Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure

TL;DR

This paper develops a vertex operator framework for the $t$-Schur measure on partitions, deriving correlation functions, determinantal point processes, and limiting distributions, connecting classical Schur theory with $t$-deformations and probabilistic models.

Contribution

It introduces a vertex algebra approach to the $t$-Schur measure, providing explicit correlation kernels and linking to classical and $t$-deformed combinatorial identities.

Findings

Correlation functions form a determinantal point process.

Explicit correlation kernel derived for the $t$-Schur measure.

Limiting distribution for the longest ascent pair in random permutations obtained.

Abstract

We study the $t$-Schur measure on partitions, defined by $ \mathbb{P}(\lambda)=Z^{-1}S_\lambda(x;t)s_\lambda(y) $, where $S_\lambda(x;t)$ denotes the $t$-Schur symmetric functions and $s_\lambda(y)$ the ordinary Schur functions, and $Z$ is the normalising constant. Using vertex operator calculus, we realise $S_\lambda(x;t)$ in the charged free-fermion Fock space, yielding a $t$-deformation of the classical boson-fermion correspondence. These realisations give vertex-algebraic proofs of the $t$-Cauchy identities and $t$-Gessel identity. Building on this framework, we compute the correlation functions of the $t$-Schur measure and show that the associated point process is determinantal, with an explicit correlation kernel. The Poissonised $t$-Plancherel measure appears as a specialisation of our construction, so its correlation functions follow as a corollary. As an application, we derive the limiting distribution for the length of the longest ascent pair in a random permutation. Our results interpolate the Schur case at $t=0$, connect to the Schur-$Q$ theory at $t=-1$, and provide a probabilistic interpretation of a natural $t$-refinement of increasing subsequences via a generalised RSK correspondence.