Multiple intersection exponents

TL;DR

This paper introduces a new mathematical framework for analyzing the decay rates of probabilities that multiple independent planar Brownian motions do not intersect, extending known results to cases with more than two processes.

Contribution

It defines a novel p-fold intersection exponent for multiple Brownian motions and provides an exact formula for specific cases, along with conjectures for others.

Findings

Exact formula for n1=1, n2=2 case

Numerical and mathematical analysis of intersection exponents

Conjectures for other configurations

Abstract

Let $p\ge2$, $n_1\le...\le n_p$ be positive integers and $B_1^1, ..., B_{n_1}^1; ...; B_1^p, ..., B_{n_p}^{p}$ be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a $p$-fold intersection exponent $\varsigma(n_1,..., n_p)$, as the exponential rate of decay of the probability that the packets $\bigcup_{j=1}^{n_i} B_j^i[0,t^2]$, $i=1,...,p$, have no joint intersection. The case $p=2$ is well-known and, following two decades of numerical and mathematical activity, Lawler, Schramm and Werner (2001) rigorously identified precise values for these exponents. The exponents have not been investigated so far for $p>2$. We present an extensive mathematical and numerical study, leading to an exact formula in the case $n_1=1$, $n_2=2$, and several interesting conjectures for other cases.