Lattice-model crossing probabilities and a system of PDEs for multiple SLE

Steven Flores, University of Helsinki


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.


Aalto Stochastics & Statistics Seminar
Mon 29 Sep 2014, 16:15
Lecture Hall M2, Otakaari 1, Espoo