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BEGIN:VEVENT
SUMMARY:Martin Vogel (Université de Strasbourg)
DTSTART:20200515T120000Z
DTEND:20200515T130000Z
DTSTAMP:20260404T094534Z
UID:MEGA/1
DESCRIPTION:Title: Spectra of Toeplitz matrices subject to small random noise.\nby Ma
rtin Vogel (Université de Strasbourg) as part of Séminaire MEGA\n\n\nAbs
tract\nThe spectra of nonselfadjoint linear operators can be very unstable
and sensitive to small perturbations. This phenomenon is usually referred
to as “pseudospectral effect”. To explore this spectral instability w
e study the spectra of small random perturbations of non-selfadjoint opera
tors in the case of Toeplitz matrices and in the case of the Toeplitz quan
tization of complex-valued functions on the torus. We will discuss recent
results by Sjöstrand\, Vogel and by Basak\, Paquette and Zeitouni\, descr
ibing the distribution of the eigenvalues in various regimes and settings.
\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roland Bauerschmidt (University of Cambridge)
DTSTART:20200515T133000Z
DTEND:20200515T143000Z
DTSTAMP:20260404T094534Z
UID:MEGA/3
DESCRIPTION:Title: Random spanning forests and hyperbolic symmetry.\nby Roland Bauers
chmidt (University of Cambridge) as part of Séminaire MEGA\n\n\nAbstract\
nWe study (unrooted) random forests on a graph where the probability of a
forest is multiplicatively weighted by a parameter $\\beta>0$ per edge. Th
is model is the $q\\to 0$ limit of the random cluster model with $p=q\\bet
a$. It is also known under different names such as the arboreal gas or the
uniform forest model. In this talk\, I will discuss the tantalizing conje
ctural behaviour of the model\, and then present our result that there is
no percolation in dimension two. This result relies on a surprising hyperb
olic symmetry and methods previously developed for linearly reinforced wal
ks. (This is joint work with Nick Crawford\, Tyler Helmuth\, and Andrew Sw
an.)\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Najnudel (University of Bristol)
DTSTART:20200605T120000Z
DTEND:20200605T130000Z
DTSTAMP:20260404T094534Z
UID:MEGA/4
DESCRIPTION:Title: The bead process for beta ensembles\nby Joseph Najnudel (Universit
y of Bristol) as part of Séminaire MEGA\n\n\nAbstract\nThe bead process i
ntroduced by Boutillier is a countable interlacing of the determinantal si
ne-kernel point processes. We construct the bead process for general sine
beta processes as an infinite dimensional Markov chain whose transition me
chanism is explicitly described. We show that this process is the microsco
pic scaling limit in the bulk of the Hermite beta corner process introduce
d by Gorin and Shkolnikov\, generalizing the process of the minors of the
Gaussian unitary and orthogonal ensembles.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theodoros Assiotis (University of Oxford)
DTSTART:20200605T133000Z
DTEND:20200605T143000Z
DTSTAMP:20260404T094534Z
UID:MEGA/5
DESCRIPTION:Title: Joint moments of the characteristic polynomial of a random unitary mat
rix\nby Theodoros Assiotis (University of Oxford) as part of Séminair
e MEGA\n\n\nAbstract\nI will speak about the joint moments of the characte
ristic polynomial of a random unitary matrix and its derivative. In joint
work with Jon Keating and Jon Warren\, by developing a connection with the
Hua-Pickrell measures and using a probabilistic approach\, we establish t
hese asymptotics for general real values of the exponents which proves a c
onjecture from the thesis of Hughes from 2001.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Camille Male (Université de Bordeaux)
DTSTART:20201113T130000Z
DTEND:20201113T140000Z
DTSTAMP:20260404T094534Z
UID:MEGA/6
DESCRIPTION:Title: Applications of Freeness over the diagonal of large random matrices.\nby Camille Male (Université de Bordeaux) as part of Séminaire MEGA\n
\n\nAbstract\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 i
ndependence. For a long time\, this notion had only a combinatorial presen
tation\, limiting its field of applicability. However\, an important break
through was achieved two years ago when we discovered a connection with th
