Walk In Random Environment Research Articles

On a class of stochastic fractional heat equations

For the fractional heat equation ∂ ∂ t u ( t , x ) = − ( − Δ ) α 2 u ( t , x ) + u ( t , x ) W ˙ ( t , x ) \frac {\partial }{\partial t} u(t,x) = -(-\Delta )^{\frac {\alpha }{2}}u(t,x)+ u(t,x)\dot W(t,x) where the covariance function of the Gaussian noise W ˙ \dot W is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that d > 2 + α d>2+\alpha is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field.

Directional transience of random walks in random environments with bounded jumps

Directional transience of random walks in random environments with bounded jumps

Quantum walk in stochastic environment.

We consider a quantized version of the Sinai-Derrida model for "random walk in random environment." The model is defined in terms of a Lindblad master equation. For a ring geometry (a chain with periodic boundary condition) it features a delocalization-transition as the bias in increased beyond a critical value, indicating that the relaxation becomes underdamped. Counterintuitively, the effective disorder is enhanced due to coherent hopping. We analyze in detail this enhancement and its dependence on the model parameters. The nonmonotonic dependence of the Lindbladian spectrum on the rate of the coherent transitions is highlighted.

Mini-Workshop: New Horizons in Motions in Random Media

The general topic of the mini-workshop New Horizons in Motions in Random Media was the study of random walks in random environments, both in their own right and in relation to stochastic homogenization and to models in statistical mechanics, in particular spin system. This is a subject at the intersection of probability, analysis and mathematical physics, and the workshop brought together leading researchers from those areas. While each of these areas has been quite active for decades with many remarkable breakthroughs obtained throughout the years, the workshop provided a unique opportunity to identify principal new objectives and initiate new collaborations.

Quenched and averaged large deviations for random walks in random environments: The impact of disorder

In 2003, Varadhan (Comm. Pure Appl. Math. 56 (2003) 1222–1245) developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on Zd. One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in d≥4, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see (Probab. Theory Related Fields 149 (2011) 463–491)), satisfies a CLT or even a law of large numbers (Electron. Commun. Probab. 7 (2002)191–197; Ann. Probab. 36 (2008) 728–738). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our approach is the introduction of an auxiliary random walk in a deterministic environment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.

The effect of disorder on quenched and averaged large deviations for random walks in random environments: Boundary behavior

For a random walk in a uniformly elliptic and i.i.d. environment on Zd with d≥4, we show that the quenched and annealed large deviation rate functions agree on any compact set contained in the boundary ∂D≔{x∈Rd:|x|1=1} of their domain which does not intersect any of the (d−2)-dimensional facets of ∂D, provided that the disorder of the environment is low enough (depending on the compact set). As a consequence, we obtain a simple explicit formula for both rate functions on any such compact set of ∂D at low enough disorder. In contrast to previous works, our results do not assume any ballistic behavior of the random walk and are not restricted to neighborhoods of any given point (on the boundary ∂D). In addition, our results complement those in Bazaes et al. (2022), where, using different methods, we investigate the equality of the rate functions in the interior of their domain. Finally, for a general parametrized family of environments, we show that the strength of disorder determines a phase transition in the equality of both rate functions, in the sense that for each x∈∂D there exists ɛx such that the two rate functions agree at x when the disorder is smaller than ɛx and disagree when it is larger. This further reconfirms the idea, introduced in Bazaes et al. (2022), that the disorder of the environment is in general intimately related with the equality of the rate functions.

Some recent advances in random walks and random environments

Recent contributions to random walks in random environments and related topics are presented. We focus on non parametric estimation for one dimensional random walks in random environment and on the Dirichlet distribution on decomposable graphs.

Large Deviations for First Hitting time of Random Walk in Random Environment (lower level)

Доказаны локальные теоремы о больших уклонениях для момента $T_{-n}$ достижения уровня $-n$, $n\in\mathbb{N}$, случайным блужданием в случайной среде. Получены точные асимптотики вероятностей больших уклонений $\mathbf{P}(T_{-n} = k)$, равномерные по таким $k\in\mathbb{N}$, что $(n-k)$ четно и отношение $k/n=k(n)/n$ принадлежит некоторому компакту.

Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium

We consider the spatially inhomogeneous Moran model with seed-banks introduced in [20]. Populations comprising active and dormant individuals are spatially structured in colonies labeled by Zd,d≥1. The population sizes are sampled from a translation-invariant, ergodic, uniformly elliptic field that constitutes a static random environment. Individuals carry one of two types: ♡ and ♠. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from the seed-bank of their own colony, and resample type by choosing a parent uniformly at random from the distinct active populations according to a symmetric migration kernel. In [20] by exploiting a dual process given by an interacting coalescing particle system, we showed that the spatial system exhibits a dichotomy between clustering (mono-type equilibrium) and coexistence (multi-type equilibrium). In this paper, we identify the domain of attraction for each mono-type equilibrium in the clustering regime for an arbitrary fixed environment. Furthermore, we show that in dimensions d≤2, when the migration kernel is recurrent, for almost all realization of the environment, the system with an initially consistent type-distribution converges weakly to a mono-type equilibrium in which the probability of fixation to the all type-♡ configuration does not depend on the environment. An explicit formula for the fixation probability is given in terms of an annealed average of the type-♡ densities in the active and the dormant population, biased by the ratio of the two population sizes at the target colony. Primary techniques employed in the proofs include stochastic duality and the environment process viewed from particle, introduced in [24] for random walk in random environment on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with the single-particle dual process to a reversible ergodic distribution, which we transfer to the spatial system of populations by using duality.

