Limit theorems for random permutations induced by Chinese restaurant processes

Abstract

We investigate the random permutation matrices induced by the Chinese restaurant processes with $(\hat{I}\pm ,\hat{I}¸)$-seating. When $\hat{I}\pm =0,\hat{I}¸>0$, the permutations are those following Ewens measures on symmetric groups, and have been extensively studied in the literature. Here, we consider $\hat{I}\pm \in (0,1)$ and $\hat{I}¸>-\hat{I}\pm$. In an accompanying paper, a functional central limit theorem is established for partial sum of weighted cycle counts in the form of ${\sum }_{j=1}^n{a}_j{C}_{n,j}$, 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.

18 pages; Section 5 revised