Probability and Mathematical Physics Seminar

Synopsis:

Ramon van Handel (Princeton University), 11:10 am:

Random covers of hyperbolic surfaces

It was shown long ago by Huber that the first nonzero eigenvalue of the Laplacian on a closed hyperbolic surface cannot exceed that of the hyperbolic plane, asymptotically as the genus goes to infinity. Whether there exists a sequence of closed hyperbolic surfaces that achieves this bound---an old conjecture of Buser---was settled a few years ago by Hide and Magee. This was done by exhibiting a sequence of finite covering spaces of a fixed base surface that have good spectral properties. In this talk, I will discuss joint work with Magee and Puder where we show that this phenomenon is in fact much more prevalent: given any closed hyperbolic surface, not only do there exist finite covers that have good spectral properties, but this is in fact the case for all but a vanishing fraction of its finite covers. The proof hinges on new developments on the notion of strong convergence in random matrix theory.

 

Vadim Gorin (UC Berkeley), 12:10 pm:

The Airy-beta line ensemble

Beta-ensembles generalize the eigenvalue distributions of self-adjoint real, complex, and quaternion matrices for beta=1, 2, and 4, respectively. These ensembles naturally extend to two dimensions by introducing operations such as corner truncation, addition, or multiplication of matrices. In this talk, we will explore the edge asymptotics of the resulting two-dimensional ensembles. I will present the Airy-beta line ensemble, a universal object that governs the asymptotics of time-evolving largest eigenvalues. This ensemble consists of an infinite collection of continuous random curves, parameterized by beta. I will share recent progress in developing a framework to describe this remarkable structure.

 

Volodymyr Riabov (ISTA), 2:10 pm:

The Zigzag Strategy for Random Band Matrices. 

Random band matrices have entries concentrated in a narrow band of width W around the main diagonal, modeling systems with spatially localized interactions. We consider one-dimensional random band matrices with bandwidth W >> N^½, general variance profile, and arbitrary entry distributions. We establish complete isotropic delocalization, quantum unique ergodicity (eigenstate thermalization), and Wigner-Dyson universality in the bulk of the spectrum. The key technical input is a family of local laws capturing the spatial decay of resolvent entries, established using a combination of Ornstein-Uhlenbeck dynamics and Green function comparison (the Zigzag strategy). Based on joint work with László Erdős.

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