It is well known that the distribution of simple random walks on Z conditioned on returning to th... more It is well known that the distribution of simple random walks on Z conditioned on returning to the origin after 2n steps does not depend on p = P (S 1 = 1), the probability of moving to the right. Moreover, conditioned on {S 2n = 0} the maximal displacement max k≤2n |S k | converges in distribution when scaled by √ n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on Z. We show that under the quenched law P ω (conditioned on the environment ω), the maximal displacement of the random walk when conditioned to return to the origin at time 2n is no longer necessarily of the order √ n. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time 2n is of order n κ/(κ+1) , where the constant κ > 0 depends on the law on environment. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time 2n is at least n 1−ε and at most n/(ln n) 2−ε for any ε > 0. As a consequence of our proofs, we obtain precise rates of decay for P ω (X 2n = 0). In particular, for certain non-nestling environments we show that P ω (X 2n = 0) = exp{−Cn − C ′ n/(ln n) 2 + o(n/(ln n) 2)} with explicit constants C, C ′ > 0. Keywords: random walk in random environment, moderate deviations. 2000 Mathematics subject classification: 60K37. Résumé. Il est bien connu que la distribution d'une marche aléatoire simple sur Z, conditionéeà retournerà l'origine au temps 2n est indépendante de p = P (S 1 = 1), la probabilité d'un pas vers la droite. De plus, conditionellementà {S 2n = 0}, le déplacement maximum max k≤2n |S k |, divisé par √ n, converge en distribution. Nous considérons le mème problème pour les marches transientes en environement aléatoire sur Z. Nous montrons que sous la loi "quenched," le déplacement maximum pour la marche conditionné a retournerà l'origine au temps 2n n'est pas toujours de l'ordre de √ n. Si l'environement a des drifts locaux positifs et negatifs alors cet ordre de grandeur est n κ/(κ+1) , où κ > 0 depend de la loi de l'environement. Mais, si l'environement n'a que des drifts locaux positifs ou nuls, alors cet ordre de grandeur est proche de n. Les preuves fournissent de plus l'ordre de grandeur de P ω (X 2n = 0). Dans le cas où les drifts locaux sont tous positifs nous montrons que P ω (X 2n = 0) = exp{−Cn−C ′ n/(ln n) 2 +o(n/(ln n) 2)}. Mots clés: marche aléatoire en environement aléatoire, déviations moderées.
This work is motivated by the study of some two-dimensional random walks in random environment (R... more This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely often, for any k. We also consider direct products of one-dimensional recurrent RWRE with another RWRE or with a RW. We point out the that models involving one-dimensional recurrent RWRE are more recurrent than the corresponding models involving simple symmetric walk.
We show that a sequence of birth-and-death chains, given by lazy random walks in a transient envi... more We show that a sequence of birth-and-death chains, given by lazy random walks in a transient environment (RWRE) on [0, n], exhibits a cutoff in the ballistic regime but does not exhibit a cutoff in the (interior of) the subballistic regime. We investigate the growth of the mixing times for this model. As an important step in the proof, we derive bounds for the quenched expectation and the quenched variance of the hitting times of the RWRE, which are of independent interest.
Let ( a k ) k ∈ N (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfyin... more Let ( a k ) k ∈ N (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfying the Hadamard gap condition a k + 1 / a k > q > 1 a_{k+1}/a_k> q >1 for all k ∈ N k\in \mathbb N , and let S n ( ω ) = ∑ k = 1 n cos ( 2 π a k ω ) , n ∈ N , ω ∈ [ 0 , 1 ] . \begin{equation*} S_n(\omega ) = \sum _{k=1}^n \cos (2\pi a_k \omega ), \qquad n\in \mathbb N, \; \omega \in [0,1]. \end{equation*} Then S n S_n is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space Ω = [ 0 , 1 ] \Omega = [0,1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for ( S n ) n ∈ N (S_n)_{n\in \mathbb {N}} has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large ...
