Generalized disconnection exponents
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Generalized disconnection exponents Wei Qian1 Received: 18 April 2019 / Revised: 18 August 2020 / Accepted: 14 September 2020 © The Author(s) 2020
Abstract We introduce and compute the generalized disconnection exponents ηκ (β) which depend on κ ∈ (0, 4] and another real parameter β, extending the Brownian disconnection exponents (corresponding to κ = 8/3) computed by Lawler, Schramm and Werner (Acta Math 187(2):275–308, 2001; Acta Math 189(2):179–201, 2002) [conjectured by Duplantier and Kwon (Phys Rev Lett 61:2514–2517, 1988)]. For κ ∈ (8/3, 4], the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity c ∈ (0, 1], which allows us to obtain the first prediction of the dimension of multiple points on the cluster boundaries of these loop-soups. In particular, according to our prediction, the dimension of double points on the cluster boundaries is strictly positive for c ∈ (0, 1) and equal to zero for the critical intensity c = 1, leading to an interesting open question of whether such points exist for the critical loop-soup. Our definition of the exponents is based on a certain general version of radial restriction measures that we construct and study. As an important tool, we introduce a new family of radial SLEs depending on κ and two additional parameters μ, ν, that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial SLEκ (ρ)s. Keywords Disconnection and intersection exponents · Hypergeometric SLE · Conformal restriction measure · Brownian loop-soup Mathematics Subject Classification 60J67 · 60J65 · 60D05
B 1
Wei Qian [email protected] CNRS and Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France
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W. Qian
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Main result on generalized disconnection exponents . . . . . . . . . . 1.3 Main result on general radial restriction measures . . . . . . . . . . . 1.4 Main results on radial hypergeometric SLE . . . . . . . . . . . . . . . 1.5 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Relation between the generalized disconnection exponents and loop-soups 2.1 Background on the Brownian disconnection exponents . . . . . . . . 2.2 Relation between Brownian motions and restriction measures . . . . . 2.3 On the generalized disconnection exponents . . . . . . . . . . . . . . 3 Radial hypergeometric SLE . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Loewner equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries on hypergeometric functions . . . . . . . . . . . . . . . 3.3 The function G: definition and properties . . . . . . . . . . . . . . . . 3.4 Geometric properties of radial hSLE . . . . . . . . . . . . . . . . . . 4 Construction of general radial restriction measures . . . . . . . . . . . . . 4.1 Method of construction . .
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