Generalized disconnection exponents

Abstract

AbstractWe introduce and compute the generalized disconnection exponents$$\eta _\kappa (\beta )$$ η κ ( β ) which depend on $$\kappa \in (0,4]$$ κ ∈ ( 0 , 4 ] and another real parameter $$\beta $$ β , extending the Brownian disconnection exponents (corresponding to $$\kappa =8/3$$ 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 $$\kappa \in (8/3,4]$$ κ ∈ ( 8 / 3 , 4 ] , the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity $$c\in (0,1]$$ 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 $$c\in (0,1)$$ c ∈ ( 0 , 1 ) and equal to zero for the critical intensity $$c=1$$ 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 $$\kappa $$ κ and two additional parameters $$\mu , \nu $$ μ , ν , that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial $$\hbox {SLE}_\kappa (\rho )s$$ SLE κ ( ρ ) s .