e notion of freeness over the diagonal. I will illustrate this connection
with three results:\n- a general asymptotic freeness theorem for a very ge
neral class of random matrices\n- a method for computing outliers in spike
d random matrix models with a variance profile\n- a characterization of th
e fluctuations of linear statistics for large Wigner matrices.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gregory Schehr (Université Paris-Saclay)
DTSTART:20201113T143000Z
DTEND:20201113T153000Z
DTSTAMP:20260404T094534Z
UID:MEGA/7
DESCRIPTION:Title: Exact persistence exponent for the 2d-diffusion equation: from random
polynomials to truncated random matrices.\nby Gregory Schehr (Universi
té Paris-Saclay) as part of Séminaire MEGA\n\n\nAbstract\nAfter an intro
duction to persistence probabilities and related first-passage time in sta
tistical physics\, I will discuss a specific example: the 2d diffusion equ
ation with 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 yet another connection\, namely to the truncated orthogonal ensemble
of random matrices.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alain Rouault (Université Paris-Saclay\, UVSQ)
DTSTART:20201113T093000Z
DTEND:20201113T110000Z
DTSTAMP:20260404T094534Z
UID:MEGA/8
DESCRIPTION:Title: Mini-cours: Analyse spectrale et grandes déviations.\nby Alain Ro
uault (Université Paris-Saclay\, UVSQ) as part of Séminaire MEGA\n\n\nAb
stract\nDans la théorie des polynômes orthogonaux\, les règles de somma
tion sont des relations remarquables entre d’une part une entropie metta
nt en jeu une mesure de référence et d’autre part une fonctionnelle de
s coefficients de récurrence. Dans ce mini-cours\, je donnerai une introd
uction historique depuis le théorème de Szegö sur le cercle jusqu’à
celui de Killip-Simon sur la droite. Je montrerai ensuite qu’il est poss
ible de retrouver ces règles de sommation et d’en établir de nouvelles
en considérant les fonctionnelles positives comme des fonctions de taux
réglant les grandes déviations de mesures spectrales (pondérées) dans
des modèles de matrices aléatoires. Cette méthode probabiliste s’avè
re particulièrement robuste et s’applique à des modèles non pris en c
ompte par l’analyse spectrale classique.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sylvia Serfaty (New York University)
DTSTART:20201211T093000Z
DTEND:20201211T110000Z
DTSTAMP:20260404T094534Z
UID:MEGA/9
DESCRIPTION:Title: Lois locales et fluctuations pour les gaz de Coulomb\nby Sylvia Se
rfaty (New York University) as part of Séminaire MEGA\n\n\nAbstract\nOn s
'intéresse à la mesure de Gibbs d'un gaz de Coulomb en dimension 2 et pl
us. On présente des ``lois locales“ permettant de contrôler la distrib
ution des points et de l'énergie jusqu'à l'échelle microscopique\, ains
i qu'un théorème central limite sur les fluctuations des statistiques li
néaires. Les ingrédients principaux sont la formulation électrique de l
'énergie et la presque additivité de l'énergie libre. Basé sur des tra
vaux avec Thomas Leblé et avec Scott Armstrong.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Krajenbrink (SISSA - Trieste)
DTSTART:20201211T130000Z
DTEND:20201211T140000Z
DTSTAMP:20260404T094534Z
UID:MEGA/10
DESCRIPTION:Title: Fredholm determinants\, exact solutions to the Kardar-Parisi-Zhang eq
uation and integro-differential Painlevé equations\nby Alexandre Kraj
enbrink (SISSA - Trieste) as part of Séminaire MEGA\n\n\nAbstract\nAs Fre
dholm determinants are more and more frequent in the context of stochastic
integrability\, I discuss in this talk the existence of a common framewor
k in many integrable systems where they appear. This consists in a hierarc
hy of equations\, akin to the Zakharov-Shabat system\, connecting an integ
ro-differential extension of the Painlevé II hierarchy\, the finite-time
solutions of the Kardar-Parisi-Zhang equation and multi-critical fermions
at finite temperature. The talk is based on the results of the paper arXiv
:2008.01509\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elliot Paquette (McGill University)
DTSTART:20201211T143000Z
