Conformal invariance of critical lattice models in two-dimensional has been vigorously studied for decades. The first example where the conformal invariance was rigorously verified was the planar uniform spanning tree (together with loop-erased random walk), proved by Lawler, Schramm and Werner around 2000. Later, the conformal invariance was also verified for Bernoulli percolation (Smirnov 2001), level lines of Gaussian free field (Schramm-Sheffield 2009), and Ising model and FK-Ising model (Chelkak-Smirnov et al 2012). In this talk, we focus on connection probabilities of these critical lattice models in polygons with alternating boundary conditions.

This talk has two parts.

In the first part, we consider critical random-cluster model with cluster weight q∈(0,4) and give conjectural formulas for connection probabilities of multiple interfaces. The conjectural formulas are proved for q=2, i.e.the FK-Ising model.

In the second part, we consider uniform spanning tree (UST) and give formulas for connection probabilities of multiple Peano curves. UST can be viewed as the limit of random-cluster model as q goes to 0. Its connection probabilities turn out to be related to logarithmic CFT.

This talk is based on joint works with Yu Feng, Mingchang Liu, and Eveliina Peltola.