I am co-organizing a virtual Probability Seminar with Wei-Kuo Chen and Arnab Sen at University of Minnesota for
the Spring 2025 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.
| 01/31/2025 | Heejune Kim (U Minnesota) |
| Title: Disorder Chaos in Short-Range, Diluted, and Lévy Spin Glasses | |
| Abstract: In a breakthrough [arXiv:2301.04112], Chatterjee proved site disorder chaos in the Edwards-Anderson (EA) short-range spin glass model utilizing the Hermite spectral method. In this talk, I will discuss the usefulness of this Hermite spectral approach by extending the validity of site disorder chaos in three related spin glass models: the mixed even p-spin short-range model, the diluted mixed p-spin model, and the Lévy model. The main novelty of our argument is played by an elementary algebraic equation for the Fourier-Hermite series coefficients for the two-spin correlation functions. It allows us to deduce necessary geometric conditions to determine the contributing coefficients in the overlap function. Based on joint works with Wei-Kuo Chen and Arnab Sen. | |
| 02/07/2025 | Si Tang (Lehigh) |
| Title: The time constant of high dimensional first passage percolation, revisited. | |
| Abstract We prove high-dimensional asymptotics for the time constants in first-passage percolation (FPP) on Zd along all diagonal-like directions v=(1, 1, .., 1, 0, 0, …, 0) of f(d) nonzero entries. We show that the behavior of the time constant is essentially the same as the axis directions if f(d)~o(d) and is characterized by the Lambert W function if f(d)~α d. The proof was based on a cluster exploration idea, which allowed us to estimate moments of non-backtracking first-passage times as well as to fix an error in [AT'16] | |
| 02/14/2025 | Yeganeh Alimohammadi (UC Berkeley) |
| Title: Epidemic Forecasting on Networks: Bridging Local Samples with Global Outcomes. | |
| Abstract: Epidemics of all kinds, from infectious diseases to technologies and ideas, spread through the hidden network of our social interactions. The structure of this underlying network determines the patterns of the epidemic spread, but mapping this network is expensive, and modeling it accurately is difficult. In this talk, I will introduce a data-driven and model-free approach to predict the time evolution of epidemics that requires surprisingly few local network samples to forecast epidemic spread accurately. I will establish theoretical guarantees for the precision of our local estimator for a general class of networks, supporting these claims with concrete empirical evidence. The technical tools discussed in the talk can provide new perspectives on various applications of network data, beyond the scope of epidemics. | |
| 02/21/2025 | Adrien Schertzer (Bonn U) |
| Title: The Two Point Function of the SK Model without External Field at High Temperature. | |
| Abstract: In this talk, I will present a strategy based on ideas related to the TAP approach that enables an exact computation of the operator norm in the Sherrington-Kirkpatrick (SK) model as N → ∞, valid in the full high- temperature regime β < 1. We show that the two point correlation matrix M=(<σiσj>)1≤i,j≤N of the SK model with zero external field satisfies Here, G denotes the GOE interaction matrix of the model. This is a joint work with C. Brennecke, C. Xu, H.-T. Yau. | |
| 02/28/2025 | Shirshendu Ganguly (UC Berkeley) |
| Title: Last passage percolation in hierarchical environments. | |
| Abstract: Last passage percolation (LPP) is a model of random geometry where the main observable is a path evolving in a random environment. When the environment distribution has light tails and a fast decay of correlation, the random fluctuations of LPP are predicted to be explained by the Kardar–Parisi–Zhang (KPZ) universality theory. However, the KPZ theory is not expected to apply in many natural settings, such as "critical" environments exhibiting a hierarchical, self-similar structure which should give rise to a fluctuation theory featuring logarithmic corrections with novel critical exponents. Predictions for these exponents are missing, even from the physics literature. In recent joint work with Victor Ginsburg and Kyeongsik Nam we initiate the study of LPP in hierarchical environments, developing a framework based on multi-scale analysis and obtaining bounds on critical exponents for two canonical examples: an i.i.d. environment with critical power-law tails, and a hierarchical approximation of the two-dimensional Gaussian Free Field. In this talk we will describe some of these results and natural directions for future research. | |
