Determinantal Correlations of Brownian Paths in the Plane with Nonintersection Condition on their Loop-Erased Parts

TL;DR

This paper studies the correlations of Brownian paths with nonintersecting loop-erased parts in the plane, showing they are described by determinantal point processes with a specific correlation kernel, linking Brownian motion, combinatorics, and random matrix theory.

Contribution

It introduces a continuum model of nonintersecting loop-erased Brownian paths and derives their correlation functions as determinantal point processes with a novel correlation kernel.

Findings

Correlation functions are given by determinants of a continuous kernel.

The correlation kernel is of Eynard-Mehta type, common in random matrix theory.

Conformal covariance of the correlation functions is established.

Abstract

As an image of the many-to-one map of loop-erasing operation $\LE$ of random walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk (LERW) model is the statistical ensemble of SAWs such that the weight of each SAW $\zeta$ is given by the total weight of all random walks $\pi$ which are inverse images of $\zeta$, $\{\pi: \LE(\pi)=\zeta \}$. We regard the Brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the LERW model. Following the theory of Fomin on nonintersecting LERWs, we introduce a nonintersecting system of $N$-tuples of LEBPs in a domain $D$ in the complex plane, where the total weight of nonintersecting LEBPs is given by Fomin's determinant of an $N \times N$ matrix whose entries are boundary Poisson kernels in $D$. We set a sequence of chambers in a planar domain and observe the first passage points at which $N$ Brownian paths $(\gamma_1,..., \gamma_N)$ first enter each chamber, under the condition that the loop-erased parts $(\LE(\gamma_1),..., \LE(\gamma_N))$ make a system of nonintersecting LEBPs in the domain in the sense of Fomin. We prove that the correlation functions of first passage points of the Brownian paths of the present system are generally given by determinants specified by a continuous function called the correlation kernel. The correlation kernel is of Eynard-Mehta type, which has appeared in two-matrix models and time-dependent matrix models studied in random matrix theory. Conformal covariance of correlation functions is demonstrated.