Monte Carlo Determination of Multiple Extremal Eigenpairs

TL;DR

This paper introduces a Monte Carlo algorithm for efficiently computing multiple extremal eigenpairs of large matrices, avoiding full vector storage and inner product calculations, demonstrated on Ising model transfer matrices.

Contribution

The paper presents a novel Monte Carlo extension of the power method, incorporating splitting, cancellation, and sewing techniques for large-scale eigenvalue problems.

Findings

Successfully computed the two largest eigenvalues of 2D Ising transfer matrices

Demonstrated efficiency and generality of the algorithm for large matrices

Potential applicability to various transfer matrix and eigenvalue problems

Abstract

We present a Monte Carlo algorithm that allows the simultaneous determination of a few extremal eigenpairs of a very large matrix without the need to compute the inner product of two vectors or store all the components of any one vector. The new algorithm, a Monte Carlo implementation of a deterministic one we recently benchmarked, is an extension of the power method. In the implementation presented, we used a basic Monte Carlo splitting and termination method called the comb, incorporated the weight cancellation method of Arnow {\it et al.}, and exploited a new sampling method, the sewing method, that does a large state space sampling as a succession of small state space samplings. We illustrate the effectiveness of the algorithm by its determination of the two largest eigenvalues of the transfer matrices for variously-sized two-dimensional, zero field Ising models. While very likely useful for other transfer matrix problems, the algorithm is however quite general and should find application to a larger variety of problems requiring a few dominant eigenvalues of a matrix.