BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Laure Dumaz (École Normale supérieure) DTSTART:20210413T140000Z DTEND:20210413T150000Z DTSTAMP:20260422T225726Z UID:POSemP/1 DESCRIPTION:Title: L ocalization of the continuous Anderson hamiltonian in 1-d and its transiti on towards delocalization\nby Laure Dumaz (École Normale supérieure) as part of Pisa Online Seminar in Probability\n\n\nAbstract\nWe 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 spe ctrum of this operator in the large L limit. We prove the joint convergenc e of the eigenvalues and of the eigenvectors and describe the limiting sha pe of the eigenvectors for all energies. When the energy is much smaller t han L\, we find that we are in the localized phase and the eigenvalues are distributed as a Poisson point process. The transition towards delocaliza tion holds for large eigenvalues of order L. In this regime\, we show the convergence at the level of operators. The limiting operator in the deloca lized phase is acting on R^2-valued functions and is of the form ``J \\par tial_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 rando m matrices. Joint works with Cyril Labbé.\n LOCATION:https://researchseminars.org/talk/POSemP/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Martin Vogel (Université de Strasbourg) DTSTART:20210413T150000Z DTEND:20210413T160000Z DTSTAMP:20260422T225726Z UID:POSemP/2 DESCRIPTION:Title: E igenvalue asymptotics and eigenvector localization for non-Hermitian noisy Toeplitz matrices\nby Martin Vogel (Université de Strasbourg) as par t of Pisa Online Seminar in Probability\n\n\nAbstract\nA most notable char acteristic of non-Hermitian matrices is that their spectra can be intrinsi cally sensitive to tiny perturbation. Although this spectral instability c auses the numerical analysis of their spectra to be extremely unreliable\, it has recently been shown to be also the source of new mathematical phen omena. I will present recent results about the eigenvalues asymptotics and eigenvector localization for deterministic non-Hermitian Toeplitz matrice s with small additive random perturbations. These results are related to r ecent developments in the theory of partial differential equations. The ta lk is based on joint work with J. Sjöstrand\, and with A. Basak and O. Ze itouni.\n LOCATION:https://researchseminars.org/talk/POSemP/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Constanza Rojas-Molina (CY Cergy Paris Université) DTSTART:20210427T140000Z DTEND:20210427T150000Z DTSTAMP:20260422T225726Z UID:POSemP/3 DESCRIPTION:Title: ( Fractional) random Schrödinger operators\, integrated density of states a nd localization\nby Constanza Rojas-Molina (CY Cergy Paris Université ) as part of Pisa Online Seminar in Probability\n\n\nAbstract\nIn this tal k we will review some recent results on random Schrödinger operators\, wh ich are used to model electronic transport in disordered quantum systems a nd to study the phenomenon of Anderson localization. After a short introdu ction to the subject\, we will focus on a particular type of random operat or driven by a fractional laplacian. The interest on the latter lies in th eir association to stable Levy processes\, random walks with long jumps an d anomalous diffusion. We will discuss in this talk the interplay between the non-locality of the fractional laplacian and the localization properti es of the random potential in the fractional Anderson model\, in both the continuous and discrete settings. In the discrete setting we study the int egrated density of states and show a fractional version of Lifshitz tails. This coincides with results obtained in the continuous setting by the pro bability community. This is based on joint work with M. Gebert (LMU Munich ).\n LOCATION:https://researchseminars.org/talk/POSemP/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Högele (Universidad de los Andes) DTSTART:20210511T140000Z DTEND:20210511T150000Z DTSTAMP:20260422T225726Z UID:POSemP/4 DESCRIPTION:Title: C utoff thermalization for Ornstein-Uhlenbeck system swith small Lévy noise in the Wasserstein distance\nby Michael Högele (Universidad de los A ndes) as part of Pisa Online Seminar in Probability\n\n\nAbstract\nThis ta lk presents recent results on cutoff thermalization (also known as the cut off phenomenon) for a general class of asymptotically exponentially stable Ornstein-Uhlenbeck systems under ε-small additive Lévy noise. The drivi ng noise processes include Brownian motion\, α-stable Lévy flights\, fin ite intensity compound Poisson processes and red noises and may be highly degenerate. Window cutoff thermalization is shown under generic mild assum ptions\, that is\, we see an asymptotically sharp ∞/0-collapse of the re normalized Wasserstein distance from the current state to the equilibrium measure μ^ε along a time window centered in a precise ε-dependent time scale t_ε . In many interesting situations such as reversible (Lévy) dif fusions it is possible to prove the existence of an explicit\, universal\, deterministic cutoff thermalization profile. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvect ors of the matrix Q. With this piece of theory at hand this article provid es a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to ε-small Brownian motion or α-stable Lévy flights. Furthermore\, we cover the highly degenerate case of a lin ear chain of oscillators in a generalized heat bath at low temperature.