Publication:
Investigation of the Product of Random Matrices and Related Evolution Models
Investigation of the Product of Random Matrices and Related Evolution Models
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Date
2023
Authors
Hirobumi Mineo, Vladimir Suvorov and David B. Saakian
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Abstract
"In this paper, we study the phase structure of the product of D * D order matrices. In
each round, we randomly choose a matrix from a finite set of d matrices and multiply it with the
product from the previous round. Initially, we derived a functional equation for the case of matrices
with real eigenvalues and correlated choice of matrices, which led to the identification of several
phases. Subsequently, we explored the case of uncorrelated choice of matrices and derived a simpler
functional equation, again identifying multiple phases. In our investigation, we observed a phase
with a smooth distribution in steady-state and phases with singularities. For the general case of
D-dimensional matrices, we derived a formula for the phase transition point. Additionally, we
solved a related evolution model. Moreover, we examined the relaxation dynamics of the considered
models. In both the smooth phase and the phase with singularities, the relaxation is exponential. The
superiority of relaxation in the smooth phase depends on the specific case"
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Keywords
Random matrix product,
evolution,
phase transitions