# UC San Diego Probability Seminar (2021-2022)

The seminar meets on Thursdays from 11 AM to 12 PM PT in AP&M 6402 with live streaming via Zoom. Please send me an email if you would like to be added to the mailing list or if you have a suggestion for a speaker. Invite links to Zoom streaming will be sent to the mailing list.

## Spring

April 28: Yizhe Zhu (UC Irvine)

Abstract: The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs have become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. We characterize the spectrum of the non-backtracking operator for the sparse random hypergraph, and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. Based on joint work with Ludovic Stephan.

May 19: Ching Wei Ho (Notre Dame)

Abstract: The limiting eigenvalue distributions of two well-known random matrix ensemble, the GUE and the Ginibre ensemble, are respectively the semicircle law (the density is semicircular in the interval $$[-2,2]$$) and the circular law (the uniform measure on the disk of radius $$1$$). These two distributions have a simple relation: the push-forward of the circular law by $$z\to 2\operatorname{Re}(z)$$ is the semicircular law. Recent results in random matrix theory show that this simple relation can be generalized into other Gaussian random matrix models; however, these results work with the (continuous) limiting eigenvalue distributions. In this talk, I will present a recent conjecture with Brian Hall. At the finite-$$N$$ level, the push-forward property can be accomplished by applying the heat operator to the random characteristic polynomial of a random matrix model. For example, if we apply the heat operator at time $$1/N$$ to the characteristic polynomial of the GUE, the roots of the resulting polynomial resemble the circular law.

May 26: Giorgio Cipolloni (Princeton)

Abstract: TBA

June 2: David Renfrew (SUNY Binghamton)

Abstract: TBA

* Available dates: