min ) with the followingadded requirement for the transitions: { x, x + 1 } / ∈ C ( t ) in the case ( A ) and { x, x − } / ∈ C ( t ) inthe case ( B ).For ( e σ min t ) t ≥ we use the same procedure as for ( σ min , C t ) t ≥ but with the addition of new deter-ministic jumps in the trajectories at times ( s i ) i ∈ I . More precisely if t = s i , e σ min t is determinedfrom e σ min t − as the unique element of Ω N,k such that Q i ( e σ min t − , e σ min t ) = 1 . (3.12)We have by definition e σ min0 = σ min , C , and it can be checked by inspection that all the transitionsare order preserving (this is a property of the graphical construction when t / ∈ { s i } Ii =1 and aconsequence of (3.9) for the special values t ∈ { s i } Ii =1 ). (cid:3) Equilibrium estimates.
Recalling (2.27) let us define κ := F ′ ( λ ) = E h ρ λ log( ρ ) i > , (3.13)and set ∆ V ω,N max = max ≤ x ≤ y ≤ N ( V ( y ) − V ( x )) . (3.14)The literature on the subject of random walks in a random environment contains very sharpinformation concerning ∆ V ω,N max , and the length of the corresponding trap (see [GK13]). Inparticular it is known under quite general assumptions that | ∆ V ω,N max − λ log N | displays randomfluctuations of order 1 and that the corresponding traps are of a length λκ log N at first order.For the sake of completeness we include a short proof of the following non-optimal resultwhich is sufficient to our purpose. Set q N := 3 u + 2 | F ( u ) | log N, (3.15)where u is the point at which F attains its minimum. Proposition 3.4.
We have lim N →∞ P (cid:20) − (cid:18) ελ (cid:19) log log N ≤ ∆ V ω,N max − λ log N ≤ ελ log log N (cid:21) = 1 . (3.16) Furthermore we have lim N →∞ P max ≤ x ≤ y ≤ Ny − x ≥ q N ( V ( y ) − V ( x )) ≥ − N = 0 . (3.17) In particular, with high probability we have ∀ x, y ∈ J , N K , (cid:8) V ( y ) − V ( x ) = ∆ V ω,N max (cid:9) ⇒ { ( y − x ) ≤ q N } . Proof.
At the cost of an additive constant on our bounds (which we omit in the proof forreadability), using our uniform ellipticity assumption we can replace V ( y ) − V ( x ) in the definitionof (3.14) by a sum of IID random variables, setting ¯ V (1) = 0 and y X z = x +1 log ρ z := ¯ V ( y ) − ¯ V ( x ) . (3.18)By definition of λ , M n = (cid:0)Q nx =1 ( ρ x ) λ (cid:1) n ≥ is a martingale for the filtration G n := σ ( ω x , x ∈ J , n K ).Using the optional stopping theorem at T A := inf { n, M n ≥ A } and using that ( A ≤ M T A ≤ A (cid:0) − αα (cid:1) λ , lim n →∞ M n = 0 , (3.19)we have for any A A (cid:18) α − α (cid:19) λ ≤ P " max n ≥ n Y x =1 ( ρ x ) λ ≥ A ≤ A . (3.20)The bound above can be used to obtain the upper bound on ∆ V ω,N max via a union bound usingtranslation invariance P (cid:20) max ≤ x ≤ y ≤ N ¯ V ( y ) − ¯ V ( x ) ≥ λ log N + ελ log log N (cid:21) SEP IN RANDOM ENVIRONMENT 15 ≤ N X x =1 P (cid:20) max y ≥ x ¯ V ( y ) − ¯ V ( x ) ≥ λ log N + ελ log log N (cid:21) ≤ N P " max n ≥ n Y x =1 ( ρ x ) λ ≥ N (log N ) ε ≤ (log N ) − ε . (3.21)Before proving the corresponding lower bound, let us move to the proof of (3.17). Again usingtranslation invariance and union bound, it is sufficient to show thatlim N →∞ N P " max n ≥ q N n X x =1 log ρ x ≥ − N = 0 . (3.22)We use Doob’s maximal inequality for the martingale e − nF ( u ) Q nx =1 ( ρ x ) u . Since F ( u ) <
0, wehave P " max n ≥ q N n Y x =1 ( ρ x ) u ≥ N − u ≤ P " max n ≥ e − nF ( u ) n Y x =1 ( ρ x ) u ≥ N − u e − q N F ( u ) ≤ N u e q N F ( u ) ≤ N − . (3.23)This is sufficient to conclude the proof of (3.17). Note that as a consequence by (3.20) and(3.23), we have for N sufficiently large P " max ≤ n ≤ q N n Y x =1 ( ρ x ) λ ≥ N (log N ) − (1+ ε ) ≥ (cid:18) α − α (cid:19) λ N − (log N ) ε . (3.24)As a consequence of independence we have P (cid:20) ∀ ( i, j ) ∈ J , ⌊ N/q N ⌋ − K × J , q N K , ¯ V ( iq N + j ) − ¯ V ( iq N ) ≤ log N − (1 + ε ) log log Nλ (cid:21) ≤ − (cid:18) α − α (cid:19) λ N − (log N ) (1+ ε ) ! ⌊ N/q N ⌋− ≤ e − c (log N ) ε (3.25)This yields the lower bound in (3.16). (cid:3) Proof of Lemma 2.3.
For ξ ∈ Ω N,k , we define the positions of its leftmost particle and rightmostempty site to be respectively L N,k ( ξ ) := inf { x ∈ J , N K : ξ ( x ) = 1 } ,R N,k ( ξ ) := sup { x ∈ J , N K : ξ ( x ) = 0 } . (3.26)Then π ωN,k (cid:16) A ∁ r (cid:17) ≤ π ωN,k ( L N,k ( ξ ) ≤ N − k − r ) + π ωN,k ( R N,k ( ξ ) ≥ N − k + r ) . Let us bound the second term, the first one can be treated in a symmetric manner. Moreover,we have π ωN,k ( R N,k ( ξ ) ≥ N − k + r ) = X x ∈ J ,N − k K y ∈ J N − k + r,N K π ωN,k ( L N,k = x, R N,k = y ) (3.27)Furthermore, we recall that ξ x,y , defined in (2.7), denotes the configuration obtained by swappingthe values at sites x, y of the configuration ξ , and observe that the map ξ → ξ x,y is injectivefrom { ξ ∈ Ω N,k : L N,k ( ξ ) = x, R N,k ( ξ ) = y } to Ω N,k defined by ξ ξ x,y . Then we have π ωN,k ( L N,k = x, R N,k = y ) = X { ξ : L N,k ( ξ )= x,R N,k ( ξ )= y } π ωN,k ( ξ x,y ) e V ω ( y ) − V ω ( x ) ≤ e V ω ( y ) − V ω ( x ) ≤ Ce ¯ V ω ( y ) − ¯ V ω ( x ) . (3.28)Now from the law of large number applied to sum of IID variables, we havelim r →∞ inf N ≥ k ∈ J ,N/ K P (cid:20) ∀ ( x, y ) ∈ J , N − k K × J N − k + r, N K , ¯ V ω ( y ) − ¯ V ω ( x ) ≤ ( y − x ) E [log ρ ]2 (cid:21) = 1 . (3.29)Moreover, since X x ∈ J ,N − k K y ∈ J N − k + r,N K e E [log ρ y − x )2 ≤ e E [log ρ ] r/ (1 − e E [log ρ ] / ) we havelim r →∞ inf N ≥ k ∈ J ,N/ K P (cid:20) π ωN,k ( R N,k ( ξ ) ≥ N − k + r ) ≤ (cid:16) − e E [log ρ (cid:17) − e E [log ρ r (cid:21) = 1 , (3.30)which concludes the proof. (cid:3) Bounds for the mixing time with arbitrary environments
Proof of Proposition 2.1.
