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 matrix (rstq-T), 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 gives rise to a circulant matrix and the diagonalization is easily achieved 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.