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

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.