We look at the variable m ( ξ ) := N X x =1 xξ ( x ) . Note that m ( ξ ) ∈ h k ( k +1)2 , k (2 N − k +1)2 i . We assume that π ωN,k (cid:18) m ( ξ ) ≥ k ( N + 1)2 (cid:19) ≥ / ξ min (we write σ min t for σ ξ min t to lighten thenotation) we have E (cid:2) m ( σ min t ) (cid:3) ≤ k ( k + 1)2 + kt. (4.1)As a consequence of Markov’s inequality, we have P (cid:20) m ( σ min t ) ≥ k ( N + 1)2 (cid:21) = P (cid:20) m ( σ min t ) − k ( k + 1)2 ≥ k ( N − k )2 (cid:21) ≤ t ( N − k ) , (4.2)which is smaller than 1 / t ≤ N/ m ( ξ ), under the equilibrium measure π ωN,k (which is denoted simplyby π in this proof for readability). Let us prove thatVar π [ m ( ξ )] ≤ N k. (4.3)To this end we introduce the filtration ( G i ) Ni =1 defined by G i := σ ( ξ ( x ) , x ∈ J , i K ), and considerthe martingale M i := E π [ m ( ξ ) | G i ] (4.4) SEP IN RANDOM ENVIRONMENT 17 where E π [ · | G i ] denotes the conditional expectation under π . We have by constructionVar π [ m ( ξ )] = N X i =1 Var( M i − M i − ) (4.5)Now, we are going to show thatVar( M i − M i − ) ≤ π ( ξ i = 1)( N − i ) (4.6)which implies (4.3). To prove (4.6) we are going to show that for any χ ∈ { , } i − with at most k − N − k − i ( χ ) = E π (cid:2) m ( ξ ) | ξ J ,i − K = χ, ξ ( i ) = 0 (cid:3) − E π (cid:2) m ( ξ ) | ξ J ,i − K = χ, ξ ( i ) = 1 (cid:3) (4.7)satisfies 0 ≤ ∆ i ( χ ) ≤ N − i. (4.8)Note that we have E π (cid:2) m ( ξ ) | ξ J ,i − K = χ (cid:3) = i − X x =1 xχ ( x ) + π ω J i,N K ,k − P i − x =1 χ ( x ) N X x = i xξ ( x ) ! , (4.9)where if I is a segment on Z and k ′ ≤ | I | , π ωI,k ′ denotes the equilibrium measure for exclusionprocess on I with k ′ particles and environment ω . For this reason it is sufficient to prove (4.7)for i = 1, and arbitrary k (not necessarily assuming k ≤ N/ N ≥ k ∈ J , N − K we have0 ≤ E π [ m ( ξ ) | ξ (1) = 0] − E π [ m ( ξ ) | ξ (1) = 1] ≤ N − . (4.10)To prove this we observe that there exists a probability Π on Ω N,k with marginals π ( · | ξ (1) = 0)and π ( · | ξ (1) = 1) such that Π N X x =1 { ξ ( x ) = ξ ( x ) } = 2 ! = 1 (4.11)(meaning that ξ ( x ) = ξ ( x ) except at two sites, 1 and another random site). With this couplingwe have E π [ m ( ξ ) | ξ (1) = 0] − E π [ m ( ξ ) | ξ (1) = 1] = Π " N X x =1 x ( ξ ( x ) − ξ ( x )) , (4.12)which yields (4.10). The coupling Π can be achieved using the graphical construction: we define( ξ t ) and ( ξ t ) starting with initial configuration J ,k +1 K and J ,k K respectively and evolvingusing the graphical construction with the edge { , } censored (recall Section 3.3). The dynamicconserves the number of discrepancy and π ( ·| ξ (1) = 0) and π ( ·| ξ (1) = 1) are the respectiveequilibrium distribution of the marginals, so that any limit point of P (cid:2) ( ξ t , ξ t ) ∈ · (cid:3) (existence isensured by compactness) provides a coupling satisfying (4.11).Now to see that (4.7) implies (4.6), we simply observe that, conditionned to the state of thefirst i = 1 vertices of the segment, ( M i − M i − ) can only assume two values which differ by anamount ∆ i ( χ ) (cf. (4.7)). The corresponding conditioned variance is equal to ∆ i ( χ ) times thatof the corresponding Bernoulli variable that is E π [( M i − M i − ) | ξ J ,i − K = χ ] = π ( ξ ( i ) = 1 | ξ J ,i − K = χ ) π ( ξ ( i ) = 0 | ξ J ,i − K = χ )∆ i ( χ ) ≤ π ( ξ ( i ) = 1 | ξ J ,i − K = χ )( N − i ) . (4.13) We then consider the average the inequality with respect to ξ J ,i − K to conclude. Now using (4.3)we can assume that for any ε there exists N ( ε ) such that for N ≥ N ( ε ) we havemin (cid:2) π ωN,k ( m ( ξ ) ≤ N k/ , π ωN,k ( m ( ξ ) ≥ N k/ (cid:3) ≤ ε/ . (4.14)Let us assume that the first of these two terms is smaller (the other case is treated symmetrically).To conclude, we must show that for t = N we have P (cid:0) m ( σ min t ) > N k/ (cid:1) ≤ ε/ . (4.15)To check this we observe that m ( σ min t ) ≤ k ( k + 1)2 + N t (4.16)where N t is the total number of particle jumps to the right. Since each particle jumps at mostwith rate one, we have for N sufficiently large P [ N t ≥ kt ] ≤ ε/ , (4.17)which allows to conclude. (cid:3) Proof of Proposition 2.2.
For the proof of Proposition 2.2, we apply the so-called flowmethod (see [LP17, Chapter 13.4]). A path Γ is a sequence of configurations ( ξ , . . . , ξ | Γ | ) whichis such that r ω ( ξ i − , ξ i ) > i ∈ J , | Γ | K . For any given ordered pair ( ξ, ξ ′ ) ∈ Ω N,k × Ω N,k , weassign a path Γ ξ,ξ ′ , whose starting point is ξ and ending point is ξ ′ .Using [LP17, Corollary 13.21], the spectral gap of the chain can be controlled by a simplequantity depending on the functional ( ξ, ξ ′ ) Γ ξ,ξ ′ . We say that an unordered pair e = { ξ, ξ ′ } ⊂ Ω N,k is an edge if q ( e ) := π ωN,k ( ξ ) r ( ξ, ξ ′ ) > q ( e ) does not depend onthe orientation). We write e ∈ Γ = ( ξ , . . . , ξ | Γ | ) if there exists i ∈ J , | Γ | K such that e = { ξ i − , ξ i } .We have then (the factor 1 / ωN,k ≥ max e q ( e ) X ( ξ,ξ ′ ) ∈ Ω Nk × Ω N,k : e ∈ Γ ξ,ξ ′ π ωN,k ( ξ ) π ωN,k ( ξ ′ ) | Γ ξ,ξ ′ | − . (4.18)In the proof we describe a choice for Γ ξ,ξ ′ which yields a relevant bound for the spectral gap.Let us fix a state ξ ∗ ∈ Ω N,k which has maximal probability, that is such that V ω ( ξ ∗ ) = min ξ ∈ Ω N,k V ω ( ξ ) (4.19)(we make an arbitrary choice if there are several minimizers). Now to build the path Γ ξ,ξ ′ weare going to build first a path from ξ to ξ ∗ and then one from ξ ∗ to ξ ′ and then concatenate thetwo.We can thus focus on the construction of Γ ξ,ξ ∗ . Let m := d H ( ξ, ξ ∗ ) := 12 N X x =1 | ξ ( x ) − ξ ∗ ( x ) | denote the Hamming distance between ξ and ξ ∗ . Our first step is to build a sequence ξ (0) , . . . , ξ ( m ) which reduces the Hamming distance in incremental steps that is such that ξ (0) = ξ and ξ ( m ) = ξ ∗ ,d H ( ξ ( i − , ξ ( i ) ) = 1 for i ∈ J , m K ,d H ( ξ ( i ) , ξ ∗ ) = m − i for i ∈ J , m K . (4.20) SEP IN RANDOM ENVIRONMENT 19
The choice we make for ξ (0) , . . . , ξ ( m ) is not relevant for the result but let us fix one for the sakeof clarity. Let the sequences ( x i ) mi =1 and ( y i ) mi =1 be defined by x i := min ( x ∈ J , N K : N X x =1 ( ξ ( x ) − ξ ∗ ( x )) + = i ) ,y i := min y ∈ J , N K : N X y =1 ( ξ ∗ ( y ) − ξ ( y )) + = i . (4.21)These sequences locate the discrepancies between ξ and ξ ∗ . Then we define ξ ( i ) inductively asbeing obtained from ξ ( i − by moving the particle at x i to y i which is equivalent to setting ξ ( i ) = ξ ∧ ξ ∗ + i X j =1 { y j } + m X j ′ = i +1 { x j ′ } . Finally, our path from ξ to ξ ∗ is defined by concatenating paths Γ ( i ) , i ∈ J , m K , linking ξ ( i − to ξ ( i ) . We define Γ ( i ) = ( ξ ( i )0 , . . . , ξ ( i ) | x i − y i | ) as a path of minimal length | x i − y i | linking ξ ( i )0 := ξ ( i − to ξ ( i ) | x i − y i | = ξ ( i ) . To define the intermediate steps, let us assume for notational simplicity (andwithout loss of generality) that x i < y i . Moreover, let ( z j ) bj =1 be defined as the decreasingsequence such that ξ ( i − | J x i ,y i K = { z j } bj =1 . We then set d j := y i − z j if j ∈ J , b K and d := 0, and define ( ξ ( i ) ℓ ) y i − x i ℓ =1 by setting if d j − < ℓ ≤ d j ξ ( i ) ℓ := ξ ( i − ℓ − { z j } + { z j + ℓ − d j − } . (4.22)In other words, we move the particle at site z j ( j ≥
1) to site z j − (with z = y i ) starting from j = 1 until j = b . We refer to Figure 4.2 for a graphical description. Lemma 4.1.
