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http://math.colorado.edu/seminars/ics/prob.ics
as the source.
Thunderbird
The Probability Seminar usually meets Thursdays at 3pm in MATH 350 and is organized by Kyle Luh and Sean O'Rourke.
Fri, Feb. 27 2:30pm (Math 3…
Lior Alon (MIT)
X
Motivated by Berry’s random wave model for chaotic domains, high-energy eigenfunctions are expected, at the scale of the wavelength, to behave like random waves. Accordingly, the size of nodal sets is determined to first order by the Weyl law, while its second-order fluctuations are expected to be universal. In analogy with spectral statistics, we refer to these fluctuations of the nodal set size as nodal statistics.
Beginning with work of Berry, and later of Blum–Gnutzmann–Smilansky and Bogomolny–Schmit, it was conjectured that in chaotic systems nodal statistics are asymptotically Gaussian. While striking mathematical results support this picture on the sphere (Nazarov–Sodin), arithmetic symmetries on the torus prevent chaoticity and lead to a breakdown of universality, as shown by Wigman and collaborators and by Marinucci, Rossi, and Peccati.
In this talk I will follow Smilansky’s insight and focus on nodal statistics in graph-based models. For quantum graphs and discrete operators on graphs, Berkolaiko’s nodal magnetic theorem relates nodal statistics to the stability of eigenvalues under magnetic perturbations. This connection yields Gaussian limiting statistics for quantum graphs with disjoint cycles (Alon–Band–Berkolaiko), and, combined with Morse inequalities, extends to random discrete operators on finite graphs with disjoint cycles and to complete graphs with strong on-site disorder (Alon–Goresky).
I will conclude with recent joint work on nodal statistics for random matrices (Alon–Mikulincer–Urschel), showing that nodal statistics for GOE matrices obey a semicircle law rather than the conjectured Gaussian behaviour. If time permits, I will briefly discuss ongoing work on the Rosenzweig–Porter model, where adding a random on-site potential to a GOE matrix leads to nodal statistics interpolating from semicircle behaviour at low disorder to Gaussian behaviour at strong disorder, suggesting a phase transition between universality classes and a connection to localization–delocalization phenomena.
Nodal statistics from quantum graphs to random matrices: universality and phase transitions
Thu, Mar. 26 3:35pm (MATH 3…
Andrew Campbell (ISTA)
X
A result dating back at least 40 years is that the roots of Hermite polynomials are asymptotically distributed according to the semicircle distribution, which is also serves the role of the normal distribution in Voiculescu's free probability theory. Hermite polynomials are also crucial to understanding the eigenvalues of the Gaussian Unitary Ensemble, a particularly important distribution of random matrices for which an infinite version is simple to define. We will discuss a class of polynomials which provide analogous approximations for other stable distributions in free probability and other infinite ensembles of random matrices. We will look at the asymptotic properties of the roots of these polynomials and how they arise naturally in the study of zeros of analytic functions. As a special case we will look at the corresponding polynomial for the Riemann zeta function. This talk is based mostly on joint work with Jonas Jalowy.
Polynomial approximations of free stable laws, infinite random matrices, and the Riemann zeta function.
Nodal statistics from quantum graphs to random matrices: universality and phase transitions