We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are gen... more We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from x to y at rate rx,y provided y is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time t converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first n particles decouple.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019
We consider one-dimensional Mott variable-range hopping. This random walk is an effective model f... more We consider one-dimensional Mott variable-range hopping. This random walk is an effective model for the phononinduced hopping of electrons in disordered solids within the regime of strong Anderson localization at low carrier density. We introduce a bias and prove the linear response as well as the Einstein relation, under an assumption on the exponential moments of the distances between neighboring points. In a previous paper (Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 1165-1203) we gave conditions on ballisticity, and proved that in the ballistic case the environment viewed from the particle approaches, for almost any initial environment, a given steady state which is absolutely continuous with respect to the original law of the environment. Here, we show that this bias-dependent steady state has a derivative at zero in terms of the bias (linear response), and use this result to get the Einstein relation. Our approach is new: instead of using e.g. perturbation theory or regeneration times, we show that the Radon-Nikodym derivative of the bias-dependent steady state with respect to the equilibrium state in the unbiased case satisfies an L p-bound, p > 2, uniformly for small bias. This L p-bound yields, by a general argument not involving our specific model, the statement about the linear response. Résumé. Nous considérons le modèle « Mott variable-range hopping ». Cette marche aléatoire décrit la conduction des electrons dans des solides désordonnés dans le régime de localisation forte d'Anderson lorsque la densité des porteurs de charge est faible. En particulier, nous considérons une marche aléatoire de Mott unidimensionelle soumise à un champ extérieur. Sous une hypothèse à propos des moments exponentiels de la distance entre les points consécutifs, nous montrons la réponse linéaire et la relation d'Einstein. Dans un travail précedent, voir (Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 1165-1203), nous avons donné des conditions pour la ballisticité de la marche. En plus, nous avons montré que l'environnement vu de la particule converge en loi (pour presque tous les points de départ) vers une mesure invariante (état stationnaire) qui est absolument continue par rapport à la loi originale de l'environnement. Ici, nous montrons que cet état stationnaire a une dérivée en zéro par rapport au bias (réponse linéaire), et nous utilisons ce résultat pour démontrer la relation d'Einstein. Notre méthode est nouvelle : au lieu d'utiliser des arguments perturbatifs ou des temps de régéneration, nous donnons une borne en L p , p > 2, pour la densité de l'état stationnaire par rapport à la mesure invariante sans biais. Cette borne est uniforme dans le biais pour des biais qui sont proches de zéro. L'argument pour déduire la réponse linéaire de cette borne est général et ne dépend pas des détails de notre modèle.
Latin American Journal of Probability and Mathematical Statistics, 2019
We consider the backbone of the infinite cluster generated by supercritical oriented site percola... more We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 `1. A directed random walk on this backbone can be seen as an "ancestral lineage" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution (under the averaged law) to the Brownian web. Hence, we prove convergence to the Brownian web for a particular system of coalescing random walks in a dynamical random environment. An important tool in the proof is a tail bound on the meeting time of two walkers on the backbone, started at the same time. Our result can be interpreted as an averaging statement about the percolation cluster: apart from a change of variance, it behaves as the full lattice, i.e. the effect of the "holes" in the cluster vanishes on a large scale.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2018
Mott variable-range hopping is a fundamental mechanism for low-temperature electron conduction in... more Mott variable-range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps. We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias. Résumé. Le « Mott variable-range hopping » est un mécanisme décrivant la conduction des electrons dans des solides désordonnés dans le régime de localisation d'Anderson. Sous l'approximation de champ moyen, le modèle se réduit à une marche aléatoire (marche aléatoire de Mott) sur un processus ponctuel. Cette marche peut sauter d'un point du processus ponctuel à n'importe quel autre, les sauts ne sont donc pas limités en taille. Nous considerons une marche aléatoire de Mott unidimensionelle soumis à un champ extérieur (équivalent à un biais à droite). Nous montrons que la marche biaisée est transiente, et nous étudions sa vitesse linéaire. Nos résultats principaux sont des conditions pour la ballisticité (vitesse strictement positif) et la sous-ballisticité (vitesse nulle). Dans le regime ballistique, nous montrons l'existence d'une mesure invariante pour l'environment vu par la particule, absolument continue par rapport à la mesure originale. Si le processus ponctuel est un processus de renouvellement, nos conditions deviennent une condition nécessaire et suffisante pour la ballisticité. Nous montrons ainsi que la vitesse de la marche n'est pas, en général, une fonction continue du biais.
