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BEGIN:VEVENT
SUMMARY:Martin Vogel (Université de Strasbourg)
DTSTART;VALUE=DATE-TIME:20200515T120000Z
DTEND;VALUE=DATE-TIME:20200515T130000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/1
DESCRIPTION:Title: Spe
ctra of Toeplitz matrices subject to small random noise.\nby Martin Vo
gel (Université de Strasbourg) as part of Séminaire MEGA\n\n\nAbstract\n
The spectra of nonselfadjoint linear operators can be very unstable and se
nsitive to small perturbations. This phenomenon is usually referred to as
“pseudospectral effect”. To explore this spectral instability we study
the spectra of small random perturbations of non-selfadjoint operators in
the case of Toeplitz matrices and in the case of the Toeplitz quantizatio
n of complex-valued functions on the torus. We will discuss recent results
by Sjöstrand\, Vogel and by Basak\, Paquette and Zeitouni\, describing t
he distribution of the eigenvalues in various regimes and settings.\n
LOCATION:https://researchseminars.org/talk/MEGA/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roland Bauerschmidt (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20200515T133000Z
DTEND;VALUE=DATE-TIME:20200515T143000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/3
DESCRIPTION:Title: Ran
dom spanning forests and hyperbolic symmetry.\nby Roland Bauerschmidt
(University of Cambridge) as part of Séminaire MEGA\n\n\nAbstract\nWe stu
dy (unrooted) random forests on a graph where the probability of a forest
is multiplicatively weighted by a parameter $\\beta>0$ per edge. This mode
l is the $q\\to 0$ limit of the random cluster model with $p=q\\beta$. It
is also known under different names such as the arboreal gas or the unifor
m forest model. In this talk\, I will discuss the tantalizing conjectural
behaviour of the model\, and then present our result that there is no perc
olation in dimension two. This result relies on a surprising hyperbolic sy
mmetry and methods previously developed for linearly reinforced walks. (Th
is is joint work with Nick Crawford\, Tyler Helmuth\, and Andrew Swan.)\n
LOCATION:https://researchseminars.org/talk/MEGA/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Najnudel (University of Bristol)
DTSTART;VALUE=DATE-TIME:20200605T120000Z
DTEND;VALUE=DATE-TIME:20200605T130000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/4
DESCRIPTION:Title: The
bead process for beta ensembles\nby Joseph Najnudel (University of Br
istol) as part of Séminaire MEGA\n\n\nAbstract\nThe bead process introduc
ed by Boutillier is a countable interlacing of the determinantal sine-kern
el point processes. We construct the bead process for general sine beta pr
ocesses as an infinite dimensional Markov chain whose transition mechanism
is explicitly described. We show that this process is the microscopic sca
ling limit in the bulk of the Hermite beta corner process introduced by Go
rin and Shkolnikov\, generalizing the process of the minors of the Gaussia
n unitary and orthogonal ensembles.\n
LOCATION:https://researchseminars.org/talk/MEGA/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theodoros Assiotis (University of Oxford)
DTSTART;VALUE=DATE-TIME:20200605T133000Z
DTEND;VALUE=DATE-TIME:20200605T143000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/5
DESCRIPTION:Title: Joi
nt moments of the characteristic polynomial of a random unitary matrix
\nby Theodoros Assiotis (University of Oxford) as part of Séminaire MEGA\
n\n\nAbstract\nI will speak about the joint moments of the characteristic
polynomial of a random unitary matrix and its derivative. In joint work wi
th Jon Keating and Jon Warren\, by developing a connection with the Hua-Pi
ckrell measures and using a probabilistic approach\, we establish these as
ymptotics for general real values of the exponents which proves a conjectu
re from the thesis of Hughes from 2001.\n
LOCATION:https://researchseminars.org/talk/MEGA/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Camille Male (Université de Bordeaux)
DTSTART;VALUE=DATE-TIME:20201113T130000Z
DTEND;VALUE=DATE-TIME:20201113T140000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/6
DESCRIPTION:Title: App
lications of Freeness over the diagonal of large random matrices.\nby
Camille Male (Université de Bordeaux) as part of Séminaire MEGA\n\n\nAbs
tract\nTraffic probability is an extension of free probability that comes
with a general notion of traffic independence. This notion encodes a large
class of relation\, in particular all non commutative notions of independ
ence. For a long time\, this notion had only a combinatorial presentation\
