Schedule

• February 5

  Matan Harel
  (Northeastern University)

  The loop O(n) model and the XOR trick

  Abstract: The loop O(n) model is a model for random configurations of non-overlapping loops on the hexagonal lattice, which contains many models of interest (such as the Ising model, self-avoiding walks, and random Lipshitz functions) as special cases. Its conjectured phase diagram is very rich, and the model is believed to undergo several different phase transitions. Over the last decade, several features of the phase diagram have been proven rigorously, mostly through the use of particular bijections or observables at critical values. We use an expansion around critical percolation to prove that, near the values that correspond to critical Bernoulli percolation, the loop O(n) model has long , infinitely-nested loops, without relying on exact solvability. This is joint work with Nick Crawford, Alexander Glazman, and Ron Peled.

• February 12

  Kevin Yang
  

  Scaling limits for random growth driven by reflecting Brownian motion

  Abstract: We discuss long-time asymptotics for a continuum version of origin-excited random walk. It is a growing submanifold in Euclidean space that is pushed outward from within by the boundary trace of a reflecting Brownian motion. We show that the leading-order behavior of the submanifold process is described by a flow-type PDE whose blow-ups correspond to changes in diffeomorphism class of the growth process. We then show that if we simultaneously smooth the submanifold as it grows, fluctuations of an associated height function are described by a regularized KPZ equation with noise modulated by a Dirichlet-to-Neumann operator. If the dimension of the manifold is 2, we show well-posedness of the singular limit of this regularized KPZ-type equation. Based on joint work with Amir Dembo.

• February 19

  Presidents Day

• February 26

  Tomas Berggren
  (MIT)

  Geometry of the doubly periodic Aztec dimer model

  Abstract: Random dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit, we see three types of macroscopic regions -- known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.




• March 11

  Jiaming Xu
  (Duke)

  Recent advances on random graph matching

  Abstract: Random graph matching aims to recover the hidden vertex correspondence between two random graphs from correlated edge connections. This is a ubiquitous problem arising in a variety of applications across diverse fields such as network privacy, computational biology, computer vision, and natural language processing. The problem is also deep and rich in theory, involving the delicate interplay of algorithms, complexity, and information limits. Recently, extensive efforts have been devoted to the study of matching two correlated Erdős–Rényi graphs and exciting progress have been made, thanks to collective efforts from a wide research community. The speaker in his talk will present an overview, recent results, and important future directions on this topic. Based on joint work with Cheng Mao (Georgia Tech), Yihong Wu (Yale), and Sophie H. Yu (Stanford).

• March 18

  Sky Cao
  (MIT)

  Random surfaces and lattice Yang-Mills

  Abstract: I will talk about recent work which studies Wilson loop expectations in lattice Yang-Mills models. In particular, I will give a representation of these expectations as sums over embedded maps. Time permitting, I will also discuss alternate derivations, interpretations, and generalizations of several recent theorems about Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov) and surface sums (Magee and Puder).

  This is joint work with Minjae Park and Scott Sheffield.

• March 25

  Spring Break

• April 1

  Guillaume Baverez
  (Humboldt University of Berlin)

  Quantisation of the Segal's semigroup from Liouville theory

  Abstract: In Segal's definition of conformal field theory (CFT), one key ingredient is to construct a representation of the semigroup of complex annuli with parametrised boundaries. In the first part of the talk, I will explain how this statement can be understood as a generalisation of the Hille-Yosida theorem, using only an analytic language. In particular, the (infinite dimensional) family of generators form a representation of the Virasoro algebra and encode the conformal symmetry of the theory. In the second part, I will give an example of this construction using the Gaussian free field, and show how it can be extended to treat the Liouville CFT. Based on joint works with Guillarmou, Kupiainen, Rhodes, Vargas.

• April 8

  Shalin Parekh
  (University of Maryland)

  Extreme value theory for random walks in random media

  Abstract: The KPZ equation is a singular stochastic PDE arising as a scaling limit of various physically and probabilistically interesting models. Often this equation describes the “crossover” between Gaussian and non-Gaussian fluctuation behavior in models of interacting particles, directed polymers, or interface growth models. In this talk, I will discuss recent progress we have made in understanding the KPZ crossover for models of random walks in dynamical random media. This talk includes joint work with Sayan Das and Hindy Drillick.

• April 15

  Patriots' Day

• April 22

  Pierre Patie
  (Cornell)

  A spectral and algebraic algorithm: the centralizer and the fixed points, scaling and universality classes

  Abstract: Over the last few decades, the exploration of scaling limits and universality classes has unveiled a spectrum of intriguing results, alongside complex and fascinating challenges. In this talk, we present a comprehensive framework designed to address these challenges in a constructive and solvable manner. It is based on an appropriate combination of group representation theory, group actions, spectral theory and operator algebras. Relying on the Stone-von Neumann Theorem, we identify a canonical setting for this framework, the so-called canonical G-module, and design a constructive algorithm. This formalism not only highlights the fundamental role played by the choice of the representation of mathematical objects but also offers constructive perspectives and connections into classical mathematical topics such as the spectral theory of self-adjoint operators, Lie point symmetry, von Neumann algebras, and the fundamental Stone-von Neumann theorem. We will illustrate this framework by describing the universality classes of the LUE ensemble, emphasizing the canonical role of its limit to the Bessel ensemble in the comprehensive framework. Finally, we shall discuss the role played by the Fourier transform, the Laplacian, and Brownian motion in this formalism.

• April 29

  Jinwoo Sung
  (University of Chicago)

  Supercritical LQG conditioned to be finite

  Abstract: Liouville quantum gravity (LQG) in the subcritical and critical phases, corresponding to 𝛾 ≤ 2 or central charge c ≤ 1, has been extensively studied as a theory of random metric measure space and conformal field theory. On the other hand, few rigorous results about LQG with parameters outside these ranges are known, mainly for the supercritical LQG metric constructed by Ding and Gwynne. In this talk, I will discuss other properties of supercritical LQG that can be investigated from its branching structure, which is due to the coupling of supercritical LQG disk with nested CLE4 by Ang and Gwynne. There are two main results: (1) the supercritical LQG area measure, which we define as a random Borel measure locally determined by a GFF and satisfying the LQG coordinate change rule, does not exist, and (2) a natural discretization of the CLE-decorated supercritical LQG disk has the continuum random tree as its scaling limit if we condition it to have a finite number of vertices. The latter behavior was predicted by Gwynne, Holden, Pfeffer, and Remy in a pioneering paper on supercritical LQG. This is joint work with Manan Bhatia and Ewain Gwynne.

• May 6

  Brice Huang
  (MIT)

  Capacity threshold for the Ising perceptron

  Abstract: We show that the capacity of the Ising perceptron is with high probability upper bounded by the constant $\alpha \approx 0.833$ conjectured by Krauth and Mézard, under the condition that an explicit two-variable function $S(\lambda_1,\lambda_2)$ is maximized at (1,0). The earlier work of Ding and Sun proves the matching lower bound subject to a similar numerical condition, and together these results give a conditional proof of the conjecture of Krauth and Mézard.

• May 13

  Byron Chin
  (MIT)

  Structure of lower tails in sparse random graphs

  Abstract: In 2011, Chatterjee and Varadhan proved a large deviations principle for the Erd\H{o}s--R\'enyi graph, $G(n,p)$, with constant edge density. One consequence of their result is a characterization of a typical random graph conditioned on having few triangles. In this talk, I will discuss recent work extending the characterization to sparse random graphs, i.e. $G(n,p)$ with vanishing edge density. The techniques connect to mean-field approximations and the hypergraph container method.