Determinantal correlations of Brownian paths in the plane with nonintersection condition on their loop-erased parts

Abstract

As an image of the many-to-one map of loop-erasing operation L 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 ζ is given by the total weight of all random walks π which are inverse images of ζ, {π : L(π) = ζ}. 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 × 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 (γ 1 , . . . , γ N ) first enter each chamber, under the condition that the loop-erased parts (L(γ 1 ), . . . , L(γ 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.

Loop-Erased Walks and Random Matrices

“…Proposition 4.7 agrees with certain asymptotics of an excursion Poisson kernel determinant in [34], in the context of rectangular domains of the complex plane.…”

“…In Sect. 4 we show that the limits (1.6) agree with eigenvalue densities of Cauchy type random matrix ensembles, for determinants of the form (1.4) (see [34] and Sect. 4.6 for similar asymptotic considerations regarding excursion Poisson kernels).…”

“…We have developed connections between loop-erased walks in two dimensions and random matrices, based on an identity of Fomin [8]. This complements earlier work of Sato and Katori [34], where an example of this type of connection was exhibited in a slightly different context, as explained in Sects. 1.3 and 4.6 .…”

“…We will see that the consideration of the above limits reveals some natural connections to random matrices, particularly Cauchy type ensembles [38]. An example of this connection was first observed by Sato and Katori [34], in the context of excursion Poisson kernel determinants , and we discuss this in Sect. 4.6.…”

“…We gratefully acknowledge the support of the European Research Council (Grant Number 669306) and CONACYT (PhD Scholarship Number 411059). We would also like to thank the anonymous referees for their careful reading and suggestions, in particular for drawing our attention to the paper [34], which have led to a much improved version of the paper.…”