Generalized disconnection exponents

We introduce and compute the generalized disconnection exponents η_κ(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_κ (ρ)s.