We consider random walks on the infinite cluster of a conditional bond percolation model on the i... more We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias λ > 0, then its asymptotic linear speed v is continuous in the variable λ > 0 and differentiable for all sufficiently small λ > 0. In the paper at hand, we complement this result by proving that v is differentiable at λ = 0. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at λ = 0 equals the diffusivity of the unbiased walk.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019
We consider biased random walk among iid, uniformly elliptic conductances on Z d , and investigat... more We consider biased random walk among iid, uniformly elliptic conductances on Z d , and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1.1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 1.3: it follows along the lines of the proof of the Einstein relation in (Ann. Probab. 45 (4) (2017) 2533-2567). On the other hand, we give a counterexample showing that for iid, uniformly elliptic conductances, the velocity is not always increasing as a function of the bias. More precisely, if d = 2 and if the conductances take the values 1 (with probability p) and κ (with probability 1 − p) and p is close enough to 1 and κ small enough, the velocity is not increasing as a function of the bias, see Theorem 1.2. Résumé. Nous étudions des marches aléatoires biaisées dans un milieu aléatoire donné par des poids iid sur les arêtes de Z d. Les poids sont bornés au-dessus et ils ont une borne inférieure qui est strictement positive. Nous nous intéressons pour la vitesse de la marche en fonction du bias. Un argument connu donne que, pour des biais suffisamment grands, la vitesse est une fonction croissante du biais. Notre résultat principal dit que si le désordre est petit, ce qui veut dire que les poids sont proches les uns aux autres, la vitesse est une fonction croissante du bias, voir Théorème 1.1. Un ingrédient crucial de la preuve est une formule pour la dérivée de la vitesse : cette dérivée peut etre écrit comme une covariance, voir Théorème 1.3. La preuve de Théorème 1.3 suis les arguments de la preuve de la relation d'Einstein dans (Ann. Probab. 45 (4) (2017) 2533-2567). Par contre, nous donnons un exemple montrant que pour des poids iid prenant les valeurs 1 (avec probabilité p) et κ (avec probabilité 1 − p), si p est suffisamment proche de 1 et κ est suffisamment petit, la vitesse n'est pas une fonction croissante du bias, voir Théorème 1.2.
We consider biased random walks on the infinite cluster of a conditional bond percolation model o... more We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed v of the walk. Namely, there exists some critical value λ c > 0 such that v > 0 if λ ∈ (0, λ c ) and v = 0 if λ ≥ λ c . We show that the speed v is continuous in λ on (0, ∞) and differentiable on (0, λ c /2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of v on (0, λ c /2), we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for λ ≥ λ c /2.
Let p ∈ [1, ∞]. Consider the projection of a uniform random vector from a suitably normalized p b... more Let p ∈ [1, ∞]. Consider the projection of a uniform random vector from a suitably normalized p ball in R n onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension n goes to ∞, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for p ∈ (1, ∞] (but not for p = 1), the corresponding rate function is "universal," in the sense that it coincides for "almost every" sequence of projection directions. We also analyze some exceptional sequences of directions in the "measure zero" set, including the sequence of directions corresponding to the classical Cramér's theorem, and show that those sequences of directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of p balls.
We consider real-valued branching random walks and prove a large deviation result for the positio... more We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a collection of a random number of dependent random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns out that the rate functions for upper large deviations coincide, but in general the rate functions for lower large deviations do not.
We consider d independent walkers on Z, m of them performing simple symmetric random walk and r =... more We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d -m of them performing recurrent RWRE (Sinai walk), in I independent random environments. We show that the product is recurrent, almost surely, if and only if m ≤ 1 or m = d = 2. In the transient case with r ≥ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r ≥ I = 1. In particular, while I does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these statements, we prove two subtle localization results for a single walker in a recurrent random environment, which are of independent interest.
We consider random walk among i.i.d., uniformly elliptic conductances on Z d , and prove the Eins... more We consider random walk among i.i.d., uniformly elliptic conductances on Z d , and prove the Einstein relation (see Theorem 1). It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle. These invariant measures are often called steady states. The Einstein relation follows at least for d ≥ 3, from an expansion of the steady states as a function of the bias (see Theorem 2), which can be considered our main result. This expansion is proved for d ≥ 3. In contrast to Guo [Ann. Probab. 44 (2016) 324-359], we need not only convergence of the steady states, but an estimate on the rate of convergence (see Theorem 4).
We consider one-dimensional random walks in random environment which are transient to the right. ... more We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time n the particle is typically at a distance of order O(n κ ) from the origin, κ ∈ (0, 1). We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time n, the particle is at a distance of order O(n ν 0 ) from the origin, ν 0 ∈ (0, κ)), and speedup (at time n, the particle is at a distance of order n ν 1 from the origin, ν 1 ∈ (κ, 1)), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time n, the particle is located around (-n ν ), thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.
