New combinatorial identity for the set of partitions and limit theorems in finite free probability theory

TL;DR

This paper introduces a refined combinatorial identity for partitions, enabling new limit theorems in finite free probability, including analogues of existing theorems and a central limit theorem linked to free semicircular distributions.

Contribution

It provides a new combinatorial identity and applies it to establish finite free analogues of limit theorems, including a central limit theorem and alternative proofs of known results.

Findings

Finite free analogue of Sakuma and Yoshida's limit theorem established.

A central limit theorem for finite free multiplicative convolution proved.

Alternative proofs for Kabluchko's limit theorems on Hermite and Laguerre polynomials provided.

Abstract

We provide a refined combinatorial identity for the set of partitions of $\{1,\dots, n\}$, which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida's limit theorem. That is, we provide the limit of $\{D_{1/m}((p_d^{\boxtimes_d m})^{\boxplus_d m})\}_{m\in \mathbb{N}}$ as $m\rightarrow\infty$ in two cases: (i) $m/d\rightarrow t$ for some $t>0$, or (ii) $m/d\rightarrow0$. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko's limit theorems concerning the unitary Hermite and the Laguerre polynomials.