Closed-form eigenvalues of randomly segmented tridiagonal quasi-Toeplitz matrices: Random Rouse block copolymer.

Area of Science:

• Statistical physics
• Polymer dynamics
• Random matrix theory

Background:

• Randomly segmented tridiagonal quasi-Toeplitz (rstq-T) matrices appear in diverse physics contexts.
• Studying Rouse polymer dynamics in random environments necessitates analyzing these matrices.
• Previous methods could not analytically diagonalize rstq-T matrices, unlike circulant matrices from homogeneous environments.

Purpose of the Study:

• To derive the exact closed-form eigenvalues for rstq-T matrices.
• To develop an analytical method for diagonalizing rstq-T matrices.
• To understand the impact of disorder on polymer dynamics modes through spectral distribution.

Main Methods:

• Exact calculation of eigenvalues for rstq-T matrices.
• Analytical derivation of the spectral distribution.
• Application of random matrix theory to polymer physics.

Main Results:

• The eigenvalues of rstq-T matrices are calculated in exact closed form.
• The spectral distribution of rstq-T matrices is analytically determined.
• This spectral distribution captures the effects of environmental disorder on polymer modes.

Conclusions:

• The study provides the first analytical solution for the diagonalization of rstq-T matrices.
• The findings offer new insights into the dynamics of polymers in disordered media.
• The derived spectral distribution is a key tool for analyzing disorder effects in such systems.