\n LOCATION:https://researchseminars.org/talk/POSemP/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandra Caraceni (University of Oxford) DTSTART:20210511T150000Z DTEND:20210511T160000Z DTSTAMP:20260422T225726Z UID:POSemP/5 DESCRIPTION:Title: P olynomial mixing time for edge flips on planar maps\nby Alessandra Car aceni (University of Oxford) as part of Pisa Online Seminar in Probability \n\n\nAbstract\nA long-standing problem proposed by David Aldous consists in giving a sharp upper bound for the mixing time of the so-called “tria ngulation walk”\, a Markov chain defined on the set of all possible tria ngulations of the regular n-gon. A single step of the chain consists in pe rforming a random edge flip\, i.e. in choosing an (internal) edge of the t riangulation uniformly at random and\, with probability 1/2\, replacing it with the other diagonal of the quadrilateral formed by the two triangles adjacent to the edge in question (with probability 1/2\, the triangulation is left unchanged).\n\nWhile it has been shown that the relaxation time f or the triangulation walk is polynomial in n and bounded below by a multip le of $n^{3/2}$\, the conjectured sharpness of the lower bound remains fir mly out of reach in spite of the apparent simplicity of the chain. For edg e flip chains on different models – such as planar maps\, quadrangulatio ns of the sphere\, lattice triangulations and other geometric graphs – e ven less is known.\n\nWe shall discuss results concerning the mixing time of random edge flips on rooted quadrangulations of the sphere obtained in joint work with Alexandre Stauffer. A “growth scheme” for quadrangulat ions\, which generates a uniform quadrangulation of the sphere by adding f aces one at a time at appropriate random locations\, can be combined with careful combinatorial constructions to build probabilistic canonical paths in a relatively novel way. This method has implications for a range of in teresting edge-manipulating Markov chains on so-called Catalan structures\ , from “leaf translations” on plane trees to “edge rotations” on g eneral planar maps.\n LOCATION:https://researchseminars.org/talk/POSemP/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Willem Van Zuijlen (WIAS (Berlin)) DTSTART:20210608T140000Z DTEND:20210608T150000Z DTSTAMP:20260422T225726Z UID:POSemP/6 DESCRIPTION:Title: T otal mass asymptotics of the parabolic Anderson model\nby Willem Van Z uijlen (WIAS (Berlin)) as part of Pisa Online Seminar in Probability\n\n\n Abstract\nWe consider the parabolic Anderson model with a white noise pote ntial in two dimensions. This model is also called the stochastic heat equ ation with a multiplicative noise. We study the large time asymptotics of the total mass of the solution. Due to the irregularity of the white noise \, in two dimensions the equation is a priori not well-posed. Using paraco ntrolled calculus or regularity structures one can make sense of the equat ion by a renormalisation\, which can be thought of as "subtracting infinit y of the potential''. To obtain the asymptotics of the total mass we use t he spectral decomposition\, an alternative Feynman-Kac type representation and heat-kernel estimates which come from joint works with Khalil Chouk\, Wolfgang König and Nicolas Perkowski.\n LOCATION:https://researchseminars.org/talk/POSemP/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Josué Corujo (Université Paris Dauphine (CEREMADE)) DTSTART:20210608T130000Z DTEND:20210608T140000Z DTSTAMP:20260422T225726Z UID:POSemP/7 DESCRIPTION:Title: S pectrum and ergodicity of a neutral multi-allelic Moran model\nby Josu é Corujo (Université Paris Dauphine (CEREMADE)) as part of Pisa Online S eminar in Probability\n\n\nAbstract\nWe will present some recent results o n the study of a neutral\nmulti-allelic Moran model\, which is a finite co ntinuous-time Markov\nprocess. For this process\, it is assumed that the i ndividuals interact\naccording to two processes: a mutation process where they mutate\nindependently of each other according to an irreducible rate matrix\, and\na Moran type reproduction process\, where two individuals ar e uniformly\nchosen\, one dies and the other is duplicated. During this ta lk we will\ndiscuss some recent results for the spectrum of the generator of the\nneutral multi-allelic Moran process\, providing explicit expressio ns for\nits eigenvalues in terms of the eigenvalues of the rate matrix tha t\ndrives the mutation process. Our approach does not require that the\nmu tation process be reversible\, or even diagonalizable. Additionally\, we\n will discuss some applications of these results to the study of the\nspeed of convergence to stationarity of the Moran process for a process\nwith g eneral mutation scheme. We specially focus on the case where the\nmutation scheme satisfies the so called "parent independent" condition\,\nwhere (a nd only where) the neutral Moran model becomes reversible. In\nthis later case we can go further and prove the existence of a cutoff\nphenomenon for the convergence to stationarity.\n\nThis presentation is based on a recen tly submitted work\, for which a\npreprint is available at https://arxiv.o rg/abs/2010.08809.\n LOCATION:https://researchseminars.org/talk/POSemP/7/ END:VEVENT END:VCALENDAR