Correlation Function of Self-Conjugate Partitions: $q$-Difference Equation and Quasimodularity

TL;DR

This paper investigates the correlation functions of self-conjugate partitions under the uniform measure, deriving $q$-difference equations, explicit formulas, and establishing quasimodularity, along with analyzing their limit shapes.

Contribution

It introduces a $q$-difference equation for the correlation functions of self-conjugate partitions and proves their quasimodularity using combinatorial methods.

Findings

Derived explicit formulas for one-point and two-point functions.

Established the quasimodularity of the $n$-point correlation functions.

Determined the limit shape of self-conjugate partitions under the uniform measure.

Abstract

In this paper, we study the uniform measure for the self-conjugate partitions. We derive the $q$-difference equation which is satisfied by the $n$-point correlation function related to the uniform measure. As applications, we give explicit formulas for the one-point and two-point functions, and study their quasimodularity. Motivated by this, we also prove the quasimodularity of the general $n$-point function using a combinatorial method. Finally, we derive the limit shape of self-conjugate partitions under the Gibbs uniform measure and compare it to the leading asymptotics of the one-point function.