Moderate deviations for diffusion in time dependent random media

Abstract

The position $x(t)$ of a particle diffusing in a one-dimensional uncorrelated and time dependent random medium is simply Gaussian distributed in the typical direction, i.e. along the ray $x=v_{0}t$, where $v_0$ is the average drift. However, it has been found that it exhibits at large time sample to sample fluctuations characteristic of the KPZ universality class when observed in an atypical direction, i.e. along the ray $x=vt$ with $v\ne v_0$. Here we show, from exact solutions, that in the moderate deviation regime $x-v_{0}t\propto t^{3/4}$ these fluctuations are precisely described by the finite time KPZ equation, which thus describes the crossover between the Gaussian typical regime and the KPZ fixed point regime for the large deviations. This confirms heuristic arguments given in [2]. These exact results include the discrete model known as the Beta RWRE, and a continuum diffusion. They predict the behavior of the maximum of a large number of independent walkers, which should be easier to observe (e.g. in experiments) in this moderate deviations regime.

18 pages

References in corpus (5)

• The Brownian web: Characterization and convergence
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• The Brownian net
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Consistent families of Brownian motions and stochastic flows of kernels

Cited by in corpus (9)

• The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion
• Entanglement entropy growth in stochastic conformal field theory and the KPZ class
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• First Passage Time for Many Particle Diffusion in Space-Time Random Environments
• A quenched local limit theorem for stochastic flows
• KPZ equation limit of sticky Brownian motion
• Random walk on nonnegative integers in beta distributed random environment
• Hidden diagonal integrability of $q$-Hahn vertex model and Beta polymer model
• Multiplicative SHE limit of random walks in space-time random environments