, limiting its field of applicability. However\, an important breakthrough
was achieved two years ago when we discovered a connection with the notio
n of freeness over the diagonal. I will illustrate this connection with th
ree results:\n- a general asymptotic freeness theorem for a very general c
lass of random matrices\n- a method for computing outliers in spiked rando
m matrix models with a variance profile\n- a characterization of the fluct
uations of linear statistics for large Wigner matrices.\n
LOCATION:https://researchseminars.org/talk/MEGA/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gregory Schehr (Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20201113T143000Z
DTEND;VALUE=DATE-TIME:20201113T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/7
DESCRIPTION:Title: Exa
ct persistence exponent for the 2d-diffusion equation: from random polynom
ials to truncated random matrices.\nby Gregory Schehr (Université Par
is-Saclay) as part of Séminaire MEGA\n\n\nAbstract\nAfter an introduction
to persistence probabilities and related first-passage time in statistica
l physics\, I will discuss a specific example: the 2d diffusion equation w
ith random initial conditions. The persistence probability in this problem
turns out to be related to the probability of no real root for Kac random
polynomials. I will show that this probability can be computed by using y
et another connection\, namely to the truncated orthogonal ensemble of ran
dom matrices.\n
LOCATION:https://researchseminars.org/talk/MEGA/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alain Rouault (Université Paris-Saclay\, UVSQ)
DTSTART;VALUE=DATE-TIME:20201113T093000Z
DTEND;VALUE=DATE-TIME:20201113T110000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/8
DESCRIPTION:Title: Min
i-cours: Analyse spectrale et grandes déviations.\nby Alain Rouault (
Université Paris-Saclay\, UVSQ) as part of Séminaire MEGA\n\n\nAbstract\
nDans la théorie des polynômes orthogonaux\, les règles de sommation so
nt des relations remarquables entre d’une part une entropie mettant en j
eu une mesure de référence et d’autre part une fonctionnelle des coeff
icients de récurrence. Dans ce mini-cours\, je donnerai une introduction
historique depuis le théorème de Szegö sur le cercle jusqu’à celui d
e Killip-Simon sur la droite. Je montrerai ensuite qu’il est possible de
retrouver ces règles de sommation et d’en établir de nouvelles en con
sidérant les fonctionnelles positives comme des fonctions de taux réglan
t les grandes déviations de mesures spectrales (pondérées) dans des mod
èles de matrices aléatoires. Cette méthode probabiliste s’avère part
iculièrement robuste et s’applique à des modèles non pris en compte p
ar l’analyse spectrale classique.\n
LOCATION:https://researchseminars.org/talk/MEGA/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sylvia Serfaty (New York University)
DTSTART;VALUE=DATE-TIME:20201211T093000Z
DTEND;VALUE=DATE-TIME:20201211T110000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/9
DESCRIPTION:Title: Loi
s locales et fluctuations pour les gaz de Coulomb\nby Sylvia Serfaty (
New York University) as part of Séminaire MEGA\n\n\nAbstract\nOn s'intér
esse à la mesure de Gibbs d'un gaz de Coulomb en dimension 2 et plus. On
présente des ``lois locales“ permettant de contrôler la distribution d
es points et de l'énergie jusqu'à l'échelle microscopique\, ainsi qu'un
théorème central limite sur les fluctuations des statistiques linéaire
s. Les ingrédients principaux sont la formulation électrique de l'énerg
ie et la presque additivité de l'énergie libre. Basé sur des travaux av
ec Thomas Leblé et avec Scott Armstrong.\n
LOCATION:https://researchseminars.org/talk/MEGA/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Krajenbrink (SISSA - Trieste)
DTSTART;VALUE=DATE-TIME:20201211T130000Z
DTEND;VALUE=DATE-TIME:20201211T140000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/10
DESCRIPTION:Title: Fr
edholm determinants\, exact solutions to the Kardar-Parisi-Zhang equation
and integro-differential Painlevé equations\nby Alexandre Krajenbrink
(SISSA - Trieste) as part of Séminaire MEGA\n\n\nAbstract\nAs Fredholm d
eterminants are more and more frequent in the context of stochastic integr
ability\, I discuss in this talk the existence of a common framework in ma
ny integrable systems where they appear. This consists in a hierarchy of e
quations\, akin to the Zakharov-Shabat system\, connecting an integro-diff
erential extension of the Painlevé II hierarchy\, the finite-time solutio
ns of the Kardar-Parisi-Zhang equation and multi-critical fermions at fini
