Bootstrapping cascaded random matrix models: correlations in permutations of matrix products

TL;DR

This paper introduces a bootstrapping method to efficiently analyze correlations in cascaded random matrix models used for studying wave scattering in thick disordered media, balancing computational speed and accuracy.

Contribution

It proposes a dual pool bootstrapping approach that accounts for permutation-induced correlations in cascaded matrices, providing analytic formulas for their frequency and impact.

Findings

Extra variance in estimates arises from permuted matrix products.

Correlations depend on permutation cycle structures.

Analytic enumeration of correlation frequencies is achieved.

Abstract

Random matrix theory is a useful tool in the study of the physics of multiple scattering systems, often striking a balance between computation speed and physical rigour. Propagation of waves through thick disordered media, as arises in for example optical scattering or electron transport, typically necessitates cascading of multiple random matrices drawn from an underlying ensemble for thin media, greatly increasing computational burden. Here we propose a dual pool based bootstrapping approach to speed up statistical studies of scattering in thick random media. We examine how potential matrix reuse in a pool based approach can impact statistical estimates of population averages. Specifically, we discuss how both bias and additional variance in the sample mean estimator are introduced through bootstrapping. In the diffusive scattering regime, the extra estimator variance is shown to originate from samples in which cascaded transfer matrices are permuted matrix products. Through analysis of the combinatorics and cycle structure of permutations we quantify the resulting correlations. Proofs of several analytic formulae enumerating the frequency with which correlations of different strengths occur are derived. Extension to the ballistic regime is briefly considered.