LA Probability Forum

UCLA MSR 6627

4pm: Cesar Cuenca: Discrete N-particle ensembles at high temperature through Jack symmetric functions

Abstract: I will speak about random discrete N-particle systems with the inverse temperature parameter theta. We find necessary and sufficient conditions for the Law of Large Numbers as the size N of the system tends to infinity simultaneously with the inverse temperature going to zero. We obtain the LLN for Markov chains of N nonintersecting particles and the LLN for the multiplication of Jack symmetric functions, as the inverse temperature tends to zero. We express the answer in terms of novel one-parameter deformations of cumulants and discuss their relation to (quantized) free probability. Finally, we discuss a crystallization phenomenon and describe it in terms of the countable spectra of certain Jacobi operators. The talk is based on joint work with Maciej Dolega.

5pm: Andreas Klippel: Long-Range Order in the Monomer Double-Dimer Model with Long-Range Interactions

Abstract: The dimer model and its associated double-dimer model are fundamental objects in probability theory, statistical mechanics, and combinatorics. While their behavior in planar settings is by now well understood, much less is known in higher dimensions and in the presence of a positive density of monomers, leading to the so-called monomer double-dimer model.
We study these models on Zd-like graphs (d≥1) that allow long-range edges whose weights decay with distance. For a large class of such interactions, we prove that the monomer double-dimer model exhibits long-range order. As a consequence, monomer correlations in the dimer model remain uniformly positive, and loops in the double-dimer model become macroscopic.
In this talk, I will introduce the models and outline the main ideas of the proof. We will see that the model admits a natural correspondence with a spin system, which allows us to transfer results obtained via reflection positivity and thereby establish long-range order. This is joint work with Lorenzo Taggi and Wei Wu.

6pm: Hindy Drillick: Extremal scaling limits for random walks in space-time random environments

Abstract: In this talk, we will consider random walks in a space-time random environment, which can be thought of as a discrete model for diffusing particles in a time-dependent random medium. We will study the scaling limits of these models in certain moderate deviation scaling regimes and show that they are described by stochastic PDEs. The solutions to these SPDEs are Gaussian processes up to a dimension-dependent critical scale. In d=1 we prove that the critical fluctuations are given by the KPZ equation. In d=2, we conjecture that the scaling limit at criticality is given by the 2d critical stochastic heat flow recently constructed by Caravenna, Sun, and Zygouras. This is based on joint works with Sayan Das and Shalin Parekh.

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