For the path collection (cid:0) Γ ξ,ξ ′ (cid:1) constructed above, we have B := max e q ( e ) X ( ξ,ξ ′ ) ∈ Ω Nk × Ω N,k : e ∈ Γ ξ,ξ ′ π ωN,k ( ξ ) π ωN,k ( ξ ′ ) | Γ ξ,ξ ′ | ≤ α − N | Ω N,k | (cid:18) − αα (cid:19) N/ . (4.23)Let us now conclude the proof of Proposition 2.2. By (4.18) and Lemma 4.1, we havegap ωN,k ≥ αN − | Ω N,k | − (cid:18) − αα (cid:19) − N/ . (4.24)Observe that max ξ,ξ ′ ∈ Ω N,k (cid:0) V ω ( ξ ) − V ω ( ξ ′ ) (cid:1) ≤ N k log 1 − αα , and then min ξ ∈ Ω N,k π ωN,k ( ξ ) ≥ | Ω N,k | − (cid:18) − αα (cid:19) − Nk . (4.25)By (2.17), we have for ε ∈ (0 , / t N,k,ω mix ( ε ) ≤ α − N | Ω N,k | (cid:18) − αα (cid:19) N (cid:18) log | Ω N,k | + N k log 1 − αα − log ε (cid:19) . (4.26) (cid:3) Proof of Lemma 4.1.
A first observation is that by construction, our paths are of length smallerthan N . Let e be an edge and ( ξ, ξ ′ ) such that e ∈ Γ ξ,ξ ′ . By symmetry and taking away thefactor 1 /
2, we can always assume that e belongs to the first part of the path linking ξ to ξ ∗ .After replacing | Γ ξ,ξ ′ | by the upper bound and summing over all ξ ′ , we obtain that the quantitywe want to bound is exactly12 q ( e ) X ( ξ,ξ ′ ) ∈ Ω N,k × Ω N,k : e ∈ Γ ξ,ξ ′ π ωN,k ( ξ ) π ωN,k ( ξ ′ ) | Γ ξ,ξ ′ | ≤ N X ξ ∈ Ω N,k : e ∈ Γ ξ,ξ ∗ π ωN,k ( ξ ) q ( e ) . (4.27)Now let χ ( e, ξ ) denote the first end of e which is visited by the path going from ξ to ξ ∗ . Nowsimply observing that q ( e ) is at least α times the smallest probability π ωN,k of its two end points,we have π ωN,k ( ξ ) q ( e ) ≤ sup ξ ′ ∈ Γ ξ,ξ ∗ α − e V ( ξ ′ ) − V ( ξ ) . (4.28)Hence using the bound in the sum in (4.27) we obtain thatlog B ≤ log α − N | Ω N,k | + sup ξ ∈ Ω N,k ξ ′ ∈ Γ ξ,ξ ∗ V ( ξ ′ ) − V ( ξ ) . (4.29)To conclude we only need to prove that for every ξ ∈ Ω N,k and ξ ′ ∈ Γ ξ,ξ ∗ we have V ( ξ ′ ) − V ( ξ ) ≤ N (cid:18) − αα (cid:19) . (4.30)This follows simply by inspection from the following observation which follows from our con-struction and our assumptions.(i) In one step of Γ ξ,ξ ∗ , V varies at most by log (cid:0) − αα (cid:1) in absolute value.(ii) Along the sequence ( ξ ( i ) ) mi =1 , V ( ξ ( i ) ) is non-increasing.(iii) Each concatenated path Γ ( i ) has a length smaller than N (hence each ξ ( i ) ℓ is within N/ ξ ( i ) or ξ ( i − ) so that we havemax ≤ ℓ ≤| x i − y i | ( V ( ξ ( i ) ℓ ) − V ( ξ )) ≤ max ≤ ℓ ≤| x i − y i | (cid:16) V ( ξ ( i ) ℓ ) − V ( ξ ( i ) ) ∧ V ( ξ ( i − ) (cid:17) ≤ N − αα . ξ ( i − x i x i +1 x i +2 z z z z z
12 345 6 7 89 ξ ( i ) y i (L) (R) ℓV ( ξ ( i ) ℓ ) Figure 4.
A bold circle represents a particle, and a particle at the same site for the configu-rations ξ ( i − and ξ ( i ) is colored black. Otherwise, it is red or blue. (L) A graphical descriptionof the movements of the particle at site x i of ξ ( i − to the empty site y i and the numbers abovethe arrows are the relative order of the movements. (R) We draw the graph of ( ℓ, V ( ξ ( i ) ℓ )) ℓ . (cid:3) SEP IN RANDOM ENVIRONMENT 21 Lower bounds on the mixing time
Theorem 2.5 contains three separate lower bounds. The first one is a consequence of Propo-sition 2.1. In this section, we are going to prove the two remaining bounds which are restatedbelow as Propositions 5.1 and 5.2 respectively. The proof of these propositions rely on the twomechanisms exposed in Subsection 2.5: The potential barrier created by rare fluctuations of V ω (cf. Proposition 3.4) has the effect of trapping individual particles and slowing down the particleflow.5.1. A lower bound from the position of the first particle.Proposition 5.1.
We have with high probability t N,k,ω mix ≥ [ N (log N ) − ] λ (5.1) Proof.
