Schedule

• Sep 8

  Pu Yu (NYU)
  3:15pm

  Convergence of circle packing for the mated-CRT map and triangulation from UST weighted maps

  Abstract: Liouville quantum gravity (LQG) is conjectured to describe scaling limits of random planar maps. One way to phrase the convergence of random planar maps is through conformal embeddings. This has been done by Holden and Sun (2019) for the uniform triangulation under Cardy embedding, and by Gwynne, Miller and Sheffield (2017) as well as Bertacco, Gwynne and Sheffield (2023) for the mated-CRT map under Tutte and Smith embedding. In this talk, I will discuss the convergence of the mated-CRT map to LQG and the triangulation from uniform spanning tree weighted maps through Mullin bijection to \sqrt{2}-LQG under circle packing. Based on joint work with Nina Holden.

• Sep 15

  Andres Contreras Hip (University of Chicago)

  Gaussian curvature for Liouville Quantum Gravity and random planar maps

  Abstract: Liouville quantum gravity is a canonical model for random surfaces conjectured to be the scaling limit of various planar maps. Since curvature is a central concept in Riemannian geometry, it is natural to ask whether this can be extended to LQG surfaces. In this talk, we introduce a notion of Gaussian curvature for LQG surfaces, despite their low regularity, and study the relations with its discrete counterparts. We conjecture that this definition of Gaussian curvature is the scaling limit of the discrete curvature. In support of this conjecture, we prove that the discrete curvature on the $\epsilon$-CRT map with a Poisson vertex set integrated with a smooth test function is of order $\epsilon^{o(1)},$ and show the convergence of the total discrete curvature on a CRT map cell when scaled by $\epsilon^{1/4}.$ Joint work with E. Gwynne.

• Sep 22

  Gabriel Raposo (UC Berkeley)

  Fluctuations for standard Young Tableaux

  Abstract: We will introduce the Young generating function and use it to characterize the law of large numbers and the central limit theorem behaviors for random partitions. As an application of these results, we present a framework to obtain conditional Gaussian Free Field fluctuations for height functions associated with random standard Young tableau. To prove these results we develop algebraic formulas for operators on the Gelfand–Tsetlin algebra of the symmetric group.

• Sep 29

  Eilon Solan (Tel-Aviv University)

  Equilibrium in Multiplayer Stopping Games

  Abstract: Stopping games generalize optimal stopping to settings with multiple decision makers. We work in discrete time on a filtered probability space. There are $N$ decision makers. For each nonempty subset $S \subseteq \{1,...,N\}$ there is an $\mathbb{R}^N$-valued stochastic process ($X^S_t$). At each stage, each decision maker, given their current information, chooses whether to stop or to continue. The game terminates (for everyone) at the first stage in which at least one decision maker stops; if the set of stoppers at that stage is $S$, then decision maker $i$ receives the $i$-th coordinate of $X^S_t$. Each player aims to maximize the expectation of their payoff.
  
  An $\epsilon$ equilibrium is a profile of (possibly randomized) stopping times such that no decision maker can gain more than $\epsilon$ by deviating while the others keep their stopping times fixed. When $N \leq 3$, an $\epsilon$-equilibrium is known to exist (assuming the payoff processes ($X^S_t$) are integrable). $When N \geq 4$, the existence of $\epsilon$-equilibria is an open problem.
  
  I will describe the model, survey known results, and present intriguing examples.

• Oct 6

  Oriol Sole Pi (MIT)

  Graph structure and soficity

  Abstract: A random rooted graph is said to be sofic if it is the Benjamini-Schramm limit of a sequence of finite graphs. Sofic graphs are known to possess a certain property known as unimodularity. In a recent breakthrough, Bowen, Chapman, Lubotzky and Vidick have shown that not all unimodular graphs are sofic. In this talk, I will give an overview of what is known in the other direction: Which additional conditions on the graph are known to imply soficity? Then, I will discuss a novel result in this direction: For any finite graph H, every one-ended, unimodular graph which does not have H as a minor must be sofic.

• Oct 13

  Indigenous People's day

• Oct 20

  Gefei Cai (Peking)

  Disconnection and non-intersection probabilities of Brownian motion on an annulus

  Abstract: We derive an exact formula for the probability that a Brownian path on an annulus does not disconnect the two boundary components of the annulus. The leading asymptotic behavior of this probability is governed by the disconnection exponent obtained by Lawler-Schramm-Werner (2001) using the connection to Schramm-Loewner evolution (SLE). The derivation of our formula is based on this connection and the coupling with Liouville quantum gravity (LQG), from which we can exactly compute the conformal moduli of random annular domains defined by SLE curves. Using a similar approach, we also derive exact formulas for the non-intersection probabilities of independent Brownian paths on an annulus, as well as extend the result to the case of Brownian loop soup. Based on joint work with X. Fu, X. Sun, and Z. Xie, and upcoming work with Z. Xie.

• Oct 27

  Melanie Matchett Wood (Harvard)

  Universality for distributions of groups and algebraic objects

  Abstract: Arithmetic statistics has led to the study of many distributions of algebraic objects, including distributions of abelian groups, non-abelian groups, modules and more. We discuss universality results showing that many different constructions of random abelian groups lead to the same asymptotic distributions, and discuss the conjectural landscape for universality for distributions of non-abelian random groups and more general algebraic objects.

• Nov 3

  Daniel Lacker (Columbia University)

  Sharp quantitative propagation of chaos for mean field and non-exchangeable systems

  Abstract: The propagation of chaos phenomenon states roughly that a large system of weakly interacting particles will remain approximately independent for all times if initialized as such. This can be quantified in terms of the distance between low-dimensional marginal distributions and suitably chosen product measures. This talk will discuss some recent sharp quantitative results of this nature, both for classical mean field diffusions and for more recently studied non-exchangeable systems based on dense graphs. These results are driven by a new "local" relative entropy method, in which low-dimensional marginals are estimated iteratively by adding one coordinate at a time, leading to surprising improvements on prior results obtained by "global" arguments such as subadditivity inequalities. In the non-exchangeable setting, we exploit an unexpected connection with first-passage percolation.

• Nov 10

  Holiday

• Nov 17

  Oren Yakir (MIT)

  Charge fluctuations in the hierarchical Coulomb gas

  Abstract: The two-dimensional Coulomb gas model describes electrically charged particles embedded in a uniform background of the opposite charge, interacting through a logarithmic potential. A celebrated prediction from the physics literature, made by Jancovici, Lebowitz and Manificat in 1993, describes probabilities of observing large charge fluctuations in the system. From the mathematical standpoint, the JLM law is only proved in a very special 'solvable' case, with only partial results known in general. A few years ago, Chatterjee introduced a hierarchical version of the Coulomb gas inspired by Dyson's hierarchical model for the Ising ferromagnet. In the talk, I will present a joint work with Alon Nishry, in which we prove that the JLM law holds for this hierarchical model.

• Nov 24

  Juan Carlos Pardo Millan (CIMAT)

• Dec 1

• Dec 8