Percolation in a Class of Band Structured Random Matrices

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.