Mapping spatial persistent large deviations of nonequilibrium surface growth processes onto the temporal persistent large deviations of stochastic random walk processes

TL;DR

This paper establishes a theoretical and numerical connection between spatial persistent large deviations in surface growth processes and temporal persistent large deviations in stochastic random walks, confirming predictions for the decay exponents.

Contribution

It maps spatial persistent large deviations onto temporal ones for surface growth, and verifies the theoretical exponents through numerical simulations, including discrete models.

Findings

Numerical simulations confirm the theoretical relationship between spatial and temporal exponents.

The spatial persistent large deviations exponents match the predicted power-law decay.

Finite-size effects hinder the asymptotic behavior in discrete models of the Mullins-Herring class.

Abstract

Spatial persistent large deviations probability of surface growth processes governed by the Edwards-Wilkinson dynamics, $P_x(x,s)$, with $-1 \leq s \leq 1$ is mapped isomorphically onto the temporal persistent large deviations probability $P_t(t,s)$ associated with the stochastic Markovian random walk problem. We show using numerical simulations that the infinite family of spatial persistent large deviations exponents $\theta_x(s)$ characterizing the power law decay of $P_x(x,s)$ agrees, as predicted on theoretical grounds by Majumdar and Bray [Phys. Rev. Lett. {\bf 86}, 3700 (2001)] with the numerical measurements of $\theta_t(s)$, the continuous family of exponents characterizing the long time power law behavior of $P_t(t,s)$. We also discuss the simulations of the spatial persistence probability corresponding to a discrete model in the Mullins-Herring universality class, where our discrete simulations do not agree well with the theoretical predictions perhaps because of the severe finite-size corrections which are known to strongly inhibit the manifestation of the asymptotic continuum behavior in discrete models involving large values of the dynamical exponent and the associated extremely slow convergence to the asymptotic regime.