Probability Seminar: Difference between revisions
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== January 25, 2024: Tatyana Shcherbina (UW-Madison) == | == 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: [https://lopat.to/index.html Patrick Lopatto (Brown)] == | == February 1, 2024: [https://lopat.to/index.html Patrick Lopatto (Brown)] == | ||
Revision as of 19:29, 19 January 2024
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: TBA
TBA
February 15, 2024: Brian Rider (Temple)
TBA
February 22, 2024: TBA
TBA
February 29, 2024: TBA
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: TBA
TBA
April 11, 2024: Bjoern Bringman (Princeton)
TBA
April 18, 2024: TBA
TBA
April 25, 2024: Colin McSwiggen (NYU)
TBA
May 2, 2024: TBA
TBA