LA Probability Forum

4pm: Ron Peled: 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.

5pm: Shalin Parekh: Universality of the 2d SHF

Abstract: The (2+1)-dimensional critical stochastic heat flow (SHF) was recently constructed as a scaling limit of discrete polymer models by Caravenna-Sun-Zygouras, with a more axiomatic characterization given in a later work of Tsai. It is an exciting object to study because it has a nontrivial correlation structure despite being "scaling-critical" in the language of singular SPDEs. To observe the SHF, the model parameters need to be tuned in some extremely precise way that is difficult to write down for most models of interest. Our main contribution is to overcome this, and show that this critical tuning can be written down abstractly for a very general class of models via the largest eigenvalue of a certain linear operator related to the gap process of the 2-point motion. The class of models we consider includes polymers, random walks in random environments, and various mollified stochastic heat equations. Our methods combine the renewal approach of Caravenna-Sun-Zygouras with the axiomatic characterization of Tsai. Joint work with Hindy Drillick and Jonathan Hou. 

6pm: Rahul Rajkumar: A One-Parameter Family of Random Walks on the Two-Dimensional p-adic Vector Space

Abstract: The study of Qp^d-valued stochastic processes is an active area of research in p-adic mathematical physics and related fields. While in the Rd setting, lots is known about scaling limits of random walks, only recently have we shown that a family of p-adic Levy processes associated to analogues of the heat equation is a scaling limit of random walks. By using the additional structure of a p-adic field, this family includes processes with a restricted class of anisotropies. We enlarge this class by constructing a one-parameter family of random walks on Qp^2 and determining their scaling limits. This talk is based on joint work with David Weisbart.

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