Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments

Abstract

We consider the transition probabilities for random walks in $1+1$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.

15 pages, revised version

References in corpus (3)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
• Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

Cited by in corpus (15)

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• The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Large Deviations For Sticky Brownian Motions
• Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer
• Moderate deviations for diffusion in time dependent random media
• Large deviations and wandering exponent for random walk in a dynamic beta environment
• First Passage Time for Many Particle Diffusion in Space-Time Random Environments
• KPZ equation limit of sticky Brownian motion
• A quenched local limit theorem for stochastic flows
• Random walk on nonnegative integers in beta distributed random environment
• Entanglement structure in the volume-law phase of hybrid quantum automaton circuits
• Tracy-Widom asymptotics for a river delta model
• Moments of the SHE under delta initial measure
• Multiplicative SHE limit of random walks in space-time random environments