Percolation in a Class of Band Structured Random Matrices

TL;DR

This paper investigates a class of band-structured random matrices modeling bio-polymers, revealing a percolation transition that affects the distribution of the largest eigenvalue, shifting from multi-peaked to Tracy-Widom distribution.

Contribution

It introduces a new class of random matrix ensembles related to bio-polymer models and analyzes their spectral transition behavior.

Findings

Largest eigenvalue distribution depends on a percolation transition.

Below the threshold, the distribution is multi-peaked.

Above the threshold, the distribution follows Tracy-Widom law.

Abstract

We define a class of random matrix ensembles that pertain to random looped polymers. Such random looped polymers are a possible model for bio-polymers such as chromatin in the cell nucleus. It is shown that the distribution of the largest eigenvalue $\lambda_{max}$ depends on a percolation transition in the entries of the random matrices. Below the percolation threshold the distribution is multi-peaked and changes above the threshold to the Tracy-Widom distribution. We also show that the distribution of the eigenvalues is neither of the Wigner form nor gaussian.