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:20260404T110832Z UID:POSemP/1 DESCRIPTION:Title: Localization of the continuous Anderson hamiltonian in 1-d and its t ransition towards delocalization\nby Laure Dumaz (École Normale supé rieure) as part of Pisa Online Seminar in Probability\n\n\nAbstract\nWe co nsider the continuous Schrödinger operator - d^2/d^x^2 + B’(x) on the i nterval [0\,L] where the potential B’ is a white noise. We study the ent ire spectrum of this operator in the large L limit. We prove the joint con vergence of the eigenvalues and of the eigenvectors and describe the limit ing shape of the eigenvectors for all energies. When the energy is much sm aller than L\, we find that we are in the localized phase and the eigenval ues are distributed as a Poisson point process. The transition towards del ocalization holds for large eigenvalues of order L. In this regime\, we sh ow 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 limitin g random matrices. Joint works with Cyril Labbé.\n LOCATION:https://stable.researchseminars.org/talk/POSemP/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Martin Vogel (Université de Strasbourg) DTSTART:20210413T150000Z DTEND:20210413T160000Z DTSTAMP:20260404T110832Z UID:POSemP/2 DESCRIPTION:Title: Eigenvalue asymptotics and eigenvector localization for non-Hermitia n noisy Toeplitz matrices\nby Martin Vogel (Université de Strasbourg) as part of Pisa Online Seminar in Probability\n\n\nAbstract\nA most notab le characteristic of non-Hermitian matrices is that their spectra can be i ntrinsically sensitive to tiny perturbation. Although this spectral instab ility causes the numerical analysis of their spectra to be extremely unrel iable\, it has recently been shown to be also the source of new mathematic al phenomena. I will present recent results about the eigenvalues asymptot ics and eigenvector localization for deterministic non-Hermitian Toeplitz matrices with small additive random perturbations. These results are relat ed to recent developments in the theory of partial differential equations. The talk is based on joint work with J. Sjöstrand\, and with A. Basak an d O. Zeitouni.\n LOCATION:https://stable.researchseminars.org/talk/POSemP/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Constanza Rojas-Molina (CY Cergy Paris Université) DTSTART:20210427T140000Z DTEND:20210427T150000Z DTSTAMP:20260404T110832Z UID:POSemP/3 DESCRIPTION:Title: (Fractional) random Schrödinger operators\, integrated density of s tates and localization\nby Constanza Rojas-Molina (CY Cergy Paris Univ ersité) as part of Pisa Online Seminar in Probability\n\n\nAbstract\nIn t his talk we will review some recent results on random Schrödinger operato rs\, which are used to model electronic transport in disordered quantum sy stems and to study the phenomenon of Anderson localization. After a short introduction to the subject\, we will focus on a particular type of random operator driven by a fractional laplacian. The interest on the latter lie s in their association to stable Levy processes\, random walks with long j umps and anomalous diffusion. We will discuss in this talk the interplay b etween the non-locality of the fractional laplacian and the localization p roperties of the random potential in the fractional Anderson model\, in bo th the continuous and discrete settings. In the discrete setting we study the integrated density of states and show a fractional version of Lifshitz tails. This coincides with results obtained in the continuous setting by the probability community. This is based on joint work with M. Gebert (LMU Munich).\n LOCATION:https://stable.researchseminars.org/talk/POSemP/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Högele (Universidad de los Andes) DTSTART:20210511T140000Z DTEND:20210511T150000Z DTSTAMP:20260404T110832Z UID:POSemP/4 DESCRIPTION:Title: Cutoff thermalization for Ornstein-Uhlenbeck system swith small Lév y noise in the Wasserstein distance\nby Michael Högele (Universidad d e los Andes) as part of Pisa Online Seminar in Probability\n\n\nAbstract\n This talk presents recent results on cutoff thermalization (also known as the cutoff phenomenon) for a general class of asymptotically exponentially stable Ornstein-Uhlenbeck systems under ε-small additive Lévy noise. Th e driving noise processes include Brownian motion\, α-stable Lévy flight s\, finite intensity compound Poisson processes and red noises and may be highly degenerate. Window cutoff thermalization is shown under generic mil d assumptions\, that is\, we see an asymptotically sharp ∞/0-collapse of the renormalized Wasserstein distance from the current state to the equil ibrium measure μ^ε along a time window centered in a precise ε-dependen t time scale t_ε . In many interesting situations such as reversible (Lé vy) diffusions it is possible to prove the existence of an explicit\, univ ersal\, deterministic cutoff thermalization profile. The existence of this limit is characterized by the absence of non-normal growth patterns in te rms of an orthogonality condition on a computable family of generalized ei genvectors of the matrix Q. With this piece of theory at hand this article provides 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 o f a linear chain of oscillators in a generalized heat bath at low temperat ure.\n LOCATION:https://stable.researchseminars.org/talk/POSemP/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandra Caraceni (University of Oxford) DTSTART:20210511T150000Z DTEND:20210511T160000Z DTSTAMP:20260404T110832Z UID:POSemP/5 DESCRIPTION:Title: Polynomial mixing time for edge flips on planar maps\nby Alessan dra Caraceni (University of Oxford) as part of Pisa Online Seminar in Prob ability\n\n\nAbstract\nA long-standing problem proposed by David Aldous co nsists in giving a sharp upper bound for the mixing time of the so-called “triangulation walk”\, a Markov chain defined on the set of all possib le triangulations of the regular n-gon. A single step of the chain consist s in performing a random edge flip\, i.e. in choosing an (internal) edge o f the triangulation uniformly at random and\, with probability 1/2\, repla cing it with the other diagonal of the quadrilateral formed by the two tri angles adjacent to the edge in question (with probability 1/2\, the triang ulation is left unchanged).\n\nWhile it has been shown that the relaxation time for the triangulation walk is polynomial in n and bounded below by a multiple of $n^{3/2}$\, the conjectured sharpness of the lower bound rema ins firmly out of reach in spite of the apparent simplicity of the chain. For edge flip chains on different models – such as planar maps\, quadran gulations of the sphere\, lattice triangulations and other geometric graph s – even less is known.\n\nWe shall discuss results concerning the mixin g time of random edge flips on rooted quadrangulations of the sphere obtai ned in joint work with Alexandre Stauffer. A “growth scheme” for quadr angulations\, which generates a uniform quadrangulation of the sphere by a dding faces one at a time at appropriate random locations\, can be combine d with careful combinatorial constructions to build probabilistic canonica l paths in a relatively novel way. This method has implications for a rang e of interesting edge-manipulating Markov chains on so-called Catalan stru ctures\, from “leaf translations” on plane trees to “edge rotations ” on general planar maps.\n LOCATION:https://stable.researchseminars.org/talk/POSemP/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Willem Van Zuijlen (WIAS (Berlin)) DTSTART:20210608T140000Z DTEND:20210608T150000Z DTSTAMP:20260404T110832Z UID:POSemP/6 DESCRIPTION:Title: Total mass asymptotics of the parabolic Anderson model\nby Wille m Van Zuijlen (WIAS (Berlin)) as part of Pisa Online Seminar in Probabilit y\n\n\nAbstract\nWe consider the parabolic Anderson model with a white noi se potential in two dimensions. This model is also called the stochastic h eat equation with a multiplicative noise. We study the large time asymptot ics of the total mass of the solution. Due to the irregularity of the whit e noise\, in two dimensions the equation is a priori not well-posed. Using paracontrolled calculus or regularity structures one can make sense of th e equation by a renormalisation\, which can be thought of as "subtracting infinity of the potential''. To obtain the asymptotics of the total mass w e use the spectral decomposition\, an alternative Feynman-Kac type represe ntation and heat-kernel estimates which come from joint works with Khalil Chouk\, Wolfgang König and Nicolas Perkowski.\n LOCATION:https://stable.researchseminars.org/talk/POSemP/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Josué Corujo (Université Paris Dauphine (CEREMADE)) DTSTART:20210608T130000Z DTEND:20210608T140000Z DTSTAMP:20260404T110832Z UID:POSemP/7 DESCRIPTION:Title: Spectrum and ergodicity of a neutral multi-allelic Moran model\n by Josué Corujo (Université Paris Dauphine (CEREMADE)) as part of Pisa O nline Seminar in Probability\n\n\nAbstract\nWe will present some recent re sults on the study of a neutral\nmulti-allelic Moran model\, which is a fi nite continuous-time Markov\nprocess. For this process\, it is assumed tha t the individuals interact\naccording to two processes: a mutation process where they mutate\nindependently of each other according to an irreducibl e rate matrix\, and\na Moran type reproduction process\, where two individ uals are uniformly\nchosen\, one dies and the other is duplicated. During this talk we will\ndiscuss some recent results for the spectrum of the gen erator of the\nneutral multi-allelic Moran process\, providing explicit ex pressions for\nits eigenvalues in terms of the eigenvalues of the rate mat rix that\ndrives the mutation process. Our approach does not require that the\nmutation process be reversible\, or even diagonalizable. Additionally \, we\nwill discuss some applications of these results to the study of the \nspeed of convergence to stationarity of the Moran process for a process\ nwith general mutation scheme. We specially focus on the case where the\nm utation scheme satisfies the so called "parent independent" condition\,\nw here (and only where) the neutral Moran model becomes reversible. In\nthis later case we can go further and prove the existence of a cutoff\nphenome non for the convergence to stationarity.\n\nThis presentation is based on a recently submitted work\, for which a\npreprint is available at https:// arxiv.org/abs/2010.08809.\n LOCATION:https://stable.researchseminars.org/talk/POSemP/7/ END:VEVENT END:VCALENDAR