Random Walk on Nonnegative Integers in Beta Distributed Random Environment

We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(\mu,\mu)$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $\eta$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter $\eta$ affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on $\mathbb{Z}$.

Large Deviations for Random Walk in Random Environment

Рассматриваются локальные теоремы о больших уклонениях для момента $T_n$ достижения уровня $n\in\mathbb{N}$ случайным блужданием в случайной среде. Получены точные асимптотики вероятностей ${\mathbf P}(T_n=k)$ в диапазоне параметра $k$, соответствующем зоне больших уклонений.

An invariance principle for ergodic scale-free random environments

There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant {\em modulo scaling}. For our purposes, an ``environment'' consists of an infinite random planar map embedded in $\mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an {\em infinite} energy embedding, it suffices to show that one can perturb it to make the energy finite.

Wave Propagation for Reaction-Diffusion Equations on Infinite Random Trees

The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees $\vec{d}$ and the random branch lengths $\vec{\ell}$ of the tree. This speed is slower than that of the same equation on the real line $\mathbb{R}$, and we estimate this slow down in terms of $\vec{d}$ and $\vec{\ell}$. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [Multi-skewed Brownian motion and diffusion in layered media, Proc. Am. Math. Soc., Vol. 139, No. 10, pp.3739-3752, 2011], with skewness and interface sets that encode the metric structure $(\vec{d}, \vec{\ell})$ of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of $2\times 2$ random matrices parametrized by $\vec{d}$ and $\vec{\ell}$ and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.

Ballistic random walks in random environment as rough paths: convergence and area anomaly

Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the `area anomaly', i.e. a deterministic linear correction for the second level iterated integral of the rescaled path, is identified in terms of a stochastic area on a regeneration interval. The main theorem is formulated in more general settings, namely for any discrete process with uniformly bounded increments which admits a regeneration structure where the regeneration times have finite moments. Here the largest finite moment translates into the degree of regularity of the rough path topology. In particular, the convergence holds in the $\alpha$-Holder rough path topology for all $\alpha<1/2$ whenever all moments are finite, which is the case for the class of ballistic random walks in random environment. The latter may be compared to a special class of random walks in Dirichlet environments for which the regularity $\alpha<1/2$ is bounded away from $1/2$, explicitly in terms of the corresponding trap parameter.

Invariance principles for random walks in random environment on trees

In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. Such a theorem finds particularly nice applications if the resistance metric measure space is a metric measure tree. To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees. First, we take a critical Galton-Watson tree conditioned on its total progeny and a non-lattice branching random walk on $\mathbb{R}^d$ indexed by it. Then, conditional on that, we consider a biased random walk on the range of the preceding. Here, by non-lattice we mean that distinct branches of the tree do not intersect once mapped in $\mathbb{R}^d$. This excludes the possibility that the random walk on the range may jump from one branch to the other without returning to the recent common ancestor. We prove, after introducing the bias parameter $\beta^{n^{-1/4}}$, for some $\beta>1$, that the biased random walk on the range of a large critical non-lattice branching random walk converges to a Brownian motion in a random Gaussian potential on Aldous' continuum random tree (CRT). Our second new result introduces the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.

Subsequential tightness for branching random walk in random environment

We consider branching random walk in random environment (BRWRE) and prove the existence of deterministic subsequences along which their maximum, centered at its mean, is tight. This partially answers an open question in arXiv:1711.00852. The method of proof adapts an argument developed by Dekking and Host for branching random walks with bounded increments. The question of tightness without the need for subsequences remains open.

Reconstructing a recurrent random environment from a single trajectory of a Random Walk in Random Environment with errors

We consider one infinite path of a Random Walk in Random Environment (RWRE, for short) in an unknown environment. This environment consists of either i.i.d. site or bond randomness. At each position the random walker stops and tells us the environment it sees at the point where it is, without telling us, where it is. These observations χ′ are spoiled by reading errors that occur with probability p<1. We show: If the RWRE is recurrent and satisfies the standard assumptions on such RWREs, then with probability one in the environment, the errors, and the random walk we are able reconstruct the law of the environment. For most situations this result is even independent of the value of p. If the distribution of the environment has a non-atomic part, we can even reconstruct the environment itself, up to translation.

Localization at the boundary for conditioned random walks in random environment in dimensions two and higher

We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on Zd, in dimensions two and higher. If d=2 or 3, we prove localization for (almost) all walks. In contrast, for d≥4, there is a phase transition for environments of the form ωe(x,e)=α(e)(1+eξ(x,e)), where {ξ(x)}x∈Zd is an i.i.d. sequence of random variables, and e represents the amount of disorder with respect to a simple random walk.

Rough invariance principle for delayed regenerative processes

We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough path version of Donsker’s Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition.

The quenched central limit theorem for a model of random walk in random environment

In the present paper we provide a proof of the quenched central limit theorem for the random walk in random environment model introduced by Boldrighini, Minlos, and Pellegrinotti in [3]. <br><br>У цій статті дано доведення квенч-центральної граничної теореми для випадкових блукань у моделі з випадковим середовищем, запропонованій Болдрігіні, Мінлосом і Пеллегринотті [3].