Speaker: Eveliina Peltola
Title: Crossing probabilities in critical 2D models: continuous and discrete aspects

Abstract:
For a number of lattice models in 2D statistical physics, it has been proven that the scaling limit of an interface at criticality (with suitable boundary conditions) is a Schramm Loewner evolution (SLE). Similarly, collections of several interfaces converge to families of interacting random curves, multiple SLEs. Connection probabilities of these interfaces encode crossing probabilities in the lattice models, which should also be related to correlation functions of appropriate fields in the corresponding conformal field theory (CFT).

I discuss crossing probabilities both in exactly solvable cases (UST, LERW, double-dimers, GFF) where no a priori information of the interfaces is used (in particular, for double-dimer interfaces, the convergence has not even been proven), and, as a non-exactly solvable example, in the critical Ising model. In the latter, there are no explicit formulas for the crossing probabilities, but their scaling limits can still be completely understood in terms of so-called pure partition functions of multiple SLEs. In particular, all of the expected CFT properties: conformal invariance, null-field equations, and fusion rules, are satisfied.


I will mention several joint works, with subsets of
Vincent Beffara (Université Grenoble Alpes, Institut Fourier),
Alex Karrila (Aalto University), Kalle Kytölä (Aalto University), and
Hao Wu (Yau Mathematical Sciences Center, Tsinghua University).