Walk In Random Environment Research Articles

Stochastic domination in space‐time for the contact process

AbstractLiggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d. Bernoulli product measure. In particular, they proved this for and (for infection rate sufficiently large) d‐ary homogeneous trees Td. In this paper, we prove some space‐time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on Td with . We show that, for large infection rate, there exists a subset Δ of the vertices of Td, containing a “positive fraction” of all the vertices of Td, such that the following holds: The contact process on Td observed on Δ stochastically dominates an independent spin‐flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on observed on certain d‐dimensional space‐time slabs stochastically dominates an i.i.d. Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.

Precise large deviations for random walk in random environment

We study one-dimensional nearest neighbour random walk in site-dependent random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.

Slowdown estimates for one-dimensional random walks in random environment with holding times

We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.

Limit theory for random walks in degenerate time-dependent random environments

We study continuous-time (variable speed) random walks in random environments on Zd, d≥2, where, at time t, the walk at x jumps across edge (x,y) at time-dependent rate at(x,y). The rates, which we assume stationary and ergodic with respect to space–time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer–Sjöstrand representation of gradient models with certain non-strictly convex potentials.

Quenched Large Deviations for Simple Random Walks on Percolation Clusters Including Long-Range Correlations

We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster (SRWPC) on $\mathbb Z^d$ ($d\geq 2$). The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and {the vacant set of random interlacements} (for $d\geq 3$) and the level sets of the Gaussian free field ($d\geq 3$). Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the {\it{pair empirical measures}} of the environment Markov chain in the non-elliptic case of SRWPC . Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main difficulty in our set up lies in the inherent non-ellipticity as well as the lack of {\it{translation-invariance}} stemming from conditioning on the fact that the origin belongs to the infinite cluster. We develop a unifying approach for proving quenched large deviations for SRWPC based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster.

Range of a random walk in random environment

Consider a random walk in random environment $\{X_n\}_{n\ge0}.$ In each step, the walk jumps at most a distance $2$ to the right or a distance $1$ to the left. For the walk transient to the right, let $N(x)=\#\{i\in[0,x]:$\linebreak $ \exists\, n\ge0, X_n= i \}$ be the number of sites in $[0,x]$ ever being visited by the walk. It is proved that almost surely $\lim_{x\rightarrow\infty}\frac{N(x)}{x}=\theta$ for some $0＜\theta＜1.$ The result shows that the range of the walk covers only a linear proportion of the lattice of the positive half line. For the nearest neighbor random walk in random or non-random environment, this phenomenon could not appear in any circumstance.

Variational formulas and disorder regimes of random walks in random potentials

We give two variational formulas (qVar1) and (qVar2) for the quenched free energy of a random walk in random potential (RWRP) when (i) the underlying walk is directed or undirected, (ii) the environment is stationary and ergodic, and (iii) the potential is allowed to depend on the next step of the walk which covers random walk in random environment (RWRE). In the directed i.i.d. case, we also give two variational formulas (aVar1) and (aVar2) for the annealed free energy of RWRP. These four formulas are the same except that they involve infima over different sets, and the first two are modified versions of a previously known variational formula (qVar0) for which we provide a short alternative proof. Then, we show that (qVar0) always has a minimizer, (aVar2) never has any minimizers unless the RWRP is an RWRE, and (aVar1) has a minimizer if and only if the RWRP is in the weak disorder regime. In the latter case, the minimizer of (aVar1) is unique and it is also the unique minimizer of (qVar1), but (qVar2) has no minimizers except for RWRE. In the case of strong disorder, we give a sufficient condition for the nonexistence of minimizers of (qVar1) and (qVar2) which is satisfied for the log-gamma directed polymer with a sufficiently small parameter. We end with a conjecture which implies that (qVar1) and (qVar2) have no minimizers under very strong disorder.

Combining Losing Games into a Winning Game

Parrondo's paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec's theorem. The regime switching mechanism gives our models a richer and more complex asymptotic behavior than the simple random walks in random environments appearing in the literature, in terms of transience and recurrence.

Random walks in Dirichlet environment: an overview

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on $\Bbb{Z}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights $(\alpha_i)_{i=1, \ldots, 2d}$, one for each direction of $\Bbb{Z}^d$. In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we give a state of the art on this model and several sketches of proofs presenting the core of the arguments. We also present new computation of the large deviation rate function for one dimensional RWDE.

Scaling limit of the recurrent biased random walk on a Galton–Watson tree

We show that the trace of the null recurrent biased random walk on a Galton–Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton–Watson tree.

Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments

In this work, we discuss certain ballistic random walks in random environments on $\mathbb{Z}^d$, and prove the equivalence between the static and dynamic points of view in dimension $d\geq4$. Using this equivalence, we also prove a version of a local limit theorem which relates the local behavior of the quenched and annealed measures of the random walk by a prefactor.

