Large deviations and nontrivial exponents in coarsening systems

TL;DR

This paper analyzes the large deviation statistics of mean magnetization in simple coarsening systems, revealing algebraic decay and nontrivial exponents, with implications for understanding persistence phenomena.

Contribution

It provides analytical and numerical insights into large deviations and persistence in coarsening systems like the diffusion equation, Ising chain, and voter model, highlighting nontrivial exponents.

Findings

Persistent large deviations decay algebraically with a continuously varying exponent.

The mean magnetization reaches a limit law at large times, studied via the independent interval approximation.

Large deviations in the voter model are algebraic, with persistent deviations resembling usual persistence probabilities.

Abstract

We investigate the statistics of the mean magnetisation, of its large deviations and persistent large deviations in simple coarsening systems. We consider more specifically the case of the diffusion equation, of the Ising chain at zero temperature and of the two dimensional voter model. For the diffusion equation, at large times, the mean magnetisation has a limit law, which is studied analytically using the independent interval approximation. The probability of persistent large deviations, defined as the probability that the mean magnetisation was, for all previous times, greater than some level $x$, decays algebraically at large times, with an exponent $\theta(x)$ continuously varying with $x$. When $x=1$, $\theta(1)$ is the usual persistence exponent. Similar behaviour is found for the Glauber-Ising chain at zero temperature. For the two dimensional Voter model, large deviations of the mean magnetisation are algebraic, while persistent large deviations seem to behave as the usual persistence probability.