Seminars & Colloquia Calendar
Spectral Rigidity of Random Schrödinger Operators via Feynman-Kac Formulas
Pierre Yves Gaudreau Lamarre - Princeton University
Location: Hill 705
Date & time: Thursday, 24 October 2019 at 2:00PM - 3:00PM
Abstract: A point process is number rigid if, for every bounded set K, the number of points in K is completely determined by the configuration of points outside of K. The notion of number rigidity was formally introduced by Ghosh and Peres in 2012. Since then, number rigidity was shown to be connected to a number of interesting features of point processes, and a problem of great interest is to uncover the mechanisms that give rise to this property. Due to their prominent role in mathematical physics, it is natural to ask whether the eigenvalue point processes of random Schrödinger operators (RSOs) are number rigid. In this talk, I will discuss the first technique for proving number rigidity of the spectrum of general RSOs. Our method makes use of Feynman-Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times.
I will then show how we can use our method to prove number rigidity for a class of one-dimensional continuous RSOs of the form ?(1/2)?+V+?, where V is a deterministic potential and ? is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential V, and the singularity of the noise ?.
This talk is based on [arXiv:1908.08422], which is Joint work with Promit Ghosal and Yuchen Liao."