Analyticity of intersection exponents for planar Brownian motion

TL;DR

This paper proves that the intersection exponents for planar Brownian motion are real analytic functions, leading to precise calculations of these exponents and confirming a conjecture about the Hausdorff dimension of the outer boundary.

Contribution

It establishes the analyticity of intersection exponents for planar Brownian motion and extends previous results to determine these exponents for a broader range of parameters.

Findings

Proved the real analyticity of intersection exponents in (0,∞).

Calculated the Hausdorff dimension of the outer boundary as 4/3.

Confirmed Mandelbrot's conjecture on the outer boundary dimension.

Abstract

We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let $B$ and $B'$ be independent planar Brownian motions started from distinct points, and define the exponent $\xi (1, \lambda)$ by $$ E[P[B[0,t] \cap B'[0,t] = \emptyset | B[0,t]]^\lambda] \approx t^{-\xi(1, \lambda)/2}, t \to \infty. $$ Then the mapping $\lambda \mapsto \xi (1, \lambda)$ is real analytic in $(0,\infty)$. The same result is proved for the exponents $\xi (k, \lambda)$ where $k$ is a positive integer. In combination with the determination of $\xi (k, \lambda)$ for integer $k \ge 1$ and real $\lambda \ge 1$ in our previous papers, this gives the value of $\xi (k, \lambda)$ also for $\lambda \in (0,1)$ and the disconnection exponents $\lim_{\lambda \searrow 0} \xi (k, \lambda)$. In particular, it shows that $\lim_{\lambda \searrow 0} \xi(2, \lambda) = 2/3$ and concludes the proof of the following result that had been conjectured by Mandelbrot: the Hausdorff dimension of the outer boundary of $B[0,1]$ is 4/3 almost surely.