Numerical Estimation of Limiting Large-Deviation Rate Functions

TL;DR

This paper introduces a method combining rare-event importance sampling with multi-histogram reweighting to estimate large-deviation rate functions in the infinite system size limit, demonstrated on binomial variables and Erdős-Rényi graphs.

Contribution

The paper presents a novel approach for system-size extrapolation of large-deviation rate functions using combined importance sampling and reweighting techniques, applicable across various sampling algorithms.

Findings

Method accurately estimates infinite-size rate functions.

Phase transitions in biased ensembles can cause deviations.

Validated on analytical models with known solutions.

Abstract

For statistics of rare events in systems obeying a large-deviation principle, the rate function is a key quantity. When numerically estimating the rate function one is always restricted to finite system sizes. Thus, if the interest is in the limiting rate function for infinite system sizes, first, several system sizes have to be studied numerically. Here, rare-event algorithms using biased ensembles give access to the low-probability region. Second, some kind of system-size extrapolation has to be performed. Here we demonstrate how rare-event importance sampling schemes can be combined with multi-histogram reweighting, which allows for rather general applicability of the approach, independent of specific sampling algorithms. We study two ways of performing the system-size extrapolation, either directly acting on the empirical rate functions, or on the scaled cumulant generating functions, to obtain the infinite-size limit. The presented method is demonstrated for a binomial distributed variable and the largest connected component in Erd\"os-R\'enyi random graphs. Analytical solutions are available in both cases for direct comparison. It is observed in particular that phase transitions appearing in the biased ensembles can lead to systematic deviations from the true result.