Free fermionic probability theory and K-theoretic Schubert calculus

TL;DR

This paper links particle process transition kernels to refined symmetric Grothendieck functions using free fermion encoding and combinatorial proofs, advancing the understanding of algebraic structures in probabilistic models.

Contribution

It establishes a novel connection between particle process kernels and K-theoretic symmetric functions via free fermion methods and combinatorial tableaux.

Findings

Transition kernels are expressed by refined symmetric Grothendieck functions.

Particle dynamics are encoded using free fermion basis and deformed Schur operators.

A direct combinatorial proof relates particle motions to tableaux.

Abstract

For each of the four particle processes given by Dieker and Warren [arXiv:0707.1843], we show the $n$-step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.