Statistical mechanics of Monte Carlo sampling and the sign problem

TL;DR

This paper models Monte Carlo sampling and the sign problem using a glass model with complex fields, identifying phases that influence convergence and proposing strategies to optimize importance sampling.

Contribution

It introduces a glass model framework with complex parameters to analyze Monte Carlo sampling and the sign problem, providing bounds and optimization methods.

Findings

Identification of three phases: liquid, frozen, and chaotic.

A lower bound on computational time based on free energy differences.

Guidelines for optimizing importance-sampling strategies.

Abstract

Monte Carlo sampling of any system may be analyzed in terms of an associated glass model -- a variant of the Random Energy Model -- with, whenever there is a sign problem, complex fields. This model has three types of phases (liquid, frozen and `chaotic'), as is characteristic of glass models with complex parameters. Only the liquid one yields the correct answers for the original problem, and the task is to design the simulation to stay inside it. The statistical convergence of the sampling to the correct expectation values may be studied in these terms, yielding a general lower bound for the computer time as a function of the free energy difference between the true system, and a reference one. In this way, importance-sampling strategies may be optimized.