Free product of generalized Gaussian processes, random matrices and positive definite functions on permutation groups (Mathematical Aspects of Quantum Fields and Related Topics)

Abstract

The paper deals with the free product 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 Vin mathcal{P}_{2}(2n). Some new combinatorial formulas are 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)=cup S(n). Connection with random matrices and positive definite functions on permutations groups are also done.