Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Random walks in space-time random media in all spatial dimensions: the full subcritical fluctuation regime

Hindy Drillick, Shalin Parekh

TL;DR

The paper advances a unified Gaussian-fluctuation theory for random walks in time-dependent random media across all spatial dimensions, identifying subcritical fluctuation regimes up to a dimension- and scale-dependent critical radius $\psi_N$. By introducing Girsanov tilts, the authors derive a discrete stochastic heat equation with martingale noise and develop Short-Range Interacting (SRI) chains to control multi-point motions, invariant measures, and additive functionals. They classify four regimes (bulk and three extremal types) and provide regime-specific SPDE limits with explicit noise coefficients tied to the environment’s two-point motion via $\pi^{\mathrm{inv}}$ and the tilted kernels; in particular, the noise takes universal forms in the bulk but becomes model-dependent in higher extremal regimes. The results unify and extend known fluctuation theories (including KPZ-type behavior in $d=1$) and lay a rigorous foundation for future exploration of critical and supercritical regimes, directed-polymers connections, and higher-dimensional phenomena. The framework offers a robust toolkit—Girsanov tilts, SRI chains, and discrete SPDEs—that can be applied to a broad class of dynamic random environments and related stochastic systems.

Abstract

In arbitrary spatial dimension $d\ge 1$, we study a generalized model of random walks in a time-varying random environment (RWRE) defined by a stochastic flow of kernels. We consider the quenched probability distribution of the random walker under a scaling where the time is of order $N$ and the spatial window is of size $N^{1/2}$. This spatial window may not necessarily be centered close to the origin. We show that as $N\to \infty$ there are Gaussian fluctuations up to a certain specific spatial centering radius $ψ_N$ in the tail of the quenched probability distribution, which we call the critical scale. This critical scale depends on the spatial dimension of the underlying random walk, specifically $ψ_N = O(N^{3/4})$ when $d=1$, $ψ_N = O( N/\sqrt{\log N})$ when $d=2$, and $ψ_N = O(N)$ when $d\ge 3$. In the particular case of centering the fluctuation window at the origin, our results recover and generalize some known fluctuation results for related models. However, farther from the origin, the previous literature is more sparse. The noise coefficient in the limiting Gaussian field is nontrivial and depends on the invariant measure of the two-point motion of the underlying RWRE model. We furthermore reconcile some of these coefficient formulas with previous works. As part of the proof, we introduce a general class of Markov chains with short-range interactions that admit nice estimates and limit formulas. One of the key technical results for such Markov chains is that in $d\ge 2$, one can propagate test functions backwards in time to obtain precise limiting moment formulas.

Random walks in space-time random media in all spatial dimensions: the full subcritical fluctuation regime

TL;DR

The paper advances a unified Gaussian-fluctuation theory for random walks in time-dependent random media across all spatial dimensions, identifying subcritical fluctuation regimes up to a dimension- and scale-dependent critical radius . By introducing Girsanov tilts, the authors derive a discrete stochastic heat equation with martingale noise and develop Short-Range Interacting (SRI) chains to control multi-point motions, invariant measures, and additive functionals. They classify four regimes (bulk and three extremal types) and provide regime-specific SPDE limits with explicit noise coefficients tied to the environment’s two-point motion via and the tilted kernels; in particular, the noise takes universal forms in the bulk but becomes model-dependent in higher extremal regimes. The results unify and extend known fluctuation theories (including KPZ-type behavior in ) and lay a rigorous foundation for future exploration of critical and supercritical regimes, directed-polymers connections, and higher-dimensional phenomena. The framework offers a robust toolkit—Girsanov tilts, SRI chains, and discrete SPDEs—that can be applied to a broad class of dynamic random environments and related stochastic systems.

Abstract

In arbitrary spatial dimension , we study a generalized model of random walks in a time-varying random environment (RWRE) defined by a stochastic flow of kernels. We consider the quenched probability distribution of the random walker under a scaling where the time is of order and the spatial window is of size . This spatial window may not necessarily be centered close to the origin. We show that as there are Gaussian fluctuations up to a certain specific spatial centering radius in the tail of the quenched probability distribution, which we call the critical scale. This critical scale depends on the spatial dimension of the underlying random walk, specifically when , when , and when . In the particular case of centering the fluctuation window at the origin, our results recover and generalize some known fluctuation results for related models. However, farther from the origin, the previous literature is more sparse. The noise coefficient in the limiting Gaussian field is nontrivial and depends on the invariant measure of the two-point motion of the underlying RWRE model. We furthermore reconcile some of these coefficient formulas with previous works. As part of the proof, we introduce a general class of Markov chains with short-range interactions that admit nice estimates and limit formulas. One of the key technical results for such Markov chains is that in , one can propagate test functions backwards in time to obtain precise limiting moment formulas.
Paper Structure (40 sections, 62 theorems, 465 equations, 1 figure)

This paper contains 40 sections, 62 theorems, 465 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Definition and assumptions for the model
  4. Examples of models satisfying the assumptions
  5. Setup and main results
  6. Classification of regimes
  7. Intuition and ideas of the proof
  8. The broader picture and some open problems
  9. Previously known fluctuations results.
  10. The Girsanov transform and the tilted $k$-point Markov chains
  11. Studying the tilted measures $\mathbf P^{( \beta,k)}$
  12. Important functions and Taylor expansions
  13. A discrete stochastic heat equation with a martingale noise
  14. Calculating the quadratic variations of the martingales $M^N$
  15. \ref{['eq:regimeA']} $| \varsigma_N| = O(N^{-1/2})$ as $N\to \infty$
  16. ...and 25 more sections

Key Result

Theorem 1.13

Let $\{K_n\}_{n\ge 1}$ denote an environment satisfying Assumption a1, and let $\mathfrak H^N$ be as in hn. Assume as in Definition goodseq that one has a good sequence of deterministic initial data $\mathfrak H^N(0,\bullet)$ converging weakly as measures to some limit $\mathfrak H_0$. There is an e Here $G_t(x) = G(t,x)= (2\pi t)^{-1/2}e^{-x^2/2t}$ is the standard heat kernel, $*$ denotes spatial

Figures (1)

  • Figure 1: An illustration of our generalized model of random walk in random environment. The random walker starts at position $x_0$ at time $r=0$, then samples $x_1$ from the random probability measure $K_1(x_0,\mathrm d x)$. After landing at position $x_1$ at time $r=1$, the random walker then samples $x_2$ from the probability measure $K_2(x_1,\mathrm d x)$, and continues in this fashion. Each plane represents a copy of the underlying state space $I$ (which in this example is $\mathbb R^2)$. The dotted arrows in the above picture represent any one of many possible jumps that were not actually executed by the random walker in this random environment, while the solid arrows represent the jumps that were actually executed.

Theorems & Definitions (170)

  • Definition 1.1: Markov Kernels
  • Definition 1.3: The microscopic model of interest
  • Example 1.4: Nearest-neighbor random walk in a dynamical random environment in $\mathbb Z^d$
  • Example 1.5: Symmetric random walk in random environment
  • Example 1.6: Diffusion in a random medium/Kraichnan flows
  • Example 1.7
  • Example 1.8: Random landscape model
  • Definition 1.10: The rescaling constants
  • Definition 1.11: Rescaled prelimiting density field
  • Definition 1.12: Good sequence of initial data
  • ...and 160 more