DTEND:20201211T153000Z
DTSTAMP:20260404T094534Z
UID:MEGA/11
DESCRIPTION:Title: The edge scaling limit of the Gaussian beta-ensemble characteristic p
olynomial\nby Elliot Paquette (McGill University) as part of Séminair
e MEGA\n\n\nAbstract\nThe Gaussian beta-ensemble (GbetaE) is a 1-parameter
generalization of the Gaussian orthogonal/unitary/symplectic ensembles wh
ich retains some integrable structure. Using this ensemble\, in Ramirez\,
Rider and Virag constructed a limiting point process\, the Airy-beta point
process\, which is the weak limit of the point process of eigenvalues in
a neighborhood of the spectral edge. They constructed a limiting Sturm—L
iouville problem\, the stochastic Airy equation with Dirichlet boundary co
nditions\, and they proved convergence of a discrete operator with spectra
given by GbetaE to this limit.\n\nJointly with Gaultier Lambert\, we give
a construction of a new limiting object\, the stochastic Airy function (S
Ai)\; we also show this is the limit of the characteristic polynomial of G
betaE in a neighborhood of the edge. It is the solution of the stochastic
Airy equation\, which is the usual Airy equation perturbed by a multiplica
tive white noise\, with specified asymptotics at time=+infinity. Its zeros
are given by the Airy-beta point process\, and the mode of convergence we
establish provides a new proof that Airy-beta is the limiting point proce
ss of eigenvalues of GbetaE. In this talk\, we survey what new information
we have on the characteristic polynomial\; we show from where the stochas
tic Airy equation arises\; we show how SAi is constructed\; and we leave s
ome unanswered questions.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Betea (KU Leuven)
DTSTART:20210115T093000Z
DTEND:20210115T110000Z
DTSTAMP:20260404T094534Z
UID:MEGA/12
DESCRIPTION:Title: Mini-course: Multi-critical Schur measures and unitary matrix models.
\nby Dan Betea (KU Leuven) as part of Séminaire MEGA\n\n\nAbstract\nW
e start by reviewing classical equalities between certain multiplicative H
aar expectations over the unitary group (partition functions for certain c
lasses of random unitary matrices)\, Toeplitz (and eventually Fredholm) de
terminants\, and extremal/edge statistics of Okounkov's Schur measure. We
pass by Heine's identity\, the Gessel identity\, the Borodin–Okounkov–
Geronimo–Case identity\, and Szego's strong theorem (if time permits). T
his brief tour aims to sketch the deep connections between random unitary
matrices and symmetric functions. Such connections were first observed by
Diaconis–Shashahani and later put to great use by Johansson\, Rains\, an
d collaborators.\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/2
012.01995)\, which shows that when the unitary matrix model potential is t
uned “multi-critically”\, all the quantities above tend to the higher-
order Tracy–Widom distributions introduced recently by Le Doussal–Maju
mdar–Schehr. This result is a gap probability result for the largest par
t of the associated random partition\, and as such extends the by now clas
sical Baik–Deift–Johansson theorem on longest increasing subsequences
of random permutations. In passing\, we try to mention some related result
s both old: limit shape results for the random partitions under considerat
ion\; the associated phase transitions of Gross–Witten and Johansson\; t
he original approach to multi-criticality of Periwal–Shevitz\; the Schro
dinger approach of Le Doussal–Majumdar–Schehr\; and some recent work o
f Cafasso–Claeys–Girotti and Krajenbrink.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emma Bailey (University of Bristol)
DTSTART:20210115T140000Z
DTEND:20210115T150000Z
DTSTAMP:20260404T094534Z
UID:MEGA/13
DESCRIPTION:Title: Characteristic polynomials of the classical compact groups.\nby E
mma Bailey (University of Bristol) as part of Séminaire MEGA\n\n\nAbstrac
t\nMoments of characteristic polynomials have connections to log-correlate
d fields\, Toeplitz and Hankel determinants\, combinatorics\, and number t
heory. In this talk\, I will introduce `moments of moments' of characteris