In the study of large deviations for random walks in random environment, a key distinction has em... more In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk {X n } on a Galton-Watson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by |X n | the distance between the node X n and the root of T. Our main result is the almost sure equality of the large deviation rate function for |X n |/n under the "quenched measure" (conditional upon T), and the rate function for the same ratio under the "annealed measure" (averaging on T according to the Galton-Watson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a specific tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when {X n } is a λ-biased walk on a Galton-Watson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a "ubiquity" lemma for Galton-Watson trees, due to .
Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2007
Let (Z n) n∈N be a d-dimensional random walk in random scenery, i.e., Z n = n−1 k=0 Y (S k) with ... more Let (Z n) n∈N be a d-dimensional random walk in random scenery, i.e., Z n = n−1 k=0 Y (S k) with (S k) k∈N 0 a random walk in Z d and (Y (z)) z∈Z d an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of P(1 n Z n > b n) for various choices of sequences (b n) n in [1, ∞). Depending on (b n) n and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [AC03] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [Ch04]. Résumé : Soit (Z n) n∈N une marche aléatoire en paysage aléatoire sur Z d ; il s'agit du processus défini par Z n = n−1 k=0 Y (S k), où (S k) k∈N 0 est une marche aléatoireà valeurs dans Z d , et le paysage aléatoire (Y (z)) z∈Z d est une famille de variables aléatoires i.i.d. independante de la marche. On suppose que S 1 est centrée et admet certains moments exponentiels finis. Nous identifions la vitesse et la fonction de taux de P(1 n Z n > b n), pour diverses suites (b n) nà valeurs dans [1, ∞[. Selon le comportement de (b n) n et de la queue de distribution du paysage aléatoire, nous découvrons différents régimes ainsi que différentes formules variationnelles pour les fonctions de taux. Contrairement au travail récent de A. Asselah and F. Castell [AC03], nousétudions le cas où le paysage aléatoire n'est pas borné. Finalement, nous observons des liens intéressants avec certaines propriétés d'auto-intersection de la marche (S k) k∈N 0 , récemmentétudiées par X. Chen [Ch04].
It is well known that the distribution of simple random walks on Z conditioned on returning to th... more It is well known that the distribution of simple random walks on Z conditioned on returning to the origin after 2n steps does not depend on p = P (S 1 = 1), the probability of moving to the right. Moreover, conditioned on {S 2n = 0} the maximal displacement max k≤2n |S k | converges in distribution when scaled by √ n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on Z. We show that under the quenched law P ω (conditioned on the environment ω), the maximal displacement of the random walk when conditioned to return to the origin at time 2n is no longer necessarily of the order √ n. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time 2n is of order n κ/(κ+1) , where the constant κ > 0 depends on the law on environment. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time 2n is at least n 1−ε and at most n/(ln n) 2−ε for any ε > 0. As a consequence of our proofs, we obtain precise rates of decay for P ω (X 2n = 0). In particular, for certain non-nestling environments we show that P ω (X 2n = 0) = exp{−Cn − C ′ n/(ln n) 2 + o(n/(ln n) 2)} with explicit constants C, C ′ > 0. Keywords: random walk in random environment, moderate deviations. 2000 Mathematics subject classification: 60K37. Résumé. Il est bien connu que la distribution d'une marche aléatoire simple sur Z, conditionéeà retournerà l'origine au temps 2n est indépendante de p = P (S 1 = 1), la probabilité d'un pas vers la droite. De plus, conditionellementà {S 2n = 0}, le déplacement maximum max k≤2n |S k |, divisé par √ n, converge en distribution. Nous considérons le mème problème pour les marches transientes en environement aléatoire sur Z. Nous montrons que sous la loi "quenched," le déplacement maximum pour la marche conditionné a retournerà l'origine au temps 2n n'est pas toujours de l'ordre de √ n. Si l'environement a des drifts locaux positifs et negatifs alors cet ordre de grandeur est n κ/(κ+1) , où κ > 0 depend de la loi de l'environement. Mais, si l'environement n'a que des drifts locaux positifs ou nuls, alors cet ordre de grandeur est proche de n. Les preuves fournissent de plus l'ordre de grandeur de P ω (X 2n = 0). Dans le cas où les drifts locaux sont tous positifs nous montrons que P ω (X 2n = 0) = exp{−Cn−C ′ n/(ln n) 2 +o(n/(ln n) 2)}. Mots clés: marche aléatoire en environement aléatoire, déviations moderées.