te temperature. The talk is based on the results of the paper arXiv:2008.0
1509\n
LOCATION:https://researchseminars.org/talk/MEGA/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elliot Paquette (McGill University)
DTSTART;VALUE=DATE-TIME:20201211T143000Z
DTEND;VALUE=DATE-TIME:20201211T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/11
DESCRIPTION:Title: Th
e edge scaling limit of the Gaussian beta-ensemble characteristic polynomi
al\nby Elliot Paquette (McGill University) as part of Séminaire MEGA\
n\n\nAbstract\nThe Gaussian beta-ensemble (GbetaE) is a 1-parameter genera
lization of the Gaussian orthogonal/unitary/symplectic ensembles which ret
ains some integrable structure. Using this ensemble\, in Ramirez\, Rider a
nd Virag constructed a limiting point process\, the Airy-beta point proces
s\, which is the weak limit of the point process of eigenvalues in a neigh
borhood of the spectral edge. They constructed a limiting Sturm—Liouvill
e problem\, the stochastic Airy equation with Dirichlet boundary condition
s\, and they proved convergence of a discrete operator with spectra given
by GbetaE to this limit.\n\nJointly with Gaultier Lambert\, we give a cons
truction of a new limiting object\, the stochastic Airy function (SAi)\; w
e also show this is the limit of the characteristic polynomial of GbetaE i
n a neighborhood of the edge. It is the solution of the stochastic Airy eq
uation\, which is the usual Airy equation perturbed by a multiplicative wh
ite noise\, with specified asymptotics at time=+infinity. Its zeros are gi
ven by the Airy-beta point process\, and the mode of convergence we establ
ish provides a new proof that Airy-beta is the limiting point process of e
igenvalues of GbetaE. In this talk\, we survey what new information we hav
e on the characteristic polynomial\; we show from where the stochastic Air
y equation arises\; we show how SAi is constructed\; and we leave some una
nswered questions.\n
LOCATION:https://researchseminars.org/talk/MEGA/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Betea (KU Leuven)
DTSTART;VALUE=DATE-TIME:20210115T093000Z
DTEND;VALUE=DATE-TIME:20210115T110000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/12
DESCRIPTION:Title: Mi
ni-course: Multi-critical Schur measures and unitary matrix models.\nb
y Dan Betea (KU Leuven) as part of Séminaire MEGA\n\n\nAbstract\nWe start
by reviewing classical equalities between certain multiplicative Haar exp
ectations over the unitary group (partition functions for certain classes
of random unitary matrices)\, Toeplitz (and eventually Fredholm) determina
nts\, and extremal/edge statistics of Okounkov's Schur measure. We pass by
Heine's identity\, the Gessel identity\, the Borodin–Okounkov–Geronim
o–Case identity\, and Szego's strong theorem (if time permits). This bri
ef tour aims to sketch the deep connections between random unitary matrice
s and symmetric functions. Such connections were first observed by Diaconi
s–Shashahani and later put to great use by Johansson\, Rains\, and colla
borators.\n\nWe then aim at proving a recent result of the author\, joint
with J. Bouttier and H. Walsh (arXiv'd here https://arxiv.org/abs/2012.019
95)\, which shows that when the unitary matrix model potential is tuned
“multi-critically”\, all the quantities above tend to the higher-order
Tracy–Widom distributions introduced recently by Le Doussal–Majumdar
–Schehr. This result is a gap probability result for the largest part of
the associated random partition\, and as such extends the by now classica
l Baik–Deift–Johansson theorem on longest increasing subsequences of r
andom permutations. In passing\, we try to mention some related results bo
th old: limit shape results for the random partitions under consideration\
; the associated phase transitions of Gross–Witten and Johansson\; the o
riginal approach to multi-criticality of Periwal–Shevitz\; the Schroding
er approach of Le Doussal–Majumdar–Schehr\; and some recent work of Ca
fasso–Claeys–Girotti and Krajenbrink.\n
LOCATION:https://researchseminars.org/talk/MEGA/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emma Bailey (University of Bristol)
DTSTART;VALUE=DATE-TIME:20210115T140000Z
DTEND;VALUE=DATE-TIME:20210115T150000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/13
DESCRIPTION:Title: Ch
aracteristic polynomials of the classical compact groups.\nby Emma Bai
ley (University of Bristol) as part of Séminaire MEGA\n\n\nAbstract\nMome
nts of characteristic polynomials have connections to log-correlated field
s\, Toeplitz and Hankel determinants\, combinatorics\, and number theory.
