Closed form Eigenvalues of Randomly Segmented Tridiagonal quasi-Toeplitz Matrices: Random Rouse block copolymer

TL;DR

This paper derives exact closed-form eigenvalues for a class of random matrices called rstq-T matrices, which model the dynamics of Rouse polymers in random environments, revealing how disorder affects their spectral properties.

Contribution

It provides the first analytical diagonalization and spectral distribution of rstq-T matrices, advancing understanding of disordered polymer dynamics.

Findings

Closed-form eigenvalues for rstq-T matrices derived.

Spectral distribution captures disorder effects on polymer modes.

Analytical results extend understanding of random matrix impacts in physics.

Abstract

We calculate the eigenvalues of a class of random matrices, namely the randomly segmented tridiagonal quasi-Toeplitz (rstq-T) matrix, in exact closed-form. The contexts under which these matrices arise are ubiquitous in physics. In our case, they arise when studying the dynamics of a Rouse polymer embedded in random environments. Unlike in the case of Rouse polymers in homogeneous environments, where the dynamics give rise to a circulant matrix and the diagonalization is achieved easily via a Fourier transform, analytical diagonalization of the rstq-T matrix has remained unsolved thus far. We analytically calculate the spectral distribution of the rstq-T matrix, which is able to capture the effect of disorder on the modes.