We let y ( ω ) > x ( ω ) be such that V ( y ( ω )) − V ( x ( ω )) is maximized within 1 ≤ x ≤ y ≤ N/ V ω is non-increasing on J , N/ K is unlikely, and then it can be ignored).We are going to prove that w.h.p. t N,k,ω mix ≥ e e V ( y ) − V ( x ) − . (5.2)As a consequence of Proposition 3.4 (applied to the segment J , N/ K ), we have w.h.p. V ( y ) − V ( x ) ≥ λ log N − λ log log N + log 20 , so that (5.1) follows from (5.2). Recall the notation (3.1). Then considering that ξ min should bethe worst initial condition, we observe that d ωN,k ( t ) ≥ P (cid:20) ¯ σ min t (1) ≤ N (cid:21) − π ωN,k (cid:0) ¯ ξ (1) ≤ N/ (cid:1) . (5.3)As a consequence of Lemma 2.3 we have w.h.p. π ωN,k ( ¯ ξ (1) ≤ N/ ≤ / . To have an estimate on the mixing time, we must prove that P (cid:2) ¯ σ min t (1) ≤ N (cid:3) ≥ /
2. We define τ y := inf (cid:8) t ≥ σ min t (1) = y (cid:9) . We have P (cid:20) ¯ σ min t (1) > N (cid:21) ≤ P [ τ y ≤ t ] . (5.4)We are going to show that P [ τ y ≤ t ] ≤ e ( t + 1) e V ( x ) − V ( y ) , (5.5)which is sufficient for us to conclude that (5.2) holds. Using the graphical construction (with anenlargement of the probability space to sample the initial condition) we can couple σ min t with X πt a random walk on the interval J , y K with transitions rates given by q ωy (cf. (1.1)) andstarting with an initial distribution sampled from the equilibrium measure π ωy , , in such a waythat ∀ t ≤ τ y , ¯ σ min t (1) ≤ X πt . Setting e τ y := inf { t ≥ X πt = y } , we then have P [ τ y ≤ t ] ≤ P [ e τ y ≤ t ] . (5.6)We define the occupation time u ( t ) := Z t { y } ( X πs )d s. We have E [ u ( t + 1)] ≥ P [ u ( t + 1) ≥ ≥ P [ e τ y ≤ t ] P h ∀ s ∈ [0 ,
1] : X π e τ y + s = y i ≥ e − P [ e τ ≤ t ] , (5.7)where in the last inequality we use the strong Markov property. As the process ( X πt ) t ≥ isstationary, E [ u ( t + 1)] = ( t + 1) π ωy , ( y ) ≤ ( t + 1) e V ( y ) − V ( x ) , which allows to conclude that P [ e τ ≤ t ] ≤ e ( t + 1) e V ( y ) − V ( x ) . (5.8) (cid:3) A lower bound derived from flow consideration.
Let us now derive the third boundwhich is necessary to complete the proof of Theorem 2.5.
Proposition 5.2.
There exists a positive constant c = c ( α, P ) such that w.h.p. we have t N,k mix ≥ ckN λ (log N ) − ( λ ) . (5.9)To prove the above result, we adopt the strategy developed in [Sch19, Proposition 4.2] byinvestigating the flow of particles through a slow segment of size of order (log N ) where thedrift of the random environment points to the left. This flow of particles is controlled via acomparison with a boundary driven exclusion process.In [Sch19] the slow segment is selected to be such that ω x < / every site . It has theadvantage of simplifying the computation since it allows for comparison with the homogeneousexclusion process for which computation has been performed in [BECE00]. Our approach bringsan improvement by selecting the slow segment based on the potential function V ω . The relevantquantity that limits the flow is the worst potential barrier that the particles have to overcome.Proposition 3.4 allows to identify the worst potential barrier in the system. We let x ( ω ) ≤ y ( ω )be the smallest elements of J N/ , N/ K such that V ω ( y ) − V ω ( x ) = max N/ ≤ x ≤ y ≤ N/ ( V ω ( y ) − V ω ( x )) . According to Proposition 3.4 we have w.h.p. V ( y ) − V ( x ) ≥ λ (log N − N ) and y − x ≤ q N . (5.10)In order to illustrate how the mixing time can be controlled using the flow of particles, westart with a simple lemma. Let J t denote the number of particles on the last portion of thesegment, J t := X x ≥ y +1 σ min t ( x ) . (5.11) Lemma 5.3.
For any ε > , we have with high probability for every t ≥ . d ωN,k ( t ) ≥ − E [ J t ] k − ε. (5.12) Proof.
Setting B := n ξ ∈ Ω N,k : P x ≥ y +1 ξ ( x ) < k/ o , we have d ωN,k ( t ) ≥ k P ξ min t − π ωN,k k TV ≥ P [ σ min t ∈ B ] − π ωN,k ( B ) . (5.13)By Lemma 2.3, the second term is smaller than ε with high probability. Concerning the firstterm, we have by Markov’s inequality P (cid:2) σ min t ∈ B (cid:3) = 1 − P [ J t ≥ k/ ≥ − E [ J t ] k . (5.14) SEP IN RANDOM ENVIRONMENT 23 (cid:3)
Now we can control E [ J t ] by comparing our system with one in which the particles flow faster.We consider a process on a different state space e Ω x ,y := { ξ : J x , y + 1 K → Z + : ∀ x ∈ J x , y K , ξ ( x ) ∈ { , }} . (5.15)Under this new process the particles follow the exclusion dynamics in the bulk but new rulesare added at the boundaries. If ξ ( x ) = 0 then a particle is added at site x with rate one. Atthe other end of the segment particles can jump from site y to site y + 1 without respectingthe exclusion rule ( i.e. , the site y + 1 is allowed to contain arbitrarily many particles) andparticles at site y + 1 remain there forever. We define the generator of the process to be (for f : e Ω x ,y R ) e L ωx ,y f ( ξ ) := y − X z = x r ω ( ξ, ξ z,z +1 ) (cid:2) f ( ξ z,z +1 ) − f ( ξ ) (cid:3) + ω x − { ξ ( x )=0 } (cid:2) f ( ξ + δ x ) − f ( ξ ) (cid:3) + ω y { ξ ( y )=1 } (cid:2) f ( ξ − δ y + δ y +1 ) − f ( ξ ) (cid:3) , (5.16)where r ω is defined in (2.8). We refer to Figure 5 for a graphical description. We let ( e σ ξt ) t ≥ denote the corresponding process starting from an initial condition ξ ∈ e Ω x ,y . x y y + 1 ω x − x − ω x × ω y − ω y y ω y − ω y × − ω z z Figure 5.
A graphical representation of the boundary driven process: a bold circle represents aparticle, and the number above every arrow represents the jump rate while a red ” × ” representsan inadmissible jump. In addition, the site y + 1 can accommodate infinite many particles andall particles at site y + 1 stay put. Lemma 5.4.
Let denote the configuration with all sites in J x , y + 1 K being empty, and thenwe have J t ≤ e σ t ( y + 1) , (5.17) where J t is defined in (5.11) .Proof. The process ( e σ t ) t ≥ can be constructed together with ( σ min t ) t ≥ on the same probabil-ity space using the graphical construction of Section 3.2 (with the obvious adaptation of theconstruction to fit the boundary condition for ( e σ t ) t ≥ using the same clocks ( T ( x ) n ) x,n ∈ N andauxiliary variables ( U ( x ) n ) x,n ∈ N for the two processes. It can then be checked by inspection thatfor every t ≥ ∀ x ∈ J x , y + 1 K N X z = x σ min t ( z ) ≤ y +1 X z = x e σ t ( z ) . (5.18)Since the above inequality is satisfied at t = 0, it is sufficient to check that it is conserved byany update of the two processes. The result then just corresponds to the case x = y + 1. (cid:3) Proposition 5.5.
There exists a constant C = C ( α, P ) such that for all t ≥ w.h.p. we have E [ e σ t ( y + 1)] ≤ tCN − λ (log N ) λ ) . (5.19) With Proposition 5.5 whose proof is detailed in the next subsection, we are ready for the proofof Proposition 5.2.
Proof of Proposition 5.2.
By Lemma 5.3 and Lemma 5.4, we have d ωN,k ( t ) ≥ − E [ e σ t ( y + 1)] k . (5.20)By Proposition 5.5, we take t = 18 C kN λ (log N ) − λ ) in (5.20) to conclude the proof. (cid:3) Proof of Proposition 5.5.