In this talk\, I will introduce `moments of moments' of characteristic pol
ynomials. Our results give their asymptotic behaviour\, answering a conjec
ture of Fyodorov and Keating. This talk will discuss joint work with Jon K
eating and Theo Assiotis.\n
LOCATION:https://researchseminars.org/talk/MEGA/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benedek Valkó (University of Wisconsin- Madison)
DTSTART;VALUE=DATE-TIME:20210115T153000Z
DTEND;VALUE=DATE-TIME:20210115T163000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/14
DESCRIPTION:Title: Th
e stochastic zeta function.\nby Benedek Valkó (University of Wisconsi
n- Madison) as part of Séminaire MEGA\n\n\nAbstract\nChhaibi\, Najnudel a
nd Nikhekgbali introduced a random entire function with zero set given by
the points of the Sine_2 process\, the point process limit of the circular
unitary ensemble (CUE). They showed that the function is the limit of the
normalized characteristic polynomials of the CUE. We provide new descript
ions for this random function: as a power series built from Brownian motio
n\, as a determinant connected to a random differential operator\, and as
the stationary solution of an SDE. Our approach extends to various general
izations of the CUE: the circular beta-ensemble\, and the Hua-Pickrell ens
emble.\n\nJoint with B. Virág (Toronto) and Yun Li (Wisconsin).\n
LOCATION:https://researchseminars.org/talk/MEGA/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillaume Cébron (Université de Toulouse)
DTSTART;VALUE=DATE-TIME:20210205T093000Z
DTEND;VALUE=DATE-TIME:20210205T110000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/15
DESCRIPTION:Title: Mi
ni-cours : Introduction à la théorie des trafics.\nby Guillaume Céb
ron (Université de Toulouse) as part of Séminaire MEGA\n\n\nAbstract\nLa
théorie des trafics a été formalisée en 2011 par Camille Male. La mot
ivation principale est l'étude des matrices aléatoires dont la loi est i
nvariante par permutation des vecteurs de base. Je vais introduire les con
cepts généraux de la théorie des trafics\, qui reposent sur un formalis
me faisant intervenir des graphes. Je parlerai ensuite de l'asymptotique e
n grande dimension de matrices indépendantes\, donnant lieu naturellement
à la notion d'indépendance de trafics.\n
LOCATION:https://researchseminars.org/talk/MEGA/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gaultier Lambert (University of Zurich)
DTSTART;VALUE=DATE-TIME:20210312T130000Z
DTEND;VALUE=DATE-TIME:20210312T140000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/16
DESCRIPTION:Title: Ap
plications of the theory of Gaussian multiplicative chaos to random matric
es\nby Gaultier Lambert (University of Zurich) as part of Séminaire M
EGA\n\n\nAbstract\nLog-correlated fields are a class of stochastic process
es which describe the fluctuations of some key observables in different pr
obabilistic models in dimension 1 and 2 such as random tilings\, or the ch
aracteristic polynomials of random matrices. Gaussian multiplicative chaos
is a renormalization procedure which aims at defining the exponential of
a Log-correlated field in the form of a family of random measures. These r
andom measures can be thought of as describing the extreme values of the u
nderlying field. In this talk\, I will present some applications of this t
heory to study the logarithm of the characteristic polynomial of some rand
om matrices. I will focus on the Ginibre ensemble and also mention some re
