2:00 - 3:00pm Coffee/tea at Annenberg Cafe patio

3:00 - 4:00pm Marcin Lis, Vienna University of Technology: On Pfaffians in spin models

Abstract: It is a classical result of Groeneveld, Boel and Kasteleyn that boundary spin correlations functions in Ising models on planar graphs satisfy Pfaffian relations. Here we consider the reverse question, and show that any classical ferromagnetic spin model whose correlation functions satisfy Pfaffian relations must by (up to local simplifications of the graph) an Ising model on a planar graph. Our main tool is a new (coupled) version of the Edwards—Sokal representation of the Ising model applied to two independent copies of the spin model.

Joint work with Diederik van Engelenburg.

4:00 - 5:00pm Yujin Kim, NYU: The shape of the front of multidimensional branching Brownian motion

Abstract: The extremal process of branching Brownian motion (BBM)--- i.e., the collection of particles furthest from the origin-- has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point-- the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.

5:00 - 6:00pm Moritz Voss, UCLA: Equilibrium in functional stochastic games with mean-field interaction

Abstract: We study a general class of finite-player stochastic games with mean-field interaction where the linear-quadratic cost functional includes linear operators acting on controls in L^2. We propose a new approach for deriving the Nash equilibrium of these games in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations which can be solved. Moreover, by deriving stability results for the system of Fredholm equations, we obtain the convergence of the finite-player Nash equilibrium to the mean-field equilibrium in the infinite player limit. Our general framework includes examples of stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.

This is joint work with Eduardo Abi Jaber (Ecole Polytechnique) and Eyal Neuman (Imperial College London). The paper is available at https://ssrn.com/abstract=4470883.