Analyticity of intersection exponents for planar Brownian motion

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dc.creatorLawler, Gregory F.
dc.creatorSchramm, Oded
dc.creatorWerner, Wendelin
dc.date2000-05-31
dc.date.accessioned2026-07-07T10:22:29Z
dc.date.available2026-07-07T10:22:29Z
dc.descriptionWe 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 $ξ(1, λ)$ by $$ E[P[B[0,t] \cap B'[0,t] = \emptyset | B[0,t]]^λ] \approx t^{-ξ(1, λ)/2}, t \to \infty. $$ Then the mapping $λ\mapsto ξ(1, λ)$ is real analytic in $(0,\infty)$. The same result is proved for the exponents $ξ(k, λ)$ where $k$ is a positive integer. In combination with the determination of $ξ(k, λ)$ for integer $k \ge 1$ and real $λ\ge 1$ in our previous papers, this gives the value of $ξ(k, λ)$ also for $λ\in (0,1)$ and the disconnection exponents $\lim_{λ\searrow 0} ξ(k, λ)$. In particular, it shows that $\lim_{λ\searrow 0} ξ(2, λ) = 2/3$ 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.
dc.identifierhttps://arxiv.org/abs/math/0005295
dc.identifierhttp://arxiv.org/abs/math/0005295
dc.identifierActaMath.189:179-201,2002
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/175530
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject60J65; 82B26
dc.titleAnalyticity of intersection exponents for planar Brownian motion
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