This work is motivated by the study of some two-dimensional random walks in random environment (R... more This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely often, for any k. We also consider direct products of one-dimensional recurrent RWRE with another RWRE or with a RW. We point out the that models involving one-dimensional recurrent RWRE are more recurrent than the corresponding models involving simple symmetric walk.
We show that a sequence of birth-and-death chains, given by lazy random walks in a transient envi... more We show that a sequence of birth-and-death chains, given by lazy random walks in a transient environment (RWRE) on [0, n], exhibits a cutoff in the ballistic regime but does not exhibit a cutoff in the (interior of) the subballistic regime. We investigate the growth of the mixing times for this model. As an important step in the proof, we derive bounds for the quenched expectation and the quenched variance of the hitting times of the RWRE, which are of independent interest.
Let ( a k ) k ∈ N (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfyin... more Let ( a k ) k ∈ N (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfying the Hadamard gap condition a k + 1 / a k > q > 1 a_{k+1}/a_k> q >1 for all k ∈ N k\in \mathbb N , and let S n ( ω ) = ∑ k = 1 n cos ( 2 π a k ω ) , n ∈ N , ω ∈ [ 0 , 1 ] . \begin{equation*} S_n(\omega ) = \sum _{k=1}^n \cos (2\pi a_k \omega ), \qquad n\in \mathbb N, \; \omega \in [0,1]. \end{equation*} Then S n S_n is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space Ω = [ 0 , 1 ] \Omega = [0,1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for ( S n ) n ∈ N (S_n)_{n\in \mathbb {N}} has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large ...
We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are gen... more We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from x to y at rate rx,y provided y is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time t converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first n particles decouple.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019
We consider one-dimensional Mott variable-range hopping. This random walk is an effective model f... more We consider one-dimensional Mott variable-range hopping. This random walk is an effective model for the phononinduced hopping of electrons in disordered solids within the regime of strong Anderson localization at low carrier density. We introduce a bias and prove the linear response as well as the Einstein relation, under an assumption on the exponential moments of the distances between neighboring points. In a previous paper (Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 1165-1203) we gave conditions on ballisticity, and proved that in the ballistic case the environment viewed from the particle approaches, for almost any initial environment, a given steady state which is absolutely continuous with respect to the original law of the environment. Here, we show that this bias-dependent steady state has a derivative at zero in terms of the bias (linear response), and use this result to get the Einstein relation. Our approach is new: instead of using e.g. perturbation theory or regeneration times, we show that the Radon-Nikodym derivative of the bias-dependent steady state with respect to the equilibrium state in the unbiased case satisfies an L p-bound, p > 2, uniformly for small bias. This L p-bound yields, by a general argument not involving our specific model, the statement about the linear response. Résumé. Nous considérons le modèle « Mott variable-range hopping ». Cette marche aléatoire décrit la conduction des electrons dans des solides désordonnés dans le régime de localisation forte d'Anderson lorsque la densité des porteurs de charge est faible. En particulier, nous considérons une marche aléatoire de Mott unidimensionelle soumise à un champ extérieur. Sous une hypothèse à propos des moments exponentiels de la distance entre les points consécutifs, nous montrons la réponse linéaire et la relation d'Einstein. Dans un travail précedent, voir (Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 1165-1203), nous avons donné des conditions pour la ballisticité de la marche. En plus, nous avons montré que l'environnement vu de la particule converge en loi (pour presque tous les points de départ) vers une mesure invariante (état stationnaire) qui est absolument continue par rapport à la loi originale de l'environnement. Ici, nous montrons que cet état stationnaire a une dérivée en zéro par rapport au bias (réponse linéaire), et nous utilisons ce résultat pour démontrer la relation d'Einstein. Notre méthode est nouvelle : au lieu d'utiliser des arguments perturbatifs ou des temps de régéneration, nous donnons une borne en L p , p > 2, pour la densité de l'état stationnaire par rapport à la mesure invariante sans biais. Cette borne est uniforme dans le biais pour des biais qui sont proches de zéro. L'argument pour déduire la réponse linéaire de cette borne est général et ne dépend pas des détails de notre modèle.
