Abstract:
The dimer model, introduced by Kasteleyn, Temperley and Fisher in the 1960s, is one of the most classical models of statistical mechanics. It is well known since the work of Kenyon, Okunkov and Sheffield that the dimer model can display coexistence of multiple phases known as solid, liquid, and gaseous respectively. In the liquid phase one expects conformal invariance / GFF behaviour, whereas near the liquid-gas boundary one expects "massive theories" in the scaling limit.
After introducing the model and recalling some classical results I will report on recent and ongoing progress on these questions. The setup will be the dimer model on the square (or hexagonal) lattice with doubly periodic weights, in the near-critical scaling regime and with Temperleyan boundary conditions. In joint work with Haundschmid-Sibitz we proved:
- convergence of branches in the associated Temperleyan tree to the so-called massive SLE_2 of Makarov and Smirnov,
- convergence of the height function towards a random field which is not conformally invariant but covariant.
- We had also conjectured that this limiting field is a specific variant of the sine-Gordon model (from quantum field theory) at its free fermion point. This conjecture is currently being proved in joint work with Mason and Rey. I will describe some of our results in this direction.
Bio-Sketch
Nathanaël Berestycki holds the chair of Stochastics at the University of Vienna since 2018. Prior to this (and after PhD studies at Ecole Normale Supérieure in Paris and Cornell University, as well as a postdoctoral fellowship at the University of British Columbia in Vancouver) he was a Professor of Probability at the University of Cambridge. He has been an associate editor of numerous journals, including the Annals of Probability, and was an invited speaker at the International Congress of Mathematical Physics (ICMP) in 2024.
Time&Venue
Apr 1 10 am-noon, Lecture 1, Location: 77201.
Apr 3 10 am-noon, Lecture 2, Location: Quan 29