Note that e σ t ( y + 1) is a superadditive ergodic sequence.To see this we let ϑ s denote the time shift operator on the graphical construction variables.Recalling (3.4) we set ϑ s (( T ( x ) i , U ( x ) i ) x ∈ Z ,i ≥ ) := (cid:16) T ( x ) i + i ( x,s ) − s, U ( x ) i + i ( x,s ) (cid:17) x ∈ Z ,i ≥ . (5.21)Now we observe that the graphical construction preserves the order on e Ω x ,y defined by ξ ξ ′ if ∀ x ≥ x , y +1 X z = x ξ ( z ) ≤ y +1 X z = x ξ ′ ( z ) . (5.22)Hence comparing the dynamic in the interval [ s, s + t ] with that starting from at time s , weobtain e σ s + t ( y + 1) ≥ e σ s ( y + 1) + ( ϑ s ◦ e σ ) t ( y + 1) . (5.23)Since the shift operator ϑ s on ( T, U ) is ergodic, we obtain from Kingman’s subbadditive ergodicTheorem [Kin73] (continuous time version) that E (cid:2)e σ t ( y + 1) (cid:3) ≤ t (cid:20) lim s →∞ s e σ s ( y + 1) (cid:21) . (5.24)Letting N s := P y x = x e σ s ( x ) denote the number of mobile particles in the system (particles atsite y + 1 which have stopped moving are not counted), we have e σ t ( y + 1) = X s ∈ (0 ,t ] {N s < N s − } . (5.25)Letting ( T n ) n ≥ denote the sequence of time at which N t < N t − (in increasing order), we havelim s →∞ s e σ s ( y + 1) = lim n →∞ n T n . (5.26)Similarly to (5.23), using preservation of order and the fact that σ s [ σ s ( y + 1) + y − x + 1] { y +1 } , we have for every s > e σ s + t ( y + 1) ≤ e σ s ( y + 1) + ( ϑ s ◦ e σ ) t ( y + 1) + y − x + 1 . (5.27)Now as a consequence of (5.27) T l + y − x +2 ≥ T l + ϑ T l ◦ T . (5.28) SEP IN RANDOM ENVIRONMENT 25
Since T l is a stopping time with respect to ( F t ) t ≥ (recall (3.5)), by the strong Markov property ϑ T l ◦ T is independent of T and has the same distribution. Iterating the process we obtain that T ( r − y − x +2)+1 ≥ T (1)1 + · · · + T ( r )1 (5.29)where ( T ( a )1 ) ra =1 is a sequence of IID copies of T . This yields thatlim inf n →∞ T n n ≥ y − x + 2 E [ T ] . (5.30)Finally let us compare ( e σ t ) t ≥ with ( e σ ′ t ) t ≥ starting from another initial condition. Now wespecify the initial condition. Let us first choose the number of particle by settingΛ( ω ) := { x ∈ J x , y K : V ( x ) ≤ [ V ( y ) + V ( x )] / } ,k ′ ( ω ) := ω ) . (5.31)We let ( e σ ′ t ) t ≥ be the dynamic with generator (5.16) and initial configuration e σ ′ is obtainedby setting e σ ′ ( y + 1) = 0 and sampling π ω [ x ,y ] ,k ′ (the invariant probability measure for theexclusion process on the segment J x , y K with k ′ particles) to set the values of ( e σ ′ ( x )) x ∈ J x ,y K .Using monotonicity again we have T ≥ inf { t ≥ e σ ′ t ( y + 1) = 1 } ≥ inf { t ≥ e σ ′ t ( x ) = 0 or e σ ′ t ( y ) = 1 } =: T ′ . (5.32)Now let us observe that until time T ′ , the process ( e σ ′ t ) t ≥ (or rather, its restriction to J x , y K )coincides with the exclusion process on the segment J x , y K with k ′ particles. Using this we canprove the following (the proof is postponed to the end of the section). Lemma 5.6.
We have E (cid:2) T ′ (cid:3) ≥ e ( y − x ) e V ( y − V ( x . (5.33)Let us now conclude the proof of Proposition 5.5. Combing (5.24), (5.26), (5.30) and (5.32),we have E (cid:2)e σ t ( y + 1) (cid:3) ≤ t (cid:20) lim s →∞ s e σ s ( y + 1) (cid:21) ≤ t ( y − x + 2) E [ T ] ≤ t ( y − x + 2) E [ T ′ ] . (5.34)Using Lemma 5.6, we obtain E (cid:2)e σ t ( y + 1) (cid:3) ≤ t e ( y − x + 2) e − V ( y − V ( x . (5.35)By (5.10), we have w.h.p. E (cid:2)e σ t ( y + 1) (cid:3) ≤ t e ( q N + 2) N − λ (log N ) .λ . (5.36) (cid:3) Proof.
Proof of Lemma 5.6 With a small abuse of notation, in this proof ( e σ ′ t ) t ≥ denotes theexclusion process on the segment J x , y K with k ′ particles starting from stationarity. Since E [ T ′ ] ≥ t P [ T ′ > t ], our goal is to provide a lower bound on P [ T ′ > t ]. We define B := (cid:8) ξ ∈ Ω J x ,y K ,k ′ : ξ ( x ) = 0 (cid:9) , B := (cid:8) ξ ∈ Ω J x ,y K ,k ′ : ξ ( y ) = 1 (cid:9) . (5.37)Using the strong Markov property at T ′ and the fact that jumping rates for particles are boundedfrom above by one at every site, we have P (cid:2) ∀ t ∈ [ T ′ , T ′ + 1] , e σ ′ t ∈ B ∪ B (cid:3) ≥ e − . Using independence as in (5.7), we have P (cid:2) T ′ ≤ t (cid:3) ≤ e ( t + 1) π ω [ x ,y ] ,k ′ ( B ∪ B ) . (5.38)We now head to provide an upper bound on π ω [ x ,y ] ,k ′ ( B ). Recalling the definition of Λ in (5.31),we observe that when ξ ∈ B , since x ∈ Λ and there are k ′ particles, there must be a particlein Λ ∁ := J x , y K \ Λ. Let R ( ξ ) be the position of the rightmost such particle R ( ξ ) := sup n z ∈ Λ ∁ : ξ ( z ) = 1 o , and set for z ∈ Λ ∁ B ,z := { ξ ∈ B : R ( ξ ) = z } . By moving the particle from site z to site x as in (3.28), we obtain π ω [ x ,y ] ,k ′ ( B ,z ) = X ξ ∈B ,z π ω [ x ,y ] ,k ′ ( ξ x ,z ) e − V ( z )+ V ( x ) ≤ e − V ( z )+ V ( x ) ≤ e − V ( y − V ( x , and then π ω [ x ,y ] ,k ′ ( B ) = X z ∈ Λ ∁ π ω [ x ,y ] ,k ′ ( B ,z ) ≤ ( y − x ) e − V ( y − V ( x . (5.39)Similarly, we can obtain π ω [ x ,y ] ,k ′ ( B ) ≤ ( y − x ) e − V ( y − V ( x . (5.40)Combining (5.39) with (5.40), in (5.38) we take t = 14 e ( y − x ) e V ( y − V ( x − E (cid:2) T ′ (cid:3) ≥ (cid:18) e ( y − x ) e V ( y − V ( x − (cid:19) ≥ e ( y − x ) e V ( y − V ( x . (5.41) (cid:3) Upper bound on the mixing time
This section is dedicated to the proof of Theorem 2.4. First in Section 6.1 we are going toreduce the problem to the estimation of the hitting time of ξ max . Afterwards using Proposition3.2 and Proposition 3.3 we are going to provide estimate of this hitting time using a modifiedcensored dynamics. First in Section 6.2 we treat the case of k ≤ q N which is a bit simpler andtreat the more general case q N < k ≤ N/ Deducing the mixing time from the hitting time of the maximal configuration.
Let us first show that the study of the mixing time can be reduced to that of the probability ofhitting the configuration ξ max starting from the other extremal configuration ξ min . Proposition 6.1.
We have for every t > and n ∈ N d ωN,k ( nt ) ≤ (1 − P t ( ξ min , ξ max )) n . (6.1) SEP IN RANDOM ENVIRONMENT 27
Proof.