Latin American Journal of Probability and Mathematical Statistics, 2019
We consider the backbone of the infinite cluster generated by supercritical oriented site percola... more We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 `1. A directed random walk on this backbone can be seen as an "ancestral lineage" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution (under the averaged law) to the Brownian web. Hence, we prove convergence to the Brownian web for a particular system of coalescing random walks in a dynamical random environment. An important tool in the proof is a tail bound on the meeting time of two walkers on the backbone, started at the same time. Our result can be interpreted as an averaging statement about the percolation cluster: apart from a change of variance, it behaves as the full lattice, i.e. the effect of the "holes" in the cluster vanishes on a large scale.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2018
Mott variable-range hopping is a fundamental mechanism for low-temperature electron conduction in... more Mott variable-range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps. We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias. Résumé. Le « Mott variable-range hopping » est un mécanisme décrivant la conduction des electrons dans des solides désordonnés dans le régime de localisation d'Anderson. Sous l'approximation de champ moyen, le modèle se réduit à une marche aléatoire (marche aléatoire de Mott) sur un processus ponctuel. Cette marche peut sauter d'un point du processus ponctuel à n'importe quel autre, les sauts ne sont donc pas limités en taille. Nous considerons une marche aléatoire de Mott unidimensionelle soumis à un champ extérieur (équivalent à un biais à droite). Nous montrons que la marche biaisée est transiente, et nous étudions sa vitesse linéaire. Nos résultats principaux sont des conditions pour la ballisticité (vitesse strictement positif) et la sous-ballisticité (vitesse nulle). Dans le regime ballistique, nous montrons l'existence d'une mesure invariante pour l'environment vu par la particule, absolument continue par rapport à la mesure originale. Si le processus ponctuel est un processus de renouvellement, nos conditions deviennent une condition nécessaire et suffisante pour la ballisticité. Nous montrons ainsi que la vitesse de la marche n'est pas, en général, une fonction continue du biais.
We consider random walks on the infinite cluster of a conditional bond percolation model on the i... more We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias λ > 0, then its asymptotic linear speed v is continuous in the variable λ > 0 and differentiable for all sufficiently small λ > 0. In the paper at hand, we complement this result by proving that v is differentiable at λ = 0. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at λ = 0 equals the diffusivity of the unbiased walk.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019
We consider biased random walk among iid, uniformly elliptic conductances on Z d , and investigat... more We consider biased random walk among iid, uniformly elliptic conductances on Z d , and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1.1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 1.3: it follows along the lines of the proof of the Einstein relation in (Ann. Probab. 45 (4) (2017) 2533-2567). On the other hand, we give a counterexample showing that for iid, uniformly elliptic conductances, the velocity is not always increasing as a function of the bias. More precisely, if d = 2 and if the conductances take the values 1 (with probability p) and κ (with probability 1 − p) and p is close enough to 1 and κ small enough, the velocity is not increasing as a function of the bias, see Theorem 1.2. Résumé. Nous étudions des marches aléatoires biaisées dans un milieu aléatoire donné par des poids iid sur les arêtes de Z d. Les poids sont bornés au-dessus et ils ont une borne inférieure qui est strictement positive. Nous nous intéressons pour la vitesse de la marche en fonction du bias. Un argument connu donne que, pour des biais suffisamment grands, la vitesse est une fonction croissante du biais. Notre résultat principal dit que si le désordre est petit, ce qui veut dire que les poids sont proches les uns aux autres, la vitesse est une fonction croissante du bias, voir Théorème 1.1. Un ingrédient crucial de la preuve est une formule pour la dérivée de la vitesse : cette dérivée peut etre écrit comme une covariance, voir Théorème 1.3. La preuve de Théorème 1.3 suis les arguments de la preuve de la relation d'Einstein dans (Ann. Probab. 45 (4) (2017) 2533-2567). Par contre, nous donnons un exemple montrant que pour des poids iid prenant les valeurs 1 (avec probabilité p) et κ (avec probabilité 1 − p), si p est suffisamment proche de 1 et κ est suffisamment petit, la vitesse n'est pas une fonction croissante du bias, voir Théorème 1.2.
We consider biased random walks on the infinite cluster of a conditional bond percolation model o... more We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed v of the walk. Namely, there exists some critical value λ c > 0 such that v > 0 if λ ∈ (0, λ c ) and v = 0 if λ ≥ λ c . We show that the speed v is continuous in λ on (0, ∞) and differentiable on (0, λ c /2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of v on (0, λ c /2), we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for λ ≥ λ c /2.
