Schedule

• September 13

  Jimmy He (MIT)

  Random walks on finite fields with deterministic jumps

  Abstract: Recently, Chatterjee and Diaconis showed that most bijections, if applied between steps of a Markov chain, cause the resulting chain to mix much faster. However, explicit examples of this speedup phenomenon are rare. I will discuss recent work studying such walks on finite fields where the bijection is algebraically defined. This work gives a large collection of examples where this speedup phenomenon occurs. These walks can be seen as a non-linear analogue of the Chung-Diaconis-Graham process, where the bijection is multiplication by a non-zero element of the finite field. This work is partially joint with Huy Pham and Max Xu.

• September 20

  *** Special Seminar on Zoom! ***

  Pierre Yves Gaudreau Lamarre (University of Chicago)

  Number rigidity in the spectrum of random Schrödinger operators

  Abstract:In this talk, I will discuss recent progress in the understanding of the structure in the spectrum of random Schrödinger operators. More specifically, I will introduce the concept of number rigidity in point processes and discuss recent efforts to understand its occurrence in the spectrum of random Schrödinger operators. Based on joint works with Promit Ghosal (MIT), Wenxuan Li (UChicago), and Yuchen Liao (Warwick).

• September 27

  Lisa Sauermann (MIT)

  On the extension complexity of random polytopes.

  Abstract: Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random polytopes. For a fixed dimension d, we consider random d-dimensional polytopes obtained as the convex hull of independent random points either in the unit ball ball or on the unit sphere. In both cases, we prove that the extension complexity is typically on the order of the square root of number of vertices of the polytope. Joint work with Matthew Kwan and Yufei Zhao.

• October 4

  *** Special Seminar on Zoom! ***

  Nick Cook (Duke University)

  Large deviations and regularity method for sparse random hypergraphs.

  Abstract:The "infamous upper tail" problem for subgraph counts in Erdős–Rényi graphs has received considerable attention since it was popularized by Janson and Rucinski, and has connections with questions in graph limit theory and statistical physics. I will survey work in this area and discuss a new approach for the more general setting of hypergraphs, based on an extension of the regularity method to sparse hypergraphs. In particular, we develop a sparse counting lemma and decomposition theorem for tensors under a novel class of norms that generalize the matrix cut norm. Based on joint work with Amir Dembo and Huy Tuan Pham.

• October 11

  Indigenous peoples day, no seminar

• October 18

  Matan Harel (Northeastern University)

  Quantitative estimates on the effect of random disorder on low-dimensional lattice models.

  Abstract: In their seminal work, Imry and Ma predicted that the addition of an arbitrarily small random external field to a low-dimensional statistical physics model causes the usual first-order phase transition to be `rounded-off.' This phenomenon was proven rigorously by Aizenman and Wehr in 1989 for a vastly general class of spin systems and random perturbations. Recently, the effect was quantified for the random-field Ising model, proving that it exhibits exponential decay of correlations at all temperatures. Unfortunately, the analysis relies on the monotonicity (FKG) properties which are not present in many other classical models of interest. This talk will present quantitative versions of the Aizenman-Wehr theorems for general spin systems with random disorder, including Potts, spin O(n), spin glasses, and random surface models. This is joint work with Paul Dario and Ron Peled.

• October 25

  Erik Bates (Wisconsin)

  Empirical measures, geodesic lengths, and a variational formula in first-passage percolation.

  Abstract:We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to [n\xi], where \xi is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

• November 1

  *** Special Seminar on Zoom! ***

  Marcelo Hilario

  Random walks on dynamic random environments with non-uniform mixing.

  Abstract: In this talk, we will discuss recent results on the limiting behavior of random walks in dynamic random environments. We will mainly discuss the case when the random walk evolves on one-dimensional random environments given by conservative interacting particle systems such as the simple symmetric exclusion process. Its transitions probabilities will depend on the current occupation environment nearby. Conservation of particles leads to poor mixing conditions and we explain how renormalization techniques can be useful to obtain the law of large numbers, large deviation estimates, and sometimes central limit theorems. The talk is based on several joint works with Oriane Blondel, Frank den Hollander, Daniel Kious, Renato dos Santos, Vladas Sidoravicius and Augusto Teixeira.

• November 8

  Seminar in room: 2-147

  Alex Dunlap (Courant)

  Fluctuations of solutions to the KPZ equation on a large torus.

  Abstract: I will discuss proofs of optimal (up to constants) variance bounds on the solutions to the KPZ equation on a torus, as the time scale and the size of the torus are taken to infinity together, in the super-relaxation regime and part of the relaxation regime. The arguments are based on stochastic analysis and do not use a connection to a discrete system. Joint work with Yu Gu and Tomasz Komorowski.

• November 15

  Seminar on Zoom!

