Description:

Title: The critical fugacity of the hard-core model in high dimensions

Abstract: In the hard-core model on a graph G, one samples a random independent subset A of G with probability proportional to λ^|A|, where λ>0 is a parameter, termed the fugacity. The seminal work of Dobrushin (1968) established a phase transition for the hard-core model on Z^d (in the sense of Gibbs measures): At fugacity λ < 1/(2d-1), the model is disordered in the sense that the random set A has half of its vertices on each partite class, while at sufficiently high fugacity, the model exhibits long-range order in the sense that the set A has density strictly larger than one half on one of the two partite classes.
Following Dobrushin, the question of determining the minimal fugacity λ_c(d) at which long-range order arises, and its asymptotics as d → ∞ has remained a challenge of enduring interest. In a breakthrough work, Galvin–Kahn (2004) proved that λ_c(d) < d^{-1/4 + o(1)}, thus showing that the critical fugacity decays to zero with the dimension. Their bound was improved by Samotij–Peled (2014) who showed  λ_c(d) <  d^{-1/3 + o(1)}.
Galvin–Kahn suggested that λ_c(d) = d^{-1 + o(1)}. In this talk I will discuss the proof of this fact, obtained jointly with Daniel Hadas.