Coloured corner processes from asymptotics of LLT polynomials

TL;DR

This paper studies the asymptotic behavior of probability measures derived from LLT symmetric polynomials, revealing a split into continuous GUE corners processes and a discrete combinatorial distribution with interesting properties.

Contribution

It introduces a novel asymptotic analysis of LLT polynomial-based measures, showing their convergence to a combination of GUE corners processes and a new discrete distribution.

Findings

Asymptotic measures split into continuous and discrete parts.

Continuous part consists of multiple GUE corners processes.

Discrete part is an explicit finite distribution with rational weights.

Abstract

We consider probability measures arising from the Cauchy summation identity for the LLT (Lascoux--Leclerc--Thibon) symmetric polynomials of rank $n \geq 1$. We study the asymptotic behaviour of these measures as one of the two sets of polynomials in the Cauchy identity stays fixed, while the other one grows to infinity. At $n=1$, this corresponds to an analogous limit of the Schur process, which is known to be given by the Gaussian Unitary Ensemble (GUE) corners process. Our main result states that, for $n>1$, our measures asymptotically split into two parts: a continuous one and a discrete one. The continuous part is a product of $n$ GUE corners processes; the discrete part is an explicit finite distribution on interlacing $n$-colourings of $n$ interlacing triangles, which has weights that are rational functions in the LLT parameter $q$. The latter distribution has a number of interesting (partly conjectural) combinatorial properties, such as $q$-nonnegativity and enumerative phenomena underlying its support. Our main tools are two different representations of the LLT polynomials, one as partition functions of a fermionic lattice model of rank $n$, and the other as finite-dimensional contour integrals, which were recently obtained in arXiv:2012.02376, arXiv:2101.01605.