  Han Huang (Georgia Tech)

  When can we recover an Erdos-Renyi graph from its 1-neighbors?

  Abstract: Suppose we have a graph G. For each vertex v of G, we observed the local structure of the G at this vertex v. Precisely, we have the induced subgraph containing all vertices at a distance at most 1 to v (including v), but the labels of the neighbors of v have been removed. Now, can we reconstruct graph G based on these 1-neighbor structures at each vertex? This question was proposed by Mossel and Ross, which was motivated by DNA shotgun assembly.
  
  To reconstruct the graph, the local structures need to have sufficient complexity. Previously, Gaudio and Mossel show that for the Erdos Renyi graph G(n,p), one cannot reconstruct the graph from its local structures when p = o(n^{-1/2}). For such values of p, unique reconstruction is not possible because the number of typical realizations of Erdos Renyi Graphs is much more than the number of typical realizations of the overall local structures. Recently, with Tikhomirov, we managed to confirm that the transition for the unique reconstruction of G(n,p) graphs happens at the level when p is at n^{-1/2} up to a polylog n factor.

• November 22

  Seminar on Zoom!

  Ellen Powell (Durham)

  Brownian excursions, conformal loop ensembles and critical Liouville quantum gravity

  Abstract: In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be described by the mating of two continuum random trees. In this talk I will discuss the counterpart of their result for critical LQG and SLE. More precisely, I will explain how, as we approach criticality from the subcritical regime, the space-filling SLE degenerates to the uniform CLE_4 exploration introduced by Werner and Wu, together with a collection of independent coin tosses indexed by the branch points of the exploration. Furthermore, although the pair of continuum random trees collapse to a single continuum random tree in the limit we can apply an appropriate affine transform to the encoding Brownian motions before taking the limit, and get convergence to a Brownian half-plane excursion. I will try to explain how observables of interest in the critical CLE decorated LQG picture are encoded by a growth fragmentation naturally embedded in the Brownian excursion. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun.

• November 29

  Seminar in room: 2-147

  Lingfu Zhang (Princeton)

  The environment seen from a geodesic in last-passage percolation

  Abstract: In exponential directed last-passage percolation, each vertex in Z^2 is assigned an i.i.d. exponential weight, and the geodesic between a pair of vertices refers to the up-right path connecting them, with the maximum total weight along the path. It is a natural question to ask what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present some new results on this. We show convergence of the distribution of the ‘environment’ as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure, as the geodesic length goes to infinity. In addition, we obtain an explicit description of the limiting environment, which depends on the direction of the geodesic. This in principle enables one to compute all the local statistics of the geodesic, and I will talk about some surprising and interesting examples. This is based on joint work with James Martin and Allan Sly.

• December 6

  Seminar in room: 2-147

  Yury Polyansky (MIT)

  Uniqueness of BP fixed point for Ising models.

  Abstract: In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed point (also known as Bethe fixed point, cavity equation etc). In this work we prove there is at most one non-trivial fixed point for Ising models with zero or random (but ``unbiased'') external fields.
  
  As a concrete example, consider a sample A of Ising model on a rooted tree (regular, Galton-Watson, etc). Let B be a noisy version of A obtained by independently perturbing each spin as follows: $B_v$ equals to $A_v$ with some small probability $\delta$ and otherwise taken to be a uniform +-1 (alternatively, 0). We show that the distribution of the root spin $A_\rho$ conditioned on values $B_v$ of all vertices $v$ at a large distance from the root is independent of $\delta$ and coincides with $\delta=0$. Previously this was only known for sufficiently ``low-temperature'' models. Our proof consists of constructing a metric under which the BP operator is a contraction (albeit non-multiplicative). I hope to convince you our proof is technically rather simple.
  
  This simultaneously closes the following 5 conjectures in the literature:
  

  1. uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm'2014);
  2. optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu'2015);
  3. independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly'2016);
  4. boundary irrelevance in BOT (Abbe-Cornacchia-Gu-P.'2021);
  5. characterization of entropy of community labels given the graph in 2-SBM (ibid).

  
  
  Joint work with Qian Yu (Princeton).

• December 13

  Seminar in room: 2-147

  Nishant Changotia (Tata Institute of Fundamental Research)

  Predictive sets

  Abstract: A stationary finite-valued process X_i; i in Z is called deterministic if we can predict X_0 by X_i for negative integers i. A set P contained in the natural numbers is called predictive if we can predict X_0 using X_i for i in the set P for all deterministic stationary finite-valued processes. In joint work with Benjamin Weiss, we will explore some necessary and some sufficient conditions for a set to be predictive. Time permitting we will discuss some links to questions in harmonic analysis and see an application of Fermat’s last theorem to a related question. Not much background will be assumed for this talk.