sults for the Gaussian unitary ensemble and circular beta ensembles.\n
LOCATION:https://researchseminars.org/talk/MEGA/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Youssef (NYU Abu Dhabi)
DTSTART;VALUE=DATE-TIME:20210312T143000Z
DTEND;VALUE=DATE-TIME:20210312T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/17
DESCRIPTION:Title: Mi
xing time of the switch chain on regular bipartite graphs\nby Pierre Y
oussef (NYU Abu Dhabi) as part of Séminaire MEGA\n\n\nAbstract\nGiven a f
ixed integer d\, we consider the switch chain on the set of d-regular bipa
rtite graphs on n vertices equipped with the uniform measure. We prove a s
harp Poincaré and log-Sobolev inequality implying that the mixing time of
the switch chain is at most O(n log^2n) which is optimal up to a logarith
mic term. This improves on earlier results of Kannan\, Tetali\, Vempala an
d Dyer et al. who obtained the bounds O(n^13 log n) and O(n^7 log n) respe
ctively. This is a joint work with Konstantin Tikhomirov.\n
LOCATION:https://researchseminars.org/talk/MEGA/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgio Cipolloni (IST Austria)
DTSTART;VALUE=DATE-TIME:20210205T130000Z
DTEND;VALUE=DATE-TIME:20210205T140000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/18
DESCRIPTION:Title: Co
rrelated DBMs and fluctuations in the circular law: CLT for i.i.d. random
matrices.\nby Giorgio Cipolloni (IST Austria) as part of Séminaire ME
GA\n\n\nAbstract\nWe consider a large non-Hermitian i.i.d. matrix X with r
eal or complex entries and show that the linear statistics of the eigenval
ues are asymptotically Gaussian for test function having 2+\\epsilon deriv
atives. Previously this result was known only for the Ginibre ensemble\, w
here explicit formulas for the correlation functions are available\, and e
nsembles close to Ginibre in the sense of moment matching\; our result hol
ds for general distribution of the matrix entries. The proof relies on two
main novel ingredients: (i) local law for product of resolvents of the He
rmitisation of X at two different spectral parameters\, (ii) coupling of s
everal dependent Dyson Brownian motions.\n
LOCATION:https://researchseminars.org/talk/MEGA/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Bufetov (University of Bonn)
DTSTART;VALUE=DATE-TIME:20210205T143000Z
DTEND;VALUE=DATE-TIME:20210205T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/19
DESCRIPTION:Title: In
teracting particle systems and random walks on Hecke algebras.\nby Ale
xey Bufetov (University of Bonn) as part of Séminaire MEGA\n\n\nAbstract\
nMulti-species versions of several interacting particle systems\, includin
g ASEP\, q-TAZRP\, and k-exclusion processes\, can be interpreted as rando
m walks on Hecke algebras. In the talk I will discuss this connection and
its probabilistic applications.\n
LOCATION:https://researchseminars.org/talk/MEGA/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Justin Salez (Université Paris-Dauphine)
DTSTART;VALUE=DATE-TIME:20210611T120000Z
DTEND;VALUE=DATE-TIME:20210611T130000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/20
DESCRIPTION:Title: Sp
arse expanders have negative curvature\nby Justin Salez (Université P
aris-Dauphine) as part of Séminaire MEGA\n\n\nAbstract\nWe prove that bou
nded-degree expanders with non-negative Ollivier-Ricci curvature do not ex
ist\, thereby solving a long-standing open problem suggested by Naor and M
ilman and publicized by Ollivier (2010). In fact\, this remains true even
if we allow for a vanishing proportion of large degrees\, large eigenvalue
s\, and negatively-curved edges. To establish this\, we work directly at t
he level of Benjamini-Schramm limits\, and exploit the entropic characteri
zation of the Liouville property on stationary random graphs to show that
non-negative curvature and spectral expansion are incompatible “at infin
ity”. We then transfer this result to finite graphs via local weak conve
rgence. The same approach applies to the Bakry-Émery curvature condition
CD(0\, ∞)\, thereby settling a recent conjecture of Cushing\, Liu and Pe
yerimhoff (2019).\n
LOCATION:https://researchseminars.org/talk/MEGA/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mylène Maïda (Université de Lille)
DTSTART;VALUE=DATE-TIME:20210312T093000Z
DTEND;VALUE=DATE-TIME:20210312T110000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/21
DESCRIPTION:Title: Mi
ni-cours: Rigidité pour les processus ponctuels\nby Mylène Maïda (U
niversité de Lille) as part of Séminaire MEGA\n\n\nAbstract\nUn processu
s ponctuel est dit rigide (ou number-rigide) si pour tout compact fixé\,
la donnée de la configuration à l'extérieur du compact prescrit presque
sûrement le nombre de points à l'intérieur. Cette propriété intrigan
te a été montrée pour certains processus déterminantaux\, des réseaux
perturbés et quelques processus apparentés. Je ferai le point sur les r
ésultats connus (pas si nombreux)\, les techniques de preuve et j'énonce
rai quelques conjectures.\n
LOCATION:https://researchseminars.org/talk/MEGA/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathieu Lewin (Université Paris-Dauphine)
DTSTART;VALUE=DATE-TIME:20210409T083000Z
DTEND;VALUE=DATE-TIME:20210409T100000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/22
DESCRIPTION:Title: Mi
ni-course: Riesz and Coulomb gases: what's known and unknown.\nby Math
ieu Lewin (Université Paris-Dauphine) as part of Séminaire MEGA\n\n\nAbs
tract\nIn this lecture I will speak about a family of random point process
es of Gibbs type\, on the whole d-dimensional space\, which includes many
examples from random matrix theory (such as GUE\, GOE and Ginibre). The po
ints are assumed to interact by pairs through the Riesz/Coulomb potentials
\, and a parameter playing the role of a temperature is used to adjust the
amount of randomness. I will try to review what is expected on physical g
round for these processes and what has been rigorously established so far.
The talk will thus be focused on open problems.\n
LOCATION:https://researchseminars.org/talk/MEGA/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antti Knowles (University of Geneva)
DTSTART;VALUE=DATE-TIME:20210409T133000Z
DTEND;VALUE=DATE-TIME:20210409T143000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/23
DESCRIPTION:Title: Th
e spectral edge of (sub-)critical Erdös-Rényi graphs.\nby Antti Know
les (University of Geneva) as part of Séminaire MEGA\n\n\nAbstract\nIt is
well known that the Erdős-Rényi graph on N vertices with edge probabili
ty d/N undergoes a dramatic change in behaviour when the mean degree d cro
sses the critical scale log(N): the degrees of the graph cease to concentr
ate about their means and the graph loses its homogeneity. We analyse the
eigenvalues and eigenvectors of its adjacency matrix in the regime where t
he mean degree d is comparable to or less than the critical scale log(N).
