Probability Seminar

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Past Seminars

Spring 2024

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

January 25, 2024: Tatyana Shcherbina (UW-Madison)

Characteristic polynomials of sparse non-Hermitian random matrices

We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of sparse non-Hermitian random matrices $X_n$ whose entries have the form $x_{jk}=d_{jk}w_{jk}$ with iid complex standard Gaussian $w_{jk}$ and normalized iid Bernoulli$(p)$ $d_{jk}$.  If $p\to\infty$, the local asymptotic behavior of the second correlation function of characteristic polynomials near $z_0\in \mathbb{C}$ coincides with those for  Ginibre ensemble of non-Hermitian matrices with iid Gaussian entries: it converges to a determinant of the Ginibre kernel in the bulk $|z_0|<1$, and it is factorized if $|z_0|>1$. It appears, however, that for the finite $p>0$, the behavior is different and it exhibits the transition between three different regimes depending on values $p$ and $|z_0|^2$.  This is the joint work with Ie. Afanasiev.  

February 1, 2024: Patrick Lopatto (Brown)

Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.

February 8, 2024: Benoit Dagallier (NYU), online talk

TBA

February 15, 2024: Brian Rider (Temple)

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February 22, 2024: TBA

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February 29, 2024: Zongrui Yang (Columbia)

TBA

March 7, 2024: Atilla Yilmaz (Temple)

TBA

March 14, 2024: Eric Foxall (UBC Okanagan)

TBA

March 21, 2024: Semon Rezchikov (Princeton)

TBA

March 28, 2024: Spring Break

TBA

April 4, 2024: Christopher Janjigian (Purdue)

TBA

April 11, 2024: Bjoern Bringman (Princeton)

TBA

April 18, 2024: TBA

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April 25, 2024: Colin McSwiggen (NYU)

TBA

May 2, 2024: TBA

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