Let p ∈ [1, ∞]. Consider the projection of a uniform random vector from a suitably normalized p b... more Let p ∈ [1, ∞]. Consider the projection of a uniform random vector from a suitably normalized p ball in R n onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension n goes to ∞, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for p ∈ (1, ∞] (but not for p = 1), the corresponding rate function is "universal," in the sense that it coincides for "almost every" sequence of projection directions. We also analyze some exceptional sequences of directions in the "measure zero" set, including the sequence of directions corresponding to the classical Cramér's theorem, and show that those sequences of directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of p balls.
We consider real-valued branching random walks and prove a large deviation result for the positio... more We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a collection of a random number of dependent random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns out that the rate functions for upper large deviations coincide, but in general the rate functions for lower large deviations do not.
We consider d independent walkers on Z, m of them performing simple symmetric random walk and r =... more We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d -m of them performing recurrent RWRE (Sinai walk), in I independent random environments. We show that the product is recurrent, almost surely, if and only if m ≤ 1 or m = d = 2. In the transient case with r ≥ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r ≥ I = 1. In particular, while I does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these statements, we prove two subtle localization results for a single walker in a recurrent random environment, which are of independent interest.
We consider random walk among i.i.d., uniformly elliptic conductances on Z d , and prove the Eins... more We consider random walk among i.i.d., uniformly elliptic conductances on Z d , and prove the Einstein relation (see Theorem 1). It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle. These invariant measures are often called steady states. The Einstein relation follows at least for d ≥ 3, from an expansion of the steady states as a function of the bias (see Theorem 2), which can be considered our main result. This expansion is proved for d ≥ 3. In contrast to Guo [Ann. Probab. 44 (2016) 324-359], we need not only convergence of the steady states, but an estimate on the rate of convergence (see Theorem 4).
We consider one-dimensional random walks in random environment which are transient to the right. ... more We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time n the particle is typically at a distance of order O(n κ ) from the origin, κ ∈ (0, 1). We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time n, the particle is at a distance of order O(n ν 0 ) from the origin, ν 0 ∈ (0, κ)), and speedup (at time n, the particle is at a distance of order n ν 1 from the origin, ν 1 ∈ (κ, 1)), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time n, the particle is located around (-n ν ), thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.
In the study of large deviations for random walks in random environment, a key distinction has em... more In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk {X n } on a Galton-Watson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by |X n | the distance between the node X n and the root of T. Our main result is the almost sure equality of the large deviation rate function for |X n |/n under the "quenched measure" (conditional upon T), and the rate function for the same ratio under the "annealed measure" (averaging on T according to the Galton-Watson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a specific tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when {X n } is a λ-biased walk on a Galton-Watson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a "ubiquity" lemma for Galton-Watson trees, due to .
Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2007
Let (Z n) n∈N be a d-dimensional random walk in random scenery, i.e., Z n = n−1 k=0 Y (S k) with ... more Let (Z n) n∈N be a d-dimensional random walk in random scenery, i.e., Z n = n−1 k=0 Y (S k) with (S k) k∈N 0 a random walk in Z d and (Y (z)) z∈Z d an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of P(1 n Z n > b n) for various choices of sequences (b n) n in [1, ∞). Depending on (b n) n and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [AC03] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [Ch04]. Résumé : Soit (Z n) n∈N une marche aléatoire en paysage aléatoire sur Z d ; il s'agit du processus défini par Z n = n−1 k=0 Y (S k), où (S k) k∈N 0 est une marche aléatoireà valeurs dans Z d , et le paysage aléatoire (Y (z)) z∈Z d est une famille de variables aléatoires i.i.d. independante de la marche. On suppose que S 1 est centrée et admet certains moments exponentiels finis. Nous identifions la vitesse et la fonction de taux de P(1 n Z n > b n), pour diverses suites (b n) nà valeurs dans [1, ∞[. Selon le comportement de (b n) n et de la queue de distribution du paysage aléatoire, nous découvrons différents régimes ainsi que différentes formules variationnelles pour les fonctions de taux. Contrairement au travail récent de A. Asselah and F. Castell [AC03], nousétudions le cas où le paysage aléatoire n'est pas borné. Finalement, nous observons des liens intéressants avec certaines propriétés d'auto-intersection de la marche (S k) k∈N 0 , récemmentétudiées par X. Chen [Ch04].
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