| 03/07/2025 | Jacob Richey (Alfréd Rényi Institute of Mathematics) |
| Title: Patterns and statistics in shifts of finite type. | |
| Abstract: Consider words over a finite alphabet that avoid a set of forbidden patterns, e.g. binary sequences with no two adjacent 1s. This set can be viewed through the lens of ergodic theory, as a dynamical system (a 'shift space'); or combinatorial probability, as an iid sequence or a Markov chain conditioned to avoid the forbidden set; or statistical physics, as a thermodynamic limit of a natural Gibbs measure. I will discuss connections between ideas from these worlds in the case where the forbidden set is of size one or two, including new results related to conjugacy of the underlying shifts, their entropies, and the asymptotic density of 1s. | |
| 03/14/2025 | Spring Break at LU and UMN |
| 03/21/2025 | Souvik Ray (UNC) |
| Title: A Notion of Stability for Solutions of Random Optimization Problems. | |
| Abstract: In this talk, we consider a notion of stability for solutions of random optimization problems based on small perturbations of the input data and inspired by the technique for proving CLT using Stein's method. This notion of stability is closely related with the size of near-optimal solution sets for those optimization problems. We establish this notion of stability for a number of settings, such as branching random walk, the Sherrington--Kirkpatrick model of mean-field spin glasses, the Edwards--Anderson model of short-range spin glasses, the Wigner and Wishart ensemble of random matrices and combinatorial optimization problems like TSP/MST/MMP on weighted complete graphs and Euclidean spaces. | |
| 03/28/2025 | Arka Adhikari (U Maryland) |
| Title: Moderate Deviations for the Capacity of the Random Walk Range in Dimension Four. | |
| Abstract: We find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen concerning the volume of the random walk range for d = 2. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, | |
| 04/04/2025 | Dylan Altschuler (CMU) |
| Title: Universal geometric (non)embedding of graphs. | |
| Abstract: Given a collection of points in a normed space, the corresponding "geometric graph" is obtained by connecting any pair of points with distance less than one. Say that a graph G is "geometrically embeddable" into a normed space X if there exist points in X whose geometric graph is isomorphic to G. Understanding geometric embeddability is a natural problem at the intersection of combinatorial geometry, metric dimension reduction, and data science. While criteria for geometric embeddings are well-studied in Euclidean space, essentially nothing is known outside this setting. We address this gap. Our result is that asymptotically almost every constant-degree regular graph G on N vertices has the following "universal" non-embedding property. There is no normed space of dimension less than admitting a geometric embedding of G. This is sharp. The proof is based on an efficient multiscale "seeded" epsilon--net argument. Only basic probability knowledge will be assumed. (Based on joint work with Konstantin Tikhomirov; arXiv:2501.09142). | |
| 04/18/2025 | Chokri Manai (TU Munich & Zeiss) |
| Title: Recent Progress in Quantum Spin Glasses. | |
| Abstract: In this talk I will give an overview over recent results on mean-field spin-glass models with a transversal magnetic field. For such models, both thermodynamic quantities, such as the free energy and its fluctuations, as well as spectral and localization properties of eigenvectors are of interest to a diverse list of communities. A full mathematical analysis of properties is completed for the simplest, yet ubiquitous Quantum Random Energy Model (QREM). For the physically most interesting model, the Quantum Sherrington-Kirkpatrick model (QSK), with a more complicated (classical) correlation structure, we give an infinte-dimensional analog of the classical Parisi formula. I will also briefly discuss the order parameters for that model and sketch a more qualitative analysis, which proves the existence of a phase transition in the QSK. | |
| 04/25/2025 | Sumit Mukherjee (Columbia) |
| Title: Limits of large contingency tables. | |
| Abstract: We explore the limiting structure of a large contingency table (in cut metric), when the row and column marginals converge in a suitable sense. Our results go beyond the classical contingency table framework, and allows for general matrices with real entries chosen from an arbitrary base measure. This talk is based on joint work with Hanbaek Lyu from University of Wisconsin-Madison. | |