Probability and Mathematical Physics Seminar

Synopsis:

Hambly and Jordan (2004) introduced the series-parallel graph, a random hierarchical lattice that is easy to define: Start with the graph consisting of one edge connecting two terminal nodes.  At each subsequent step of the construction, perform independent coin flips for each edge of the graph, and replace the edge by two edges in series if the coin is heads-up or two edges in parallel if tails.  This results in a sequence of random graphs, which can be interpreted as a resistor network.  Hambly and Jordan showed that the logarithm of the resistance grows linearly if the coins are biased to land more often heads-up.  In this talk, I will discuss what happens in the critical case when fair coins are used.  Starting with a new recursive distributional equation (RDE) observed by Gurel-Gurevich, I develop a framework for analyzing RDE's based on parabolic PDE theory and use this to characterize the asymptotic behavior of the log. resistance.  In the sub- or supercritical case (where the coins are biased), I discuss a tantalizing connection to the Fisher-KPP equation and front propagation.

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Synopsis:

I'll talk about various limit theorems for iterated sums and integrals of the form S^{(ν)}_N(t) = N^{-ν/2} ∑_{0≤k1<...<kν≤Nt} ξ(k1) ⊗ ··· ⊗ ξ(kν), t ∈ [0, T] and S^{(ν)}_N(t) = N^{-ν/2} ∫_{0≤s1≤...≤sν≤Nt} ξ(s1)⊗ ··· ⊗ ξ(sν) ds1 ··· dsν, where {ξ(k)}_{−∞<k<∞} and {ξ(s)}_{−∞<s<∞} are centered stationary vector processes with some weak dependence properties. Collections of such iterated sums and integrals were called signatures in papers on the rough paths theory and their applications to data science, machine learning and neural networks were discussed recently. I'll speak on the law of large numbers, strong limit theorems, law of iterated logarithm and large deviations for such objects. An application to averaging will be mentioned, as well. All proofs are direct and they do not rely on the rough paths theory.

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Synopsis:

The Sherrington–Kirkpatrick (SK) model is a well-known model for spin glasses that originated in physics. In a series of ground-breaking articles by Giorgio Parisi proposed in 1979, it was predicted that the limiting free energy of the SK model is given by a variational principle, known as the Parisi formula, and its minimizer, known as the Parisi measure, will be full replica symmetry breaking (FRSB) at low temperature. The former one was confirmed in the past decade. However the existence of FRSB has not been established before. In this talk, I will show that the Parisi measure of the SK model is FRSB slightly beyond the high temperature. I will also talk about the structure of the corresponding Parisi measure.

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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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