tic polynomials. Our results give their asymptotic behaviour\, answering a
conjecture of Fyodorov and Keating. This talk will discuss joint work wit
h Jon Keating and Theo Assiotis.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benedek Valkó (University of Wisconsin- Madison)
DTSTART:20210115T153000Z
DTEND:20210115T163000Z
DTSTAMP:20260404T094534Z
UID:MEGA/14
DESCRIPTION:Title: The stochastic zeta function.\nby Benedek Valkó (University of W
isconsin- Madison) as part of Séminaire MEGA\n\n\nAbstract\nChhaibi\, Naj
nudel and Nikhekgbali introduced a random entire function with zero set gi
ven by the points of the Sine_2 process\, the point process limit of the c
ircular unitary ensemble (CUE). They showed that the function is the limit
of the normalized characteristic polynomials of the CUE. We provide new d
escriptions for this random function: as a power series built from Brownia
n motion\, as a determinant connected to a random differential operator\,
and as the stationary solution of an SDE. Our approach extends to various
generalizations of the CUE: the circular beta-ensemble\, and the Hua-Pickr
ell ensemble.\n\nJoint with B. Virág (Toronto) and Yun Li (Wisconsin).\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillaume Cébron (Université de Toulouse)
DTSTART:20210205T093000Z
DTEND:20210205T110000Z
DTSTAMP:20260404T094534Z
UID:MEGA/15
DESCRIPTION:Title: Mini-cours : Introduction à la théorie des trafics.\nby Guillau
me Cébron (Université de Toulouse) as part of Séminaire MEGA\n\n\nAbstr
act\nLa théorie des trafics a été formalisée en 2011 par Camille Male.
La motivation principale est l'étude des matrices aléatoires dont la lo
i est invariante par permutation des vecteurs de base. Je vais introduire
les concepts généraux de la théorie des trafics\, qui reposent sur un f
ormalisme faisant intervenir des graphes. Je parlerai ensuite de l'asympto
tique en grande dimension de matrices indépendantes\, donnant lieu nature
llement à la notion d'indépendance de trafics.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gaultier Lambert (University of Zurich)
DTSTART:20210312T130000Z
DTEND:20210312T140000Z
DTSTAMP:20260404T094534Z
UID:MEGA/16
DESCRIPTION:Title: Applications of the theory of Gaussian multiplicative chaos to random
matrices\nby Gaultier Lambert (University of Zurich) as part of Sémi
naire MEGA\n\n\nAbstract\nLog-correlated fields are a class of stochastic
processes which describe the fluctuations of some key observables in diffe
rent probabilistic models in dimension 1 and 2 such as random tilings\, or
the characteristic polynomials of random matrices. Gaussian multiplicativ
e chaos is a renormalization procedure which aims at defining the exponent
ial of a Log-correlated field in the form of a family of random measures.
These random measures can be thought of as describing the extreme values o
f the underlying field. In this talk\, I will present some applications of
this theory to study the logarithm of the characteristic polynomial of so
me random matrices. I will focus on the Ginibre ensemble and also mention
some results for the Gaussian unitary ensemble and circular beta ensembles
.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Youssef (NYU Abu Dhabi)
DTSTART:20210312T143000Z
DTEND:20210312T153000Z
DTSTAMP:20260404T094534Z
UID:MEGA/17
DESCRIPTION:Title: Mixing time of the switch chain on regular bipartite graphs\nby P
ierre Youssef (NYU Abu Dhabi) as part of Séminaire MEGA\n\n\nAbstract\nGi
ven a fixed integer d\, we consider the switch chain on the set of d-regul
ar bipartite graphs on n vertices equipped with the uniform measure. We pr
ove a sharp 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 l
ogarithmic term. This improves on earlier results of Kannan\, Tetali\, Vem
pala and Dyer et al. who obtained the bounds O(n^13 log n) and O(n^7 log n
) respectively. This is a joint work with Konstantin Tikhomirov.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgio Cipolloni (IST Austria)
DTSTART:20210205T130000Z
DTEND:20210205T140000Z
DTSTAMP:20260404T094534Z
UID:MEGA/18
DESCRIPTION:Title: Correlated DBMs and fluctuations in the circular law: CLT for i.i.d.
