Comparison of arm exponents in planar FK-percolation.

Resumen:  

FK-percolation is a generalisation of Bernoulli percolation that was found to be related to a wide range of other models in statistical mechanics, including the Ising model and the six-vertex model. In this talk, we will focus on the specific case of critical planar FK-percolation in the continuous phase transition regime. In this setting, the model exhibits properties similar to those of critical planar Bernoulli percolation; in particular, the Russo-Seymour-Welsh theory applies and the model is conjectured to be conformally invariant.

Conformal invariance would imply that the probabilities of arm events are governed by arm exponents. Using the FKG inequality, one can establish natural inequalities involving these exponents. We will show how to strengthen these FKG results by establishing strict inequalities. Specifically, we will use the Russo-Seymour-Welsh theory to prove a refined inequality between the probability of the alternating two-arm event and the product of the probabilities of having a primal arm and a dual arm, respectively. It follows that if the
alternating two-arm exponent α01 and the one-arm exponents α0 and α1 exist, they must satisfy the strict inequality α01 > α0 + α1. This result enables us to extend a method of Garban and Steif to prove the existence of exceptional times for Bernoulli percolation on the
square lattice (without assuming conformal invariance).

Posted on Apr 24, 2025 in Seminario de Probabilidades de Chile, Seminars