I am co-organizing a virtual Probability Seminar with Wei-Kuo Chen and Arnab Sen at University of Minnesota for
the Spring 2023 semester. Please email me to get the Zoom link for the seminar series.
The seminar is held on Fridays at 2:30pm (ET), unless otherwise noted below.
We are also running an in-person Prob/Stat seminar series for the Spring 2023 semester.
Please find the schedule here.
| 02/03/2023 | Justin Ko (ENS de Lyon) |
| Title: TAP Variational Principle for the Constrained Overlap Multiple Spherical Sherrington-Kirkpatrick Model. | |
| Abstract: In this talk, we discuss the large deviations of overlaps from spherical spin glasses. It is known that the large deviations are given by a Parisi type variational problem. For the spherical Sherrington-Kirkpatrick model, we will show that it can also be expressed in terms of a TAP variational principle. In this setting, we are able to apply results from random matrix theory such as the asymptotics of the n-dimensional spherical integrals studied by Husson and Guionnet to derive an explicit simple form of the variational principle. The derived variational formula confirms that this model is replica symmetric for all positive temperatures, a fact which is natural but not obvious from the Parisi formula for the model. This is joint work with David Belius and Leon Frober. | |
| 02/10/2023 | Jinyoung Park (NYU) |
| Title: Thresholds. | |
| Abstract: For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the "expectation-threshold") for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). In this talk, I will present recent progress on this topic. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham. | |
| 02/24/2023 | Paul Simanjuntak (U Missouri) |
| Title: A probabilistic approach to isoperimetric inequalities for dual Lp centroid bodies | |
| Abstract: An isoperimetric inequality for the Lp centroid body for p≥1 was first proved by Lutwak, Yang, and Zhang, which extends the inequality for the dual Lp centroid body by Lutwak and Zhang. We show that the volume of the dual Lp centroid body is also maximized by the Euclidean ball for certain values of p<1, which extends Lutwak and Zhang’s result on convex bodies to star-shaped sets. This result is achieved through a probabilistic approach which associates certain random star bodies to the dual Lp centroid body. In this talk, we will discuss the tools used in the randomized framework and the role of certain special distribution in giving a representation of the random body. | |
| 03/03/2023 | Jonathan Niles-Weed (NYU) |
| Title: Strong recovery of geometric planted matchings. | |
| Abstract: We consider the problem of recovering a hidden matching between two correlated sets of n Gaussian samples in Rd. We analyze the performance of the maximum likelihood estimator, establish thresholds at which the MLE almost perfectly recovers the planted matching, and, in this regime, characterize the number of errors up to sub-polynomial factors. These results extend to the geometric setting a recent line of work on recovering matchings planted in random graphs with independently-weighted edges. Joint work with D. Kunisky (Yale). | |
| 03/31/2023 | Julian Sahasrabudhe (Cambridge) |
| Title: An exponential improvement for diagonal Ramsey. | |
| Abstract: Let R(k) be the kth diagonal Ramsey number: that is, the smallest n for which every 2-colouring of the edges of Kn contains a monochromatic Kk. In recent work with Marcelo Campos, Simon Griffiths and Rob Morris, the speaker showed that R(k) < (4-c)k, for some absolute constant c>0. This is the first exponential improvement over the bound of Erdős and Szekeres, proved in 1935. In this talk I will discuss the proof. | |
| 04/07/2023 | Duncan Dauvergne (U Toronto) |
| Title: Geodesics networks in the directed landscape. | |
| Abstract: The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry. | |
| 04/14/2023 | Saraí Hernández-Torres (UNAM) |
| Title: The chemical distance in random interlacements in the low-intensity regime. | |
| Abstract: Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on Zd, with a parameter u > 0 controlling the intensity of the Poisson point process. In a natural way, the model defines a percolation on the edges of Zd with long-range correlations. We consider the time constant associated to the chemical distance in random interlacements at low intensity u > 0. It is conjectured that the time constant times u½ converges to the Euclidean norm, as u ↓ 0. In dimensions d ≥ 5, we prove a sharp upper bound and an almost sharp lower bound for the time constant as the intensity decays to zero. Joint work with Eviatar Procaccia and Ron Rosenthal. | |
| 04/21/2023 | Istvan Gyongy (U Edinburgh) Special time: 1:15pm - 2:15pm. |
| Title: On nonlinear filtering of jump diffusions. | |
| Double talks, 1 of 2 |
Abstract: We consider a general filtering model for partially observed jump diffusions, satisfying an SDE driven by Wiener processes and Poisson martingale measures. The filtering equations are derived if the coefficients satisfy natural growth conditions. Recent results on the existence of the filtering density in Lp, and on its regularity in terms of Sobolev spaces are presented, when the coefficients of the SDE are Lipschitz functions, and when the coefficients satisfy appropriate smoothness conditions, respectively. The talk is based on a joint work with Fabian Germ. |
| 04/21/2023 | Nicolas Fraiman (UNC) |
| Title: Weight distribution of Minimal Spanning Acycles. | |
| Double talks, 2 of 2 |
Abstract: A classic result by Frieze is that the total weight of the minimum spanning tree (MST) of a uniformly weighted graph converges to zeta(3). Recently, this result was extended to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analogue: the Minimum Spanning Acycle (MSA). In this talk, we look at the distribution of weights in this random MSA both in the bulk and in the extremes. We show that the rescaled empirical distribution of weights in the MSA converges to a measure based on the density of the shadow. We also show that the shifted extremal weights converge to an inhomogeneous Poisson point process. This is joint work with Sayan Mukherjee and Gugan Thoppe. |
| 04/28/2023 | Giorgio Cipolloni (Princeton) |
| Title: How do the eigenvalues of a large non-Hermitian random matrix behave? | |
| Abstract: We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. Then, motivated by the long time behaviour of the ODE ů=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X. | |
| 05/05/2023 | Reza Gheissari (Northwestern) |
| Title: Cutoff in the Glauber dynamics for the Gaussian free field. | |
| Abstract: The Gaussian free field (GFF) is a canonical model of random surfaces, generalizing the Brownian bridge to two dimensions. It arises naturally as the stationary solution to the stochastic heat equation with additive noise (SHE), and together the SHE and GFF are expected to be the universal scaling limit of many random surface evolutions arising in lattice statistical physics. We consider the mixing time (time to converge to stationarity, when started out of equilibrium) for the pre-limiting object, the discrete Gaussian free field (DGFF) evolving under the Glauber dynamics. We establish that on a box of side-length n in Z2, the Glauber dynamics for the DGFF exhibits the cutoff phenomenon, mixing exactly at time (2n2log n)⁄π2. Based on joint work with S. Ganguly. | |