Limit theorems for random permutations induced by Chinese restaurant processes

TL;DR

This paper studies the asymptotic behavior of random permutations generated by Chinese restaurant processes with parameters, extending known results for Ewens measures to a broader class, and explores spectral properties and characteristic polynomials.

Contribution

It extends limit theorems for permutations induced by Chinese restaurant processes to the case with lpha and eta parameters, including new spectral and polynomial analyses.

Findings

Functional central limit theorem for weighted cycle counts

Applications to spectrum linear statistics

Analysis of characteristic polynomials outside the unit circle

Abstract

We investigate the random permutation matrices induced by the Chinese restaurant processes with $(\alpha,\theta)$-seating. When $\alpha=0,\theta>0$, the permutations are those following Ewens measures on symmetric groups, and have been extensively studied in the literature. Here, we consider $\alpha\in(0,1)$ and $\theta>-\alpha$. In an accompanying paper, a functional central limit theorem is established for partial sum of weighted cycle counts in the form of $\sum_{j=1}^n a_jC_{n,j}$, where $C_{n,j}$ is the number of $j$-cycles of the permutation matrix of size $n$. Two applications are presented. One is on linear statistics of the spectrum, and the other is on the characteristic polynomials outside the unit circle.