Stochastic lists: Sampling multi-variable functions with population methods

TL;DR

The paper introduces stochastic lists, a population-based sampling method for efficiently representing and computing multi-variable positive functions defined by self-consistent equations, with applications in physics models.

Contribution

It presents a novel stochastic list approach for sampling complex multi-variable functions, addressing limitations of traditional methods in physics and mathematics.

Findings

Successfully applied to compute vertex corrections in Fröhlich polaron

Accurately determined ground state energy and wavefunction of 2D Heisenberg model

Demonstrated effectiveness with short lists despite extrapolation challenges

Abstract

We introduce the method of stochastic lists to deal with a multi-variable positive function, defined by a self-consistent equation, typical for certain problems in physics and mathematics. In this approach, the function's properties are represented statistically by lists containing a large collection of sets of coordinates (or "walkers") that are distributed according to the function's value. The coordinates are generated stochastically by the Metropolis algorithm and may replace older entries according to some protocol. While stochastic lists offer a solution to the impossibility of efficiently computing and storing multi-variable functions without a systematic bias, extrapolation in the inverse of the number of walkers is usually difficult, even though in practice very good results are found already for short lists. This situation is reminiscent of diffusion Monte Carlo, and is hence generic for all population-based methods. We illustrate the method by computing the lowest-order vertex corrections in Hedin's scheme for the Fr\"ohlich polaron and the ground state energy and wavefunction of the Heisenberg model in two dimensions.