We show that the eigenvalue process near the spectral edges is asymptotica
lly Poisson\, and the intensity measure is determined by the fluctuations
of the large degrees as well as the size of the 2-spheres around vertices
of large degree. We conclude that in general the laws of the largest eigen
values are not described by the classical Fisher–Tippett–Gnedenko theo
rem. As an application of our result\, we prove that the associated eigenv
ectors are are exponentially localized in unique\, disjoint balls. Togethe
r with the previously established complete delocalization of the eigenvect
ors in the middle of the spectrum\, this establishes the coexistence of a
delocalized and a localized phase in the critical Erdös-Rényi graph. Joi
nt work with Johannes Alt and Raphael Ducatez.\n
LOCATION:https://researchseminars.org/talk/MEGA/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Pain (New York University)
DTSTART;VALUE=DATE-TIME:20210409T120000Z
DTEND;VALUE=DATE-TIME:20210409T130000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/24
DESCRIPTION:Title: Op
timal local law and central limit theorem for beta-ensembles.\nby Mich
el Pain (New York University) as part of Séminaire MEGA\n\n\nAbstract\nIn
this talk\, I will present a joint work with Paul Bourgade and Krishnan M
ody. We consider beta-ensembles with general potentials (or equivalently a
log-gas in dimension 1)\, which are a generalization of Gaussian beta-ens
embles and of classical invariant ensembles of random matrices. We prove a
multivariate central limit theorem for the logarithm of the characteristi
c polynomial\, showing that it behaves as a log-correlated field. A key in
gredient is an optimally sharp local law for the the Stieljes transform of
the empirical measure which can be of independent interest. Both the proo
fs of the CLT and the local law are based essentially on loop equations te
chniques.\n
LOCATION:https://researchseminars.org/talk/MEGA/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Romain Couillet (CentraleSupélec\, Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20210507T083000Z
DTEND;VALUE=DATE-TIME:20210507T100000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/25
DESCRIPTION:Title: Wh
y Random Matrices can Change the Future of Research in AI?\nby Romain
Couillet (CentraleSupélec\, Université Paris-Saclay) as part of Séminai
re MEGA\n\n\nAbstract\nMachine learning and AI algorithms are becoming inc
reasingly more powerful but also increasingly more complex\, mathematicall
y less tractable\, and energetically less environmental friendly. In this
talk\, we will demonstrate that large dimensional statistics\, and particu
larly random matrix theory\, simultaneously (i) explains why ML algorithms
are so stable when dealing with large dimensional data\, (ii) manages to
break the difficulties that make these algorithms mathematically intractab
le (non-linearities and data modelling)\, thereby (iii) allowing for the f
irst time to get (iii-a) an inside understanding of the algorithms\, of th
eir multiple biases and\, most crucially\, of their quite counter-intuitiv
e behavior as well as (iii-b) a toolbox to easily improve the algorithms p
erformance and cost efficiency. Possibly even more surprisingly\, the univ
ersality notion in random matrix theory shows (iv) why ML algorithms appli
ed to intricate real data (in general impossible to model) behave the same
as when applied to elementary Gaussian random vector models.\n\nThe cours
e will introduce basic notions of random matrix theory by emphasizing on t
he counter-intuitive behavior of large dimensional data (so to raise aware
ness in the audience). These notions will be applied to a range of telling
applications in machine learning (spectral clustering\, semi-supervised l
earning\, transfer learning\, low-cost processing\, etc.).\n\nThe audience
can dynamically decide on which topic they'd like me to cover preferably.
A time for debate will also be given for the audience to react on the pre
sentation. An extensive coverage of the class material is available online
in the upcoming book “Romain COUILLET\, Zhenyu LIAO\, “Random Matrix
Theory for Machine Learning” https://romaincouillet.hebfree.org/book.htm
l\n
LOCATION:https://researchseminars.org/talk/MEGA/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Giraud (Université Paris Saclay)
DTSTART;VALUE=DATE-TIME:20210507T123000Z
DTEND;VALUE=DATE-TIME:20210507T133000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/26
DESCRIPTION:Title: Sp
ectral properties of structured random matrices\nby Olivier Giraud (Un
iversité Paris Saclay) as part of Séminaire MEGA\n\n\nAbstract\nMotivate
d by the problem of metal-insulator transition in the Anderson model of co
ndensed matter physics\, I will discuss some spectral properties of Hermit
ian Toeplitz\, Hankel\, and Toeplitz-plus-Hankel random matrices with inde
pendent identically distributed entries. Spectral statistics of all these
random matrices turns out to be of intermediate type\, as found for instan