We have (see for instance [LP17, Lemma 4.10]) d ωN,k ( t ) ≤ ¯ d ωN,k ( t ) := max ξ,ξ ′ k P ξt − P ξ ′ t k TV ≤ max ξ,ξ ′ P h σ ξt = σ ξ ′ t i (6.2)Using the monotonicity under the graphical construction (cf. Proposition 3.1) for all ξ ∈ Ω N,k and t ≥ σ min t ≤ σ ξt ≤ σ max t , where σ min and σ max are starting from the extremal conditions ξ min and ξ max in (3.3). As aconsequence for arbitrary ξ and ξ ′ with τ ′ := inf { t ≥ σ ξt = σ ξ ′ t } , we have ∀ t ≥ τ ′ , σ ξt = σ ξ ′ t . (6.3)On the other hand we have τ ′ ≥ τ := inf (cid:8) t ≥ σ min t = ξ max (cid:9) . (6.4)Therefore (6.2) implies that d ωN,k ( t ) ≤ P (cid:0) τ > t (cid:1) . (6.5)Using again the Markov property and the monotonicity in Proposition 3.1, we have for anypositive integer n P ( τ > nt ) ≤ P (cid:0) σ min it = ξ max , ∀ i ∈ J , n K (cid:1) ≤ P (cid:0) σ min t = ξ max (cid:1) n . (6.6) (cid:3) The case k N ≤ q N . Before stating the main result of this section, let us present a strategyto bound P t ( ξ min , ξ max ) from below. We present in the process a few key technical lemmas whoseproof is presented in the next subsection. We consider environment within the following event A N := ω : max ≤ x ≤ y ≤ Ny − x ≥ q N ( V ( y ) − V ( x )) ≤ − N . (6.7)Note that by Proposition 3.4, this is an high probability event. The event A N ensures that onsegments of length 4 q N , at equilibrium the particles concentrate on the right half of the segmentwith high probability. It also ensures that with high probability the last site is occupied by aparticle. Lemma 6.2. If ω ∈ A N , then we have for any x ∈ J , N − q N K and any k ≤ q N , π ω [ x +1 ,x +4 q N ] ,k (cid:2) ¯ ξ (1) ≤ x + 2 q N (cid:3) ≤ q N N − ,π ω [ x +1 ,x +4 q N ] ,q N [ ξ ( x + 4 q N ) = 0] ≤ q N N − . (6.8)Our second technical lemma is a direct consequence of Proposition 2.2. It allows to bound themixing time of the system for each of the intervals of length 4 q N in a quantitative way. Wedefine T = T N := 80 α − q N (cid:18) q N q N (cid:19) (cid:18) − αα (cid:19) q N log (cid:18) − αα (cid:19) . The following result is obtained by taking ε = N − in Proposition 2.2. Lemma 6.3.
Under the assumption (2.21) we have for all k ≤ q N , all ω and all x ∈ J , N − q N K d ω [ x +1 ,x +4 q N ] ,k ( T ) ≤ N − . (6.9) We are going to use the censoring inequality to guide all the particles to the right with thefollowing plan. We are going to design our censoring such that on the time interval [ iT, ( i + 1) T ),with i ∈ Z + satisfying 2( i + 2) q N < N , our k particles perform the exclusion process restrictedin the interval on the interval J iq N + 1 , i + 2) q N K (of length 4 q N ). Hence at each such timestep, particles take a time T to shift towards the right of an amount 2 q N . After the whole ⌈ N/ (2 q N ) ⌉ − J N − q N + 1 , N K . Once thisis done we conclude using censoring again by showing that the dynamics in J N − q N + 1 , N K with less than q N particles hits ξ max after time T with a positive probability. For this last stepwe need the following result which states roughly that ξ max has a positive weight with highprobability under the equilibrium measure. Lemma 6.4.
We have lim ε → inf N ≥ k ∈ J ,N/ K P (cid:2) π ωN,k ( ξ max ) > ε (cid:3) = 1 . (6.10) In particular if B N,k := n ω : π ω [ N − q N +1 ,N ] ,k ( ξ max ) ≥ q − N o , we have lim N →∞ inf k ∈ J ,q N K P [ B N,k ] = 1 . (6.11) Proposition 6.5. If k ≤ q N , if ω ∈ A N ∩ B N,k and setting t := T (cid:16)l N q N m − (cid:17) , we have P t ( ξ min , ξ max ) ≥ q N . (6.12) In particular the inequality holds with high probability.
The last part of the statement is of course a direct consequence of the first part combinedwith (6.11) and of Proposition 3.4 (which ensures that A N and B N,k are high probability events).Before proving a proof of Proposition 6.5 using the strategy exposed above, let us use it toconclude the proof of the upper bound on the mixing time.
Proof of Theorem 2.4 when k ≤ q N . By Proposition 6.1 and Proposition 6.5, we have d ωN,k (2 q N t ) ≤ (1 − P t ( ξ min , ξ max )) q N ≤ (cid:18) − q N (cid:19) q N ≤ , (6.13)which allows us to conclude the proof for the case k ≤ q N with the inequality (cid:18) q N q N (cid:19) ≤ (cid:18) (cid:19) q N . (cid:3) Now we move to prove Proposition 6.5 using the censoring inequality (Proposition 3.2). Moreprecisely, we define for i ∈ J , ⌈ N/ (2 q N ) ⌉ − K C i := n { i q N , i q N + 1 } , { ( i + 2)2 q N , ( i + 2)2 q N + 1 } o (6.14)and set C ⌈ N/ (2 q N ) ⌉− := { N − q N , N − q N + 1 } . (6.15)We define a censoring scheme by setting C ( t ) := C i for t ∈ [ iT, ( i + 1) T ) , i ∈ J , ⌈ N/ (2 q N ) ⌉ − K , (6.16)and C ( t ) = ∅ for t ≥ ⌈ N/ (2 q N ) ⌉ −
1. Let us write A fin := { ξ ∈ Ω N,k : ∀ x ∈ J , N − q N K , ξ ( x ) = 0 } . (6.17) SEP IN RANDOM ENVIRONMENT 29
Recalling the notation of Section 3.3, we let ( σ min , C t ) t ≥ denote the corresponding censoreddynamics with initial condition ξ min . Lemma 6.6. If ω ∈ A N , we have P h σ min , C ( ⌈ N/ q N ⌉− T ∈ A fin i ≥ − N − . (6.18) Proof.
For i ∈ J , ⌈ N/ q N ⌉ − K , we define A i := (cid:8) ξ ∈ Ω N,k : 2 iq N < ¯ ξ (1) ≤ ¯ ξ ( k ) ≤ i + 2) q N (cid:9) . Now we prove by induction that for all i ∈ J , ⌈ N/ q N ⌉ − K P h σ min , C iT ∈ A i i ≥ − i q N N . (6.19)From the definitions of C and ξ min , the inequality in (6.19) holds for i = 0. Assuming that(6.19) holds for i , then k particles perform the simple exclusion process restricted in the interval J iq N + 1 , i + 2) q N K . By Lemma 6.2 and Lemma 6.3 with x = 2 iq N , we have P h σ min , C ( i +1) T ∈ A i +1 i ≥ P h σ min , C iT ∈ A i i − (cid:16) π ω [2 iq N +1 , i +2) q N ] ,k (cid:0) ¯ ξ (1) ≤ i + 1) q N (cid:1) + d ω [2 iq N +1 , i +2) q N ] ,k ( T ) (cid:17) ≥ − i q N N − q N N . (6.20)This concludes the induction and the case i = ⌈ N/ q N ⌉ − ⌈ N/ q N ⌉ − q N ≥ N − q N . (cid:3) Proof of Proposition 6.5.
Using Proposition 3.2, it is sufficient to bound the corresponding prob-ability for the censored dynamics, that is, P C ( ⌈ N/ q N ⌉− T ( ξ min , ξ max ). If σ min , C ( ⌈ N/ q N ⌉− T ∈ A fin , thenthe restriction to the segment J N − q N + 1 , N K of the dynamics corresponds to an exclusionprocess with k particles on a segment of length 4 q N . Let π [ N − q N +1 ,N ] ,k and d [ N − q N +1 ,N ] ,k ( t )denote respectively the equilibrium measure and the distance to equilibrium for this dynamics,and then we have P C ( ⌈ N/ q N ⌉− T ( ξ min , ξ max ) ≥ P [ σ ξ min , C ( ⌈ N/ q N ⌉− T ∈ A fin ]( π [ N − q N +1 ,N ] ,k ( ξ max ) − d [ N − q N +1 ,N ] ,k ( T )) ≥ (1 − N − )(2 q − N − N − ) ≥ q N (6.21)where we have used the definition of B N,k (recall (6.11)) and Lemma 6.3 with x = N − q N . (cid:3) Proof of auxiliary lemmas.
Proof of Lemma 6.2.
