Multiplicative functionals on ensembles of non-intersecting paths

TL;DR

This paper develops a theoretical framework connecting path-integral kernels with determinantal point processes and non-intersecting path ensembles, revealing how extended kernels relate to Fredholm determinants in various stochastic models.

Contribution

It introduces a novel theory linking path-integral kernels to determinantal processes and demonstrates their equivalence in multiple important stochastic models.

Findings

Determinants involving path-integral kernels arise naturally in non-intersecting path ensembles.

Fredholm determinants with extended kernels are equal to those with path-integral kernels.

The theory applies to models like GUE Dyson Brownian motion, Airy processes, and Markov processes on partitions.

Abstract

The purpose of this article is to develop a theory behind the occurrence of "path-integral" kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants involving such kernels arise naturally in studying ratios of partition functions and expectations of multiplicative functionals for ensembles of non-intersecting paths on weighted graphs. Our second result shows how Fredholm determinants with extended kernels (as arise in the study of extended determinantal point processes such as the Airy_2 process) are equal to Fredholm determinants with path-integral kernels. We also show how the second result applies to a number of examples including the stationary (GUE) Dyson Brownian motion, the Airy_2 process, the Pearcey process, the Airy_1 and Airy_{2->1} processes, and Markov processes on partitions related to the z-measures.