Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups

TL;DR

This paper constructs generalized Gaussian processes linked to pair-partitions, proves positivity of certain functions, introduces new combinatorial formulas, and explores connections with free probability and positive definite functions on permutation groups.

Contribution

It provides explicit constructions of generalized Gaussian processes, new combinatorial formulas, and establishes positive definiteness of functions on permutations and their relation to free probability.

Findings

Constructed generalized Gaussian processes with explicit functions.

Proved positivity of functions on pair-partitions and permutations.

Connected combinatorial formulas with free additive convolutions.

Abstract

The main purpose of this paper of the paper is an explicite construction of generalized Gaussian process with function $t_b(V)=b^{H(V)}$, where $H(V)=n-h(V)$, $h(V)$ is the number of singletons in a pair-partition $V \in \st{P}_2(2n)$. This gives another proof of Theorem of A. Buchholtz \cite{Buch} that $t_b$ is positive definite function on the set of all pair-partitions. Some new combinatorial formulas are also presented. Connections with free additive convolutions probability measure on $\mathbb{R}$ are also done. Also new positive definite functions on permutations are presented and also it is proved that the function $H$ is norm (on the group $S(\infty)=\bigcup S(n)$.