The LA Probability Forum, Held at UCLA

The LA Probability Forum

3-4pm: Informal meet & greet at the Kerckhoff Coffee House

4pm: Brice Huang(Stanford): Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature

Abstract: We consider the Sherrington--Kirkpatrick spin glass model at the critical inverse temperature $\beta = 1$ with zero external field. We prove that the free energy of this model has variance (1/6) \log N + O(1) confirming a physics prediction of Aspelmeier (2008), and that the centered and scaled free energy satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving E<R{1,2}^2> \asymp N^{-2/3}, as conjectured by Talagrand (2011), together with a uniform exponential moment bound for N^{1/3} |R{1,2}|.

The key input is a critical reweighted moment method, in the spirit of the "small subgraph conditioning" technique from probabilistic combinatorics, but capable of capturing diverging fluctuations. Through this reweighting, we relate the critical SK model to the BBP critical edge, which determines the overlap and fluctuation scales. Based on joint work with Hang Du.

5pm: Alex Wein(UCD): Computational Complexity of Testing in the Spiked Wigner Model

Abstract: This topic lies at the intersection of probability, statistics, and computational complexity. We consider one of the most basic high-dimensional hypothesis testing problems: that of detecting the presence of a rank-1 "spike" in a random Gaussian (GOE) matrix. When the spike has structure such as sparsity, inherent statistical-computational tradeoffs are expected. I will discuss some precise results about the computational complexity, arguing that the "linear spectral statistics" achieve the best possible tradeoff between false-positive and false-negative errors among all polynomial-time algorithms, even though an exponential-time algorithm can do better. This is based on https://arxiv.org/abs/2311.00289 with Ankur Moitra which uses a version of the low-degree polynomial heuristic, as well as forthcoming work with Ansh Nagda which gives a stronger form of reduction-based hardness.

6pm: Sergey Lototsky(USC): Self-Normalized Functionals of Stationary Ornstein-Uhlenbeck Processes: Tail Bounds and Applications in Statistics

Abstract: Bernstein-type inequalities are derived for certain self-normalized linear combinations of stochastic integrals in the second chaos space. The results have immediate applications in certain statistical problems, and potential connections with the Gaussian Free Field.

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024