random matrices.\nby Giorgio Cipolloni (IST Austria) as part of Sémin
aire MEGA\n\n\nAbstract\nWe consider a large non-Hermitian i.i.d. matrix X
with real or complex entries and show that the linear statistics of the e
igenvalues are asymptotically Gaussian for test function having 2+\\epsilo
n derivatives. Previously this result was known only for the Ginibre ensem
ble\, where explicit formulas for the correlation functions are available\
, and ensembles close to Ginibre in the sense of moment matching\; our res
ult holds for general distribution of the matrix entries. The proof relies
on two main novel ingredients: (i) local law for product of resolvents of
the Hermitisation of X at two different spectral parameters\, (ii) coupli
ng of several dependent Dyson Brownian motions.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Bufetov (University of Bonn)
DTSTART:20210205T143000Z
DTEND:20210205T153000Z
DTSTAMP:20260404T094534Z
UID:MEGA/19
DESCRIPTION:Title: Interacting particle systems and random walks on Hecke algebras.\
nby Alexey Bufetov (University of Bonn) as part of Séminaire MEGA\n\n\nAb
stract\nMulti-species versions of several interacting particle systems\, i
ncluding ASEP\, q-TAZRP\, and k-exclusion processes\, can be interpreted a
s random walks on Hecke algebras. In the talk I will discuss this connecti
on and its probabilistic applications.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Justin Salez (Université Paris-Dauphine)
DTSTART:20210611T120000Z
DTEND:20210611T130000Z
DTSTAMP:20260404T094534Z
UID:MEGA/20
DESCRIPTION:Title: Sparse expanders have negative curvature\nby Justin Salez (Univer
sité Paris-Dauphine) as part of Séminaire MEGA\n\n\nAbstract\nWe prove t
hat bounded-degree expanders with non-negative Ollivier-Ricci curvature do
not exist\, thereby solving a long-standing open problem suggested by Nao
r and Milman and publicized by Ollivier (2010). In fact\, this remains tru
e even if we allow for a vanishing proportion of large degrees\, large eig
envalues\, and negatively-curved edges. To establish this\, we work direct
ly at the level of Benjamini-Schramm limits\, and exploit the entropic cha
racterization of the Liouville property on stationary random graphs to sho
w that non-negative curvature and spectral expansion are incompatible “a
t infinity”. We then transfer this result to finite graphs via local wea
k convergence. The same approach applies to the Bakry-Émery curvature con
dition CD(0\, ∞)\, thereby settling a recent conjecture of Cushing\, Liu
and Peyerimhoff (2019).\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mylène Maïda (Université de Lille)
DTSTART:20210312T093000Z
DTEND:20210312T110000Z
DTSTAMP:20260404T094534Z
UID:MEGA/21
DESCRIPTION:Title: Mini-cours: Rigidité pour les processus ponctuels\nby Mylène Ma
ïda (Université de Lille) as part of Séminaire MEGA\n\n\nAbstract\nUn p
rocessus ponctuel est dit rigide (ou number-rigide) si pour tout compact f
ixé\, la donnée de la configuration à l'extérieur du compact prescrit
presque sûrement le nombre de points à l'intérieur. Cette propriété i
ntrigante a été montrée pour certains processus déterminantaux\, des r
éseaux perturbés et quelques processus apparentés. Je ferai le point su
r les résultats connus (pas si nombreux)\, les techniques de preuve et j'
énoncerai quelques conjectures.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathieu Lewin (Université Paris-Dauphine)
DTSTART:20210409T083000Z
DTEND:20210409T100000Z
DTSTAMP:20260404T094534Z
UID:MEGA/22
DESCRIPTION:Title: Mini-course: Riesz and Coulomb gases: what's known and unknown.\n
by Mathieu Lewin (Université Paris-Dauphine) as part of Séminaire MEGA\n
\n\nAbstract\nIn this lecture I will speak about a family of random point
processes of Gibbs type\, on the whole d-dimensional space\, which include
s many examples from random matrix theory (such as GUE\, GOE and Ginibre).
