Asymptotics of Symmetric Polynomials: A Dynamical Point of view

TL;DR

This paper investigates the asymptotic behavior of symmetric polynomials like Macdonald and Jack as variables grow large, linking them to variational problems and large deviation principles for related random walks.

Contribution

It introduces a new approach to the variational principle for non-intersecting Bernoulli walks, extending it to arbitrary  and connecting it to symmetric polynomial asymptotics.

Findings

Established a large deviation principle for -Bernoulli walks.

Extended the variational principle to -walks with general domains.

Identified that rate functions are consistent across  values, scaled by 1/.

Abstract

In this paper we study the asymptotic behavior of the (skew) Macdonald and Jack symmetric polynomials as the number of variables grows to infinity. We characterize their limits in terms of certain variational problems. As an intermediate step, we establish a large deviation principle for the $\theta$ analogue of non-intersecting Bernoulli random walks. When $\theta=1$, these walks are equivalent to random Lozenges tilings of strip domains, where the variational principle (with general domains and boundary conditions) has been proven in the seminal work by Cohn, Kenyon, and Propp. Our result gives a new argument of this variational principle, and also extends it to non-intersecting $\theta$-Bernoulli random walks for any $\theta \in (0,\infty)$. Remarkably, the rate functions remain identical, differing only by a factor of $1/\theta$.