Range and critical generations of a random walk on Galton–Watson trees

In this paper we consider a random walk in random environment on a tree and focus on the frontier case for the underlying branching potential. We study the range $R_n$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of the order of $n / \log n$. This result is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\log n)^2$, which turn out to be the most visited generations. Our proof which involves a quenched analysis gives a description of the typical environments responsible for the behavior of $R_n$.

Maximum likelihood estimator consistency for recurrent random walk in a parametric random environment with finite support

We consider a one-dimensional recurrent random walk in random environment (RWRE) when the environment is i.i.d. with a parametric, finitely supported distribution. Based on a single observation of the path, we provide a maximum likelihood estimation procedure of the parameters of the environment.Unlike most of the classical maximum likelihood approach, the limit of the criterion function is in general a non degenerate random variable and convergence does not hold in probability. Not only the leading term but also the second order asymptotic is needed to fully identify the unknown parameter. We present different frameworks to illustrate these facts. We also explore the numerical performance of our estimation procedure.

Sharp ellipticity conditions for ballistic behavior of random walks in random environment

We sharpen ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ram\'{\i}rez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ram\'{\i}rez. As a corollary, we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension $d=2$ under which the corresponding random walk is ballistic.

Random-walk in Beta-distributed random environment

We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in $$\mathbb {Z}$$ which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution. We also describe a related directed polymer model, which is a limit of the q-Hahn interacting particle system. Using a Fredholm determinant representation for the quenched probability distribution function of the walker’s position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the walker’s position, with convergence to the Tracy–Widom distribution. We also show that this limit theorem can be interpreted in terms of the maximum of strongly correlated random variables: the positions of independent walkers in the same environment. The zero-temperature counterpart of the Beta RWRE can be studied in a parallel way. We also prove a Tracy–Widom limit theorem for this model.

Einstein relation for random walks in random environment

In this article, we consider the speed of the random walks in a (uniformly elliptic and i.i.d.) random environment (RWRE) under perturbation. We obtain the derivative of the speed of the RWRE w.r.t. the perturbation, under the assumption that one of the following holds: (i) the environment is balanced and the perturbation satisfies a Kalikow-type ballisticity condition, (ii) the environment satisfies Sznitman's ballisticity condition. This is a generalized version of the Einstein relation for RWRE. Our argument is based on a modification of Lebowitz-Rost's argument developed in [Stochastic Process. Appl. 54 (1994) 183-196] and a new regeneration structure for the perturbed balanced environment.

A quenched functional central limit theorem for random walks in random environments under [formula omitted

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in Rassoul-Agha and Seppäläinen (2009) and Berger and Zeitouni in Berger and Zeitouni (2008) under the assumption of large finite moments for the regeneration time. In this paper, with the extra (T)γ condition of Sznitman we reduce the moment condition to E(τ2(lnτ)1+m)<+∞ for m>1+1/γ, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

Einstein relation for reversible random walks in random environment on [formula omitted

The aim of this paper is to consider reversible random walk in a random environment in one dimension and prove the Einstein relation for this model. It says that the derivative at 0 of the effective velocity under an additional local drift equals the diffusivity of the model without drift (Theorem 1.2). Our method here is very simple: we solve the Poisson equation (Pω−I)g=f and then use the pointwise ergodic theorem in Wiener (1939) [10] to treat the limit of the solutions to obtain the desired result. There are analogous results for Markov processes with discrete space and for diffusions in random environment.

Exit laws from large balls of (an)isotropic random walks in random environment

We study exit laws from large balls in $\mathbb{Z}^d$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment. Our work includes the results on isotropic random walks in random environment of Bolthausen and Zeitouni [Probab. Theory Related Fields 138 (2007) 581-645]. Since several proofs in Bolthausen and Zeitouni (2007) were incomplete, a somewhat different approach was given in the first author's thesis [Long-time behavior of random walks in random environment (2013) Z\"{u}rich Univ.]. Here, we extend this approach to certain anisotropic walks and provide a further step towards a fully perturbative theory of random walks in random environment.

Branching random walks in random environment and super-Brownian motion in random environment

Nous etudions l’existence et la caracterisation de la limite de marches branchantes critiques dans un environnement spatio-temporel aleatoire en dimension 1 introduit par Birkner, Geiger and Kersting dans (In Interacting Stochastic Systems (2005) 269–291 Springer). Chaque particule effectue une marche aleatoire simple sur $\mathbb{Z}$ et le mecanisme de branchement depend du site indexe par l’espace et le temps. La limite de ce processus a valeur mesure est caracterisee comme l’unique solution d’un probleme de martingale non-trivial et correspond au super mouvement Brownien en environnement aleatoire par Mytnik dans (Ann. Probab. 24 (1996) 1953–1978).