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.

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.

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.

Abstract: We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis (ETH), which is a strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. Then, using this a priori bound as an input, we analyze the Stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in QUE. The main methods behind the above results are: (i) multi-resolvents local laws established via a novel bootstrap scheme; (ii) energy estimates for SEE.

Abstract: We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.

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