Generalized disconnection exponents

Abstract

We introduce and compute the generalized disconnection exponentsη<inf>κ</inf>(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 SLE <inf>κ</inf>(ρ) s.