To provide an upper bound on π ω [ x +1 ,x +4 q N ] ,k (cid:2) ¯ ξ (1) ≤ x + 2 q N (cid:3) , for ξ ∈ Ω [ x +1 ,x +4 q N ] ,k we define its rightmost empty site to be¯ R ( ξ ) := sup { y ∈ J x + 1 , x + 4 q N K : ξ ( y ) = 0 } . (6.22)As in (3.28), we have π ω [ x +1 ,x +4 q N ] ,k (cid:2) ¯ ξ (1) = z, ¯ R ( ξ ) = y (cid:3) ≤ e V ω ( y ) − V ω ( z ) ≤ N − (6.23)where we have used y − z ≥ q N and ω ∈ A N . Then we have π ω [ x +1 ,x +4 q N ] ,k (cid:2) ¯ ξ (1) ≤ x + 2 q N (cid:3) = X z ∈ J x +1 ,x +2 q N K y ∈ J x +4 q N − k +2 ,x +4 q N K π [ x +1 ,x +4 q N ] ,k (cid:2) ¯ ξ (1) = z, ¯ R ( ξ ) = y (cid:3) ≤ q N N − . (6.24)We now move to deal with π ω [ x +1 ,x +4 q N ] ,q N [ ξ ( x + 4 q N ) = 0]. For ξ ∈ Ω [ x +1 ,x +4 q N ] ,q N , we defineits leftmost particle to be¯ L ( ξ ) := inf { y ∈ J x + 1 , x + 4 q N K : ξ ( y ) = 1 } . As in (3.28), we have π ω [ x +1 ,x +4 q N ] ,q N (cid:2) ξ ( x + 4 q N ) = 0; ¯ L ( ξ ) = y (cid:3) ≤ e V ω ( x +4 q N ) − V ω ( y ) ≤ N − (6.25)where we have used y ≤ x + 3 q N and ω ∈ A N . Then π ω [ x +1 ,x +4 q N ] ,q N [ ξ ( x + 4 q N ) = 0] = X y ∈ J x +1 ,x +3 q N K π ω [ x +1 ,x +4 q N ] ,q N (cid:2) ξ ( x + 4 q N ) = 0; ¯ L ( ξ ) = y (cid:3) ≤ q N N − . (6.26) (cid:3) Proof of Lemma 6.4.
Recall the event A r in (2.25). Observe thatmax ξ ∈A r ( V ω ( ξ max ) − V ω ( ξ )) ≤ r log 1 − αα , (6.27)and then we have π ωN,k ( ξ max ) π ωN,k ( A r ) ≥ |A r | − exp (cid:18) − max ξ ∈A r ( V ω ( ξ max ) − V ω ( ξ )) (cid:19) ≥ − r e − r log − αα . (6.28)For given ε > r ( ε ) := − log 2 ε log − α ) α ! / (6.29)so that the rightmost hand-side of (6.28) is larger than 2 ε . Moreover, by (3.30) we know thatlim r →∞ inf N ≥ k ∈ J ,N/ K P h π ωN,k ( A r ) ≥ − − e E [log ρ ) − e E [log ρ r i = 1 . (6.30)Since when r is sufficiently large we have1 − − e E [log ρ ) − e E [log ρ r ≥ , then by (6.30) with r chosen as in (6.29) we obtainlim ε → inf N ≥ k ∈ J ,N/ K P (cid:2) π ωN,k ( ξ max ) ≥ ε (cid:3) = 1 . (6.31) (cid:3) SEP IN RANDOM ENVIRONMENT 31
The case k N ≥ q N . To treat the case of a larger number of particles, the small problemthere is with the strategy of the previous subsection is that it does not allow to channel all the k particles to the right at the same time. What we do instead is that we use the process totransport one particle to the right, and then use Proposition 3.3 to be able to move all otherparticles to the left and iterate the process. We largely recycle the strategy used in the previoussection. In the final step as in (6.21), we need to deal with the leftmost q N particles performingthe exclusion process restricted in the interval J N − k − q N + 1 , N − k + q N K , and then define B ′ N,k = n ω : π ω [ N − k − q N +1 ,N − k + q N ] ,q N ( ξ ′ max ) ≥ q − N o where ξ ′ max := { N − k +1 ≤ x ≤ N − k + q N } . By Lemma 6.4 we havelim N →∞ inf k ∈ J q N +1 ,N/ K P (cid:2) B ′ N,k (cid:3) = 1 . (6.32) Proposition 6.7. If k > q N and ω ∈ A N ∩ B ′ N,k , setting t := (cid:16)l N − k + q N q N m − (cid:17) ( k − q N + 1) T we have P t ( ξ min , ξ max ) ≥ q N . (6.33) Proof of Theorem 2.4 when k > q N . By Proposition 6.1 and Proposition 6.7, we have d ωN,k (2 q N t ) ≤ (1 − P t ( ξ min , ξ max )) q N ≤ (cid:18) − q N (cid:19) q N ≤ , (6.34)which allows us to conclude the proof for the case k > q N with the inequality (cid:18) q N q N (cid:19) ≤ (cid:18) (cid:19) q N . (cid:3) The remaining of the subsection is devoted to the proof of Proposition 6.7. This time we needto combine our censoring scheme with displacements of particles to the left (using Proposition3.3). Our plan is to first move (one by one) the rightmost k − q N particles to the segment J N − k + q N + 1 , N K and use censoring to block the these k − q N particles afterwards. We arethen left with the problem of moving the remaining q N particles, and this can be treated as inProposition 6.5.Let us explain our plan to move the the rightmost k − q N particles one by one with censoringand displacement. We proceed by induction (each step is going to leave aside an event of smallprobability, and our technical estimates are such that the sum over all steps of these probabilitieswill remain small). We set r = ⌈ ( N − k + q N ) / q N ⌉ −
1, and define for j ∈ J , k − q N K , i ∈ J , ⌈ ( N − k + q N ) / q N ⌉ − K , a i,j := k − q N − j + 2 q N i , C i,j := n { a i,j , a i,j + 1 } , { a i,j + 4 q N , a i,j + 4 q N + 1 } , { N − j, N − j + 1 } o , C ∗ j = n { N − q N − j, N − q N − j + 1 } , { N − j, N − j + 1 } o . (6.35)We define the censoring scheme C by setting C ( t ) = C i,j if t ∈ [( i + rj ) T, ( i + rj + 1) T ) , C ( t ) = C ∗ j if t ∈ [( r ( j + 1) − T, r ( j + 1) T ) , C ( t ) = ∅ if t ≥ r ( k − q N + 1) T. (6.36)The censored dynamic ( σ C , min t ) moves the first particle to the right in a time rT . Indeed, thesame mechanism used in the proof of Proposition 6.5 moves (w.h.p) the last q N particles in the segment J N − q N + 1 , N K by time ( r − T . Then we mix the q N particles within the segment J N − q N + 1 , N K and Lemma 6.2 ensures that after an additional time T , the last site N isoccupied by a particle.We then proceed by induction to show that for j ≤ k − q N all the sites in the segment J N − j + 1 , N K are occupied by particles by time rjT . Our censoring is designed so that aftertime rjT the number of particles in the j rightmost sites does not change.In order to facilitate the induction (this is not strictly necessary though) at each time of theform rjT =: s j we move all the leftmost N − j particles to the left on the segment J , N − j K ,so that the beginning of each induction step looks the same. We define thus Q j by setting Q j ( ξ, ξ ∗ j ) = 1 , Q j ( ξ, ξ ′ ) = 0 if ξ ′ = ξ ∗ j (6.37)where the function ξ → ξ ∗ j is defined by (recall (3.1))¯ ξ ∗ j ( ℓ ) = ( ℓ if l ≤ k − j, ¯ ξ ( ℓ ) if ℓ > k − j. (6.38)Since ξ ∗ j ≤ ξ , Q j satisfies (3.9). We let ( e σ t ) t ≥ denote the composed censored dynamics (recall(3.10)) corresponding to C , ( s j ) k − q N j =1 and ( Q j ) k − q N j =1 and starting from ξ min . We set ξ j := J ,k − j K + J N − j +1 ,N K . The following lemma formalizes in a quantitative manner the induction described above.
Lemma 6.8.
