Abstract
We 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
.
Reversed Radial SLE and the Brownian Loop Measure