ce at the metal-insulator transition\, or in certain pseudo-integrable bil
liards. Namely\, nearest-neighbor spacing distributions are characterized
by level repulsion at small distances and an exponential decrease at large
distances\, while the spectral compressibility has a non-trivial value. S
uch statistics are usually associated with multifractal eigenstates\, and
I will show that it is also the case here.\n
LOCATION:https://researchseminars.org/talk/MEGA/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tatyana Shcherbina (University of Wisconsin - Madison)
DTSTART;VALUE=DATE-TIME:20210507T140000Z
DTEND;VALUE=DATE-TIME:20210507T150000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/27
DESCRIPTION:Title: Un
iversality for random band matrices\nby Tatyana Shcherbina (University
of Wisconsin - Madison) as part of Séminaire MEGA\n\n\nAbstract\nRandom
band matrices (RBM) are natural intermediate models to study eigenvalue st
atistics and quantum propagation in disordered systems\, since they interp
olate between mean-field type Wigner matrices and random Schrodinger opera
tors. In particular\, RBM can be used to model the Anderson metal-insulato
r phase transition (crossover) even in 1d. In this talk we will discuss so
me recent progress in application of the supersymmetric method (SUSY) and
transfer matrix approach to the analysis of local spectral characteristics
of some specific types of 1d RBM.\n
LOCATION:https://researchseminars.org/talk/MEGA/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cécilia Lancien (Université de Toulouse)
DTSTART;VALUE=DATE-TIME:20210611T083000Z
DTEND;VALUE=DATE-TIME:20210611T100000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/28
DESCRIPTION:Title: Mi
ni-course: Quantum expander graphs\nby Cécilia Lancien (Université d
e Toulouse) as part of Séminaire MEGA\n\n\nAbstract\nThe goal of this lec
ture is to understand what quantum expander graphs are\, what they are use
ful for\, and how they can be constructed. We will first recall the defini
tion of classical expander graphs\, and explain how quantum analogues of t
hese objects can be defined. We will then show that\, both classically and
quantumly\, random constructions provide with high probability examples o
f expander graphs. In the quantum case\, such result is derived from a spe
ctral analysis for random matrix models with a tensor product structure. T
he presentation will be based\, among others\, on:\n\n-Random unitaries gi
ve quantum expanders. M.B.Hastings. 2007.\n\n- Quantum expanders and geome
try of operator spaces. G.Pisier. 2014\n\n- Correlation length in random M
PS and PEPS. C.Lancien and D.Peréz-García. 2019.\n\n- Characterizing exp
ansion\, classicaly and quantumly. C.Lancien. 2020.\n
LOCATION:https://researchseminars.org/talk/MEGA/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Pastur (B.Verkin Institute for Low Temperature Physics and
Engineering\, Kharkiv\, Ukraine)
DTSTART;VALUE=DATE-TIME:20210611T133000Z
DTEND;VALUE=DATE-TIME:20210611T143000Z
DTSTAMP;VALUE=DATE-TIME:20211128T093016Z
UID:MEGA/29
DESCRIPTION:Title: On
Random Matrices Arising in Deep Neural Networks\nby Leonid Pastur (B.
Verkin Institute for Low Temperature Physics and Engineering\, Kharkiv\, U
kraine) as part of Séminaire MEGA\n\n\nAbstract\nWe study the distributio
n of singular values of product of random matrices pertinent to the analys
is of deep neural networks. The matrices resemble the product of the sampl
e covariance matrices. However\, an important dierence is that the analog
the of the population covariance matrices\, assumed to be non-random or r
andom but independent of the random data matrix in statistics and random m
atrix theory\, are now certain functions of random data matrices (synaptic
weight matrices in the deep neural network terminology). For the Gaussian
synaptic weight matrices the problem has been treated in recent work [1]
and certain subsequent works by using the techniques of free probability t
heory. Since\, however\, free probability theory deals with population cov
ariance matrices which are independent of the data matrices\, its applicab
ility to this case has to be justi\ned. We use a version of the techniques
of random matrix theory to justify and generalize the results of [1] to t
he case where the entries of the synaptic weight matrices are just indepen
dent identically distributed random variables with zero mean and \nnite fo
urth moment [2]. This\, in particular\, extends the property of the so-cal
led macroscopic universality to the considered random matrices.\n\n[1] J.
Pennington\, S. Schoenholz\, and S. Ganguli\, The emergence of spectral un
iversality In: Proc. Mach. Learn. Res. (PMLR 70) 84 (2018) 1924-1932\, htt
p://arxiv.org/abs/1802.09979\n\n[2] L. Pastur and V. Slavin\, On Random Ma
trices Arising in Deep Neural Networks: General I.I.D. Case\, http://arxiv
.org/abs/2011.11439.\n
LOCATION:https://researchseminars.org/talk/MEGA/29/
END:VEVENT
END:VCALENDAR