For all j ∈ J , k − q N K , we have P (cid:2)e σ rjT = ξ j (cid:3) ≥ − jq N N − . (6.39) Proof.
The statement is trivial for j = 0. For the induction step it is sufficient to prove that P (cid:2)e σ r ( j +1) T = ξ j +1 | e σ rjT = ξ j (cid:3) ≥ − q N N − . (6.40)With our choice for C , the j particles in the interval J N − j + 1 , N K do not move between time in-stants rjT and r ( j + 1) T , it is therefore sufficient to control P he σ r ( j +1) T ( N − j ) = 1 | e σ rjT = ξ j i .Let us define B j := ξ ∈ Ω N,k : N − j X N − j − q N +1 ξ ( x ) = q N (6.41)We can repeat the proof of Lemma 6.6 to obtain that P (cid:2)e σ rjT +( r − T ∈ B j | e σ rjT = ξ ∗ j (cid:3) ≥ − ( r −
1) 4 q N N . (6.42)Now in the time interval [ rjT + ( r − T, r ( j + 1) T ), the censoring makes the restriction of thedynamics to the segment J N − j − q N + 1 , N − j K an exclusion process with q N particles. Henceusing Lemma 6.3 and the second estimate in Lemma 6.2 we have for any χ ∈ B j P (cid:2)e σ r ( j +1) T ( N − j ) = 1 | e σ rjT +( r − T = χ (cid:3) ≥ − N − (1 + 3 q N ) . (6.43)Combining (6.42) and (6.43), we obtain P (cid:2)e σ r ( j +1) T = ξ j (cid:3) ≥ P (cid:2)e σ r ( j +1) T = ξ j (cid:3) − r q N N ≥ − j + 1) q N N − . (6.44) (cid:3) SEP IN RANDOM ENVIRONMENT 33
Proof of Proposition 6.7.
Taking j = k − q N in Lemma 6.8, from now on we assume that theevent { e σ ( k − q N ) rT = ξ k − q N } holds. Then the rightmost k − q N particles are frozen in the rightmost k − q N sites for t ≥ ( k − q N ) rT , and at t = ( k − q N ) rT the leftmost q N particles are in theleftmost q N sites. Thus we can repeat the proof in Proposition 6.5 to obtain P (cid:2)e σ r ( k − q N +1) T = ξ max (cid:3) ≥ q − N (cid:18) − ( k − q N ) 4 q N N (cid:19) ≥ q N (6.45)where we have used ω ∈ B ′ N,k . We conclude the proof by Proposition 3.2 and Proposition 3.3. (cid:3)
References [AFJV15] Luca Avena, Tertuliano Franco, Milton Jara, and Florian V¨ollering. Symmetric exclusion as a randomenvironment: hydrodynamic limits.
Ann. Inst. Henri Poincar´e Probab. Stat. , 51(3):901–916, 2015.[Ald83] David Aldous. Random walks on finite groups and rapidly mixing Markov chains. In
Seminar onprobability, XVII , volume 986 of
Lecture Notes in Math. , pages 243–297. Springer, Berlin, 1983.[BBHM05] Itai Benjamini, Noam Berger, Christopher Hoffman, and Elchanan Mossel. Mixing times of the biasedcard shuffling and the asymmetric exclusion process.
Transactions of the American MathematicalSociety , 357(8):3013–3029, 2005.[BECE00] RA Blythe, MR Evans, F Colaiori, and FHL Essler. Exact solution of a partially asymmetric ex-clusion model using a deformed oscillator algebra.
Journal of Physics A: Mathematical and General ,33(12):2313, 2000.[Fag08] Alessandra Faggionato. Random walks and exclusion processes among random conductances on ran-dom infinite clusters: homogenization and hydrodynamic limit.
Electronic Journal of Probability ,13:2217–2247, 2008.[Fag20] Alessandra Faggionato. Hydrodynamic limit of simple exclusion processes in symmetric random envi-ronments via duality and homogenization. arXiv e-prints , page arXiv:2011.11361, November 2020.[GK13] Nina Gantert and Thomas Kochler. Cutoff and mixing time for transient random walks in randomenvironments.
ALEA Lat. Am. J. Probab. Math. Stat. , 10(1):449–484, 2013.[GNS20] Nina Gantert, Evita Nestoridi, and Dominik Schmid. Mixing times for the simple exclusion processwith open boundaries. arXiv e-prints , page arXiv:2003.03781, March 2020.[HKT20] Marcelo R Hil´ario, Daniel Kious, and Augusto Teixeira. Random walk on the simple symmetricexclusion process.
Communications in Mathematical Physics , 379(1):61–101, 2020.[HS15] Fran¸cois Huveneers and Fran¸cois Simenhaus. Random walk driven by simple exclusion process.
Elec-tronic Journal of Probability , 20, 2015.[Jar11] Milton Jara. Hydrodynamic limit of the exclusion process in inhomogeneous media. In
Dynamics,Games and Science II , pages 449–465. Springer, 2011.[JM20] Milton Jara and Ot´avio Menezes. Symmetric exclusion as a random environment: invariance principle.
Annals of Probability , 48(6):3124–3149, 2020.[Kin73] J. F. C. Kingman. Subadditive ergodic theory.
Ann. Probability , 1:883–909, 1973.[KKS75] Harry Kesten, Mykyta V Kozlov, and Frank Spitzer. A limit law for random walk in a randomenvironment.
Compositio Mathematica , 30(2):145–168, 1975.[Lac16a] Hubert Lacoin. The cutoff profile for the simple exclusion process on the circle.
Ann. Probab. ,44(5):3399–3430, 2016.[Lac16b] Hubert Lacoin. Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion.
The Annals of Probability , 44(2):1426–1487, 2016.[Lig12] Thomas Milton Liggett.
Interacting particle systems , volume 276. Springer Science & Business Media,2012.[LL19] Cyril Labb´e and Hubert Lacoin. Cutoff phenomenon for the asymmetric simple exclusion process andthe biased card shuffling.
The Annals of Probability , 47(3):1541–1586, 2019.[LL20] Cyril Labb´e and Hubert Lacoin. Mixing time and cutoff for the weakly asymmetric simple exclusionprocess.
Annals of Applied Probability , 30(4):1847–1883, 2020.[LP16] David A Levin and Yuval Peres. Mixing of the exclusion process with small bias.
Journal of StatisticalPhysics , 165(6):1036–1050, 2016.[LP17] David A. Levin and Yuval Peres.
Markov chains and mixing times . American Mathematical Society,Providence, RI, 2017. Second edition of [ MR2466937], With contributions by Elizabeth L. Wilmer,With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson.[Mor06] Ben Morris. The mixing time for simple exclusion.
Ann. Appl. Probab. , 16(2):615–635, 2006. [PW13] Yuval Peres and Peter Winkler. Can extra updates delay mixing?
Communications in MathematicalPhysics , 323(3):1007–1016, 2013.[Sch19] Dominik Schmid. Mixing times for the simple exclusion process in ballistic random environment.
Electronic Journal of Probability , 24, 2019.[Sin82] Yakov Grigor’evich Sinai. Limit behaviour of one-dimensional random walks in random environments.
Teoriya Veroyatnostei i ee Primeneniya , 27(2):247–258, 1982.[Sol75] Fred Solomon. Random walks in a random environment.
The annals of probability , pages 1–31, 1975.[Szn04] Alain-Sol Sznitman. Topics in random walks in random environment. In
School and conference onprobability theory: 13-17 May 2002 , volume 17, pages 203–266. The Abdus Salam International Centrefor Theoretical Physics, 2004.[Wil04] David Bruce Wilson. Mixing times of lozenge tiling and card shuffling markov chains.
The Annals ofApplied Probability , 14(1):274–325, 2004.[Zei04] Ofer Zeitouni. Part ii: Random walks in random environment. In
Lectures on Probability Theory andStatistics , pages 190–312. Springer, 2004.
Hubert LacoinIMPA, Estrada Dona Castorina, 110, Rio de Janeiro 22460-320, Brazil.
Email address : [email protected] Shangjie YangIMPA, Estrada Dona Castorina, 110, Rio de Janeiro 22460-320, Brazil.
Email address ::