Asymptotics of Jack polynomials as the number of variables goes to infinity

TL;DR

This paper investigates the asymptotic behavior of Jack polynomials as the number of variables increases, extending previous results and connecting to spherical functions of infinite-dimensional symmetric spaces.

Contribution

It generalizes known asymptotic results for Schur functions to Jack polynomials for various parameters, linking finite and infinite-dimensional symmetric spaces.

Findings

Extended asymptotic formulas for Jack polynomials as variables grow large

Connected asymptotics to spherical functions of infinite-dimensional symmetric spaces

Provided approximation results for spherical functions of specific symmetric spaces

Abstract

In this paper we study the asymptotic behavior of the Jack rational functions as the number of variables grows to infinity. Our results generalize the results of A. Vershik and S. Kerov obtained in the Schur function case (theta=1). For theta=1/2,2 our results describe approximation of the spherical functions of the infinite-dimensional symmetric spaces $U(\infty)/O(\infty)$ and $U(2\infty)/Sp(\infty)$ by the spherical functions of the corresponding finite-dimensional symmetric spaces.