2026 FALL ABSTRACTS

Sep 22    Penn: Reuben Drogin  (Yale)

Random Band Matrices and Random Permutations

Random band matrices are Hermitian matrices with random entries supported in a band of width W around the diagonal. The eigenfunctions of such matrices are expected to decay exponentially at the scale W^2 in dimension one, and exp(CW^2) in dimension two. Remarkably, the same scaling is expected for the cycle lengths in various models of random permutations where points are typically displaced by distances of order W. In this talk I will discuss some recent progress on these problems, some connections, and some open questions.

Sep 29    Penn: Dmitry Krachun  (Princeton)

Towards conformal invariance of critical planar lattice models

Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.

In this talk I will explain some ideas behind the proof of rotational invariance of the FK-percolation model and more recently the computation of critical exponents of planar FK model with q=4. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz. We will then explore the connection to the six-vertex model further to derive critical exponents of the FK model with q=4.

Based on joint works with Hong-Bin Chen, Hugo Duminil-Copin, Tiancheng He, François Jacopin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, Tatiana Tikhonovskaia, and Jiaming Xia.

Oct 06    Temple: Xin Zhang  (NYU)

Gradient flow for weak optimal transport

Weak optimal transport is a relaxation of classical optimal transport in which the cost depends nonlinearly on the coupling through its disintegration. This framework is motivated by applications in transport inequalities, entropic optimal transport, and martingale optimal transport.

In this talk, we propose a gradient flow approach to weak optimal transport in the adapted Wasserstein space. We formally derive the corresponding adapted Wasserstein derivative, project it onto the tangent space of couplings, leading to a McKean–Vlasov SDE that characterizes the flow. We establish well-posedness of this dynamic, and prove its convergence to the optimal coupling for the weak optimal transport problem. This talk is based on joint work with Nathan Sauldubois.

Oct 13    Temple: Carla Crucianelli  (Princeton)

Non-linear graphon games

When studying a non-cooperative stochastic differential game, it is classical to use the mean-field game framework. However, this approach has its limitations when the particles interact through a random graph. This is because we expect the heterogeneity of the system to be retained in the large population limit. In this talk, we consider a non-cooperative stochastic differential game with player interactions modeled through a sequence of graphs converging to a graphon. We show that under standard regularity conditions and dissipativity of the drift of the state process, when the number of players goes to infinity, the game converges to a general graphon game. The limit is quantitative and based on a new form of propagation of chaos for interacting FBSDEs on graphs. Further, when the cost function satisfies the Lasry-Lions monotonicity condition, we propose an existence and uniqueness result for graphon games. Our proof relies on investigating a system of uncountably many McKean-Vlasov type BSDEs driven by a continuum of Brownian motions. We study this BSDE in a specific probability framework called a Fubini extension and discuss its well-posedness, characterize the graphon mean-field equilibrium and show its existence and uniqueness. The existence proof is based on a new application of the continuation method. Based on joint works with Dylan Possamaï and Ludovic Tangpi.

Oct 20    Penn: Remy Mahfouf  (EPFL)

From the planar Ising model to quasiconformal mappings

We identify the scaling limit of full-plane Kadanoff-Ceva fermions on generic, non-degenerate s-embeddings. In this broad setting, the scaling limits are described in terms of solutions to conjugate Beltrami equations with prescribed singularities. For the underlying Ising model, this leads to the scaling limit of the energy-energy correlations connecting the scaling limits of (near-)critical planar Ising models to Green kernels of uniformly elliptic operators and quasiconformal mappings. For grids approximating bounded domains in the complex plane, we establish, in the scaling regime, the conformal covariance of the energy density on critical doubly periodic graph. Moreover, we prove a connection between the planar Ising energy field in smooth limiting structure and (massive)-holomorphic fermions defined on surfaces of the Minkowski space R^{(2,1)}. These results highlight that the scaling limits of generic (near-)critical Ising models naturally live on a substantially richer conformal structure than the classical Euclidean one.

Oct 27    Penn: Hindy Drillick (NYU) 

Scaling Limits of Random Walks in Random Environments

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 regimes. In earlier work, we showed that in d=1 the critical fluctuations are governed by the KPZ equation. In recent work, we proved that the scaling limit at criticality in d=2 is given by the critical 2d stochastic heat flow. This talk is based on joint works with Sayan Das, Jonathan Hou, and Shalin Parekh.

Dec 01    Penn: Matthew Rosenzweig  (Carnegie Mellon)

Title and abstract to be announced.