Title: Lattice-model crossing probabilities and a system of PDEs
for multiple SLE
Abstract: We consider a critical lattice model, such as percolation
or a random cluster model, on a very fine regular lattice in a polygon P
with 2N sides. (N is a positive integer.) If we condition all lattice
sites in every other side of P to exhibit the same state, then clusters
of nearest-neighboring sites exhibiting that state may cross the interior
of P to join those sides together. There are C_N distinct such
“crossing events,” with C_N the Nth Catalan number, and our goal is
to determine their probabilities. An example of such an event is the
event of a percolation cluster connecting the opposite sides of a
rectangle (N=2), and Cardy’s formula gives its probability.
In a foundational paper on multiple SLE, M. Bauer, D. Bernard, and
K. Kytölä conjecture a formula for these crossing probabilities in
terms of putative solutions for a certain system of PDEs. We call
these solutions “connectivity weights.” In this talk, I present a
rigorous method for completely determining the vector space S_N of
solutions for this system that are bounded by power-law growth.
By “completely determine,” we mean determine both the dimension of S_N
and a basis of functions with explicit formulas.
(Explicit formulas follow from the Coulomb gas formalism of conformal
field theory.) Our method and results give a natural candidate
definition for the connectivity weights and a means of calculating
them for the conjectured crossing probability formula. We compare
our predictions with measurements of these probabilities via
computer simulations, finding very good agreement.