Dear colleagues,

I would like to invite you to the following online seminar organized by the Probability group of the University of Pisa. The two talks will be accessible under the link

Click here to join the meetinghttps://teams.microsoft.com/l/meetup-join/19%3Aaf3d635091e049579e555a84219ab378%40thread.tacv2/1617719237635?context=%7B%22Tid%22%3A%22c7456b31-a220-47f5-be52-473828670aa1%22%2C%22Oid%22%3A%22dfd1e5f6-331d-43e0-a180-4bb6ce727fb7%22%7D

Best regards,

Giacomo

Tuesday, April 13, 16:00

Speaker: Laure Dumaz (École Normale supérieure)

Title: Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization.

Abstract: We consider the continuous Schrödinger operator - d^2/d^x^2 + B’(x) on the interval [0,L] where the potential B’ is a white noise. We study the entire spectrum of this operator in the large L limit. We prove the joint convergence of the eigenvalues and of the eigenvectors and describe the limiting shape of the eigenvectors for all energies. When the energy is much smaller than L, we find that we are in the localized phase and the eigenvalues are distributed as a Poisson point process. The transition towards delocalization holds for large eigenvalues of order L. In this regime, we show the convergence at the level of operators. The limiting operator in the delocalized phase is acting on R^2-valued functions and is of the form ``J \partial_t + 2*2 noise matrix'' (where J is the matrix ((0, -1)(1, 0))), a form appearing as a conjecture by Edelman Sutton (2006) for limiting random matrices. Joint works with Cyril Labbé.

Tuesday, April 13, 17:00

Speaker: Martin Vogel (Université de Strasbourg)

Title: Eigenvalue asymptotics and eigenvector localization for non-Hermitian noisy Toeplitz matrices

Abstract: A most notable characteristic of non-Hermitian matrices is that their spectra can be intrinsically sensitive to tiny perturbation. Although this spectral instability causes the numerical analysis of their spectra to be extremely unreliable, it has recently been shown to be also the source of new mathematical phenomena. I will present recent results about the eigenvalues asymptotics and eigenvector localization for deterministic non-Hermitian Toeplitz matrices with small additive random perturbations. These results are related to recent developments in the theory of partial differential equations. The talk is based on joint work with J. Sjöstrand, and with A. Basak and O. Zeitouni.

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Giacomo Di Gesù

Dipartimento di Matematica

Università di Pisa

Largo Bruno Pontecorvo 5

56127 - Pisa, Italy

giacomo.digesu@unipi.itmailto:giacomo.digesu@unipi.it

https://sites.google.com/site/giacomodigesu/