Importance Sampling for counting statistics in one-dimensional systems

TL;DR

This paper introduces an alternative importance sampling method with local tilt for counting statistics in one-dimensional systems, improving efficiency over traditional exponential tilt approaches.

Contribution

The paper proposes a novel importance sampling strategy with local tilt, addressing limitations of exponential tilt in counting statistics for discrete observables.

Findings

Efficient sampling demonstrated on Gaussian variables, Dyson gas, and SSEP.

Outperforms traditional exponential tilt in discrete counting problems.

Provides a practical approach for numerical analysis of counting statistics.

Abstract

In this paper, we consider the problem of numerical investigation of the counting statistics for a class of one-dimensional systems. Importance sampling, the cornerstone technique usually implemented for such problems, critically hinges on selecting an appropriate biased distribution. While exponential tilt in the observable stands as the conventional choice for various problems, its efficiency in the context of counting statistics may be significantly hindered by the genuine discreteness of the observable. To address this challenge, we propose an alternative strategy which we call importance sampling with the local tilt. We demonstrate the efficiency of the proposed approach through the analysis of three prototypical examples: a set of independent Gaussian random variables, Dyson gas, and Symmetric Simple Exclusion Process (SSEP) with a steplike initial condition.