Hierarchy of KPZ limits arising from directed random walk models in random media

TL;DR

The paper develops a broad invariance principle for the KPZ universality class by proving that, for a wide class of one-dimensional random walks in dynamical random environments, the tail fluctuations of quenched transition densities converge to the solution of a stochastic heat equation with multiplicative noise. The approach hinges on a generalized discrete Girsanov transform and a thorough analysis of the k-point motion under tilted measures, enabling identification of an environment-driven noise coefficient γ_{ ext{ext}} and a drift-recentered scaling with a hierarchy of crossover exponents (3/4, 7/8, ...). The results extend prior work to non-nearest-neighbor interactions and models with spatial correlations, and establish a robust framework for deriving KPZ fluctuations from stochastic kernels under minimal mixing assumptions. In addition to proving convergence, the work clarifies the precise tail locations where KPZ behavior emerges and provides detailed constructions and examples of models satisfying the hypotheses, linking physical predictions to rigorous probabilistic limits. Overall, the paper delivers a versatile, general pathway to KPZ in stochastic flows of kernels and highlights the universality and structural robustness of the KPZ/SHE connection in random media.

Abstract

We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicative-noise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise location in the tail. The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of space-time, i.e., there is no critical tuning of the model parameters when scaling to the stochastic PDE limit. The proof is done by pushing the methods developed in [arxiv 2304.14279, arXiv 2311.09151] to their maximum, substantially weakening the assumptions and obtaining fairly sharp conditions under which one expects to see the SHE arise in a wide variety of random walk models in random media. In particular we are able to get rid of conditions such as nearest-neighbor interaction as well as spatial independence of quenched transition kernels. Moreover, we observe an entire hierarchy of moderate deviation exponents at which the SHE can be found, confirming a physics prediction of J. Hass.

Figures (1)

• Figure 1: An illustration of our generalized model of random walk in random environment. Throughout the paper, $r \in \mathbb Z_{\ge 0}$ will be used to denote the microscopic time variable. The random walker starts at position $x=0$ at time $r=0$, then samples $x_1$ from the random probability measure $K_1(0,\mathrm d x)$. 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 vertical line represents a copy of the underlying lattice $I= \mathbb Z$ or $I=\mathbb R$, and the Markov kernel $K_i$ should be thought of as the entire collection of probability measures $\{K_i(x,\cdot)\}_{x\in I}$ which lives on the vertical line $r=i-1$. The random environment is then the whole collection of Markov kernels $\{K_i\}_{i=1}^\infty$ which in this paper are assumed to be independent of one another for distinct values of $i$, translationally invariant in law, and sampled from the same distribution on Markov kernels on $I$. 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 executed.

Theorems & Definitions (134)

• Definition 1.1
• Proposition 1.3
• Definition 1.4
• Theorem 1.5: Main result
• Remark 1.6
• Remark 1.7: Optimality questions
• Remark 1.8: Quenched tail field
• Definition 2.1: $k$-point motion
• Lemma 2.2
• Definition 2.3