The points are assumed to interact by pairs through the Riesz/Coulomb pot
entials\, and a parameter playing the role of a temperature is used to adj
ust the amount of randomness. I will try to review what is expected on phy
sical ground for these processes and what has been rigorously established
so far. The talk will thus be focused on open problems.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antti Knowles (University of Geneva)
DTSTART:20210409T133000Z
DTEND:20210409T143000Z
DTSTAMP:20260404T094534Z
UID:MEGA/23
DESCRIPTION:Title: The spectral edge of (sub-)critical Erdös-Rényi graphs.\nby Ant
ti Knowles (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 pr
obability d/N undergoes a dramatic change in behaviour when the mean degre
e d crosses the critical scale log(N): the degrees of the graph cease to c
oncentrate about their means and the graph loses its homogeneity. We analy
se the eigenvalues and eigenvectors of its adjacency matrix in the regime
where the mean degree d is comparable to or less than the critical scale l
og(N). We show that the eigenvalue process near the spectral edges is asym
ptotically Poisson\, and the intensity measure is determined by the fluctu
ations of the large degrees as well as the size of the 2-spheres around ve
rtices of large degree. We conclude that in general the laws of the larges
t eigenvalues are not described by the classical Fisher–Tippett–Gneden
ko theorem. As an application of our result\, we prove that the associated
eigenvectors are are exponentially localized in unique\, disjoint balls.
Together with the previously established complete delocalization of the ei
genvectors in the middle of the spectrum\, this establishes the coexistenc
e of a delocalized and a localized phase in the critical Erdös-Rényi gra
ph. Joint work with Johannes Alt and Raphael Ducatez.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Pain (New York University)
DTSTART:20210409T120000Z
DTEND:20210409T130000Z
DTSTAMP:20260404T094534Z
UID:MEGA/24
DESCRIPTION:Title: Optimal local law and central limit theorem for beta-ensembles.\n
by Michel Pain (New York University) as part of Séminaire MEGA\n\n\nAbstr
act\nIn this talk\, I will present a joint work with Paul Bourgade and Kri
shnan Mody. We consider beta-ensembles with general potentials (or equival
ently a log-gas in dimension 1)\, which are a generalization of Gaussian b
eta-ensembles and of classical invariant ensembles of random matrices. We
prove a multivariate central limit theorem for the logarithm of the charac
teristic polynomial\, showing that it behaves as a log-correlated field. A
key ingredient is an optimally sharp local law for the the Stieljes trans
form of the empirical measure which can be of independent interest. Both t
he proofs of the CLT and the local law are based essentially on loop equat
ions techniques.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Romain Couillet (CentraleSupélec\, Université Paris-Saclay)
DTSTART:20210507T083000Z
DTEND:20210507T100000Z
DTSTAMP:20260404T094534Z
UID:MEGA/25
DESCRIPTION:Title: Why Random Matrices can Change the Future of Research in AI?\nby
Romain Couillet (CentraleSupélec\, Université Paris-Saclay) as part of S
éminaire MEGA\n\n\nAbstract\nMachine learning and AI algorithms are becom
ing increasingly more powerful but also increasingly more complex\, mathem
atically less tractable\, and energetically less environmental friendly. I
n this talk\, we will demonstrate that large dimensional statistics\, and
particularly random matrix theory\, simultaneously (i) explains why ML alg
orithms are so stable when dealing with large dimensional data\, (ii) mana
ges to break the difficulties that make these algorithms mathematically in
tractable (non-linearities and data modelling)\, thereby (iii) allowing fo
r the first time to get (iii-a) an inside understanding of the algorithms\
, of their multiple biases and\, most crucially\, of their quite counter-i
ntuitive behavior as well as (iii-b) a toolbox to easily improve the algor
ithms performance and cost efficiency. Possibly even more surprisingly\, t
he universality notion in random matrix theory shows (iv) why ML algorithm
s applied to intricate real data (in general impossible to model) behave t
he same as when applied to elementary Gaussian random vector models.\n\nTh
e course will introduce basic notions of random matrix theory by emphasizi
ng on the counter-intuitive behavior of large dimensional data (so to rais
e awareness in the audience). These notions will be applied to a range of
telling applications in machine learning (spectral clustering\, semi-super
vised learning\, transfer learning\, low-cost processing\, etc.).\n\nThe a
udience can dynamically decide on which topic they'd like me to cover pref
erably. A time for debate will also be given for the audience to react on
the presentation. 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/b
ook.html\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Giraud (Université Paris Saclay)
DTSTART:20210507T123000Z
DTