Analyticity of intersection exponents for planar Brownian motion

Abstract

We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let $B$ and $B^{\prime }$ be independent planar Brownian motions started from distinct points, and define the exponent $\xi (1,\lambda )$ by $$E[P[B[0,t]\cap B^{\prime }[0,t]=\varnothing |B[0,t]{]}^{\lambda }]\approx t^{-\xi (1,\lambda )/2},t\to \infty .$$ 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.

References in corpus (1)

• Values of Brownian intersection exponents III: Two-sided exponents

Cited by in corpus (1)

• Discrete Loewner evolution