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SUMMARY:Laure Dumaz (École Normale supérieure)
DTSTART;VALUE=DATE-TIME:20210413T140000Z
DTEND;VALUE=DATE-TIME:20210413T150000Z
DTSTAMP;VALUE=DATE-TIME:20240423T112450Z
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;VALUE=DATE-TIME:20210413T150000Z
DTEND;VALUE=DATE-TIME:20210413T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T112450Z
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;VALUE=DATE-TIME:20210427T140000Z
DTEND;VALUE=DATE-TIME:20210427T150000Z
DTSTAMP;VALUE=DATE-TIME:20240423T112450Z
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;VALUE=DATE-TIME:20210511T140000Z
DTEND;VALUE=DATE-TIME:20210511T150000Z
DTSTAMP;VALUE=DATE-TIME:20240423T112450Z
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;VALUE=DATE-TIME:20210511T150000Z
DTEND;VALUE=DATE-TIME:20210511T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T112450Z
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;VALUE=DATE-TIME:20210608T140000Z
DTEND;VALUE=DATE-TIME:20210608T150000Z
DTSTAMP;VALUE=DATE-TIME:20240423T112450Z
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;VALUE=DATE-TIME:20210608T130000Z
DTEND;VALUE=DATE-TIME:20210608T140000Z
DTSTAMP;VALUE=DATE-TIME:20240423T112450Z
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/
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