Asymptotic results on the product of random probability matrices

TL;DR

This paper analyzes the asymptotic behavior of products of i.i.d. random probability matrices, revealing exponential convergence to a uniform row matrix and exploring how convergence rate and element distribution depend on matrix size and distribution.

Contribution

It provides exact asymptotic results for the product of random probability matrices, including the convergence rate parameter and distribution characteristics of the asymptotic matrix elements.

Findings

Both left and right products approach a matrix with identical rows exponentially.

The convergence rate parameter λ varies with matrix size and element distribution.

The asymptotic matrix elements follow a non-universal distribution, often Gaussian with mean 1/D.

Abstract

I study the product of independent identically distributed $D\times D$ random probability matrices. Some exact asymptotic results are obtained. I find that both the left and the right products approach exponentially to a probability matrix(asymptotic matrix) in which any two rows are the same. A parameter $\lambda$ is introduced for the exponential coefficient which can be used to describe the convergent rate of the products. $\lambda$ depends on the distribution of individual random matrices. I find $\lambda = 3/2$ for D=2 when each element of individual random probability matrices is uniformly distributed in [0,1]. In this case, each element of the asymptotic matrix follows a parabolic distribution function. The distribution function of the asymptotic matrix elements can be numerically shown to be non-universal. Numerical tests are carried out for a set of random probability matrices with a particular distribution function. I find that $\lambda$ increases monotonically from $\simeq 1.5$ to $\simeq 3$ as D increases from 3 to 99, and the distribution of random elements in the asymptotic products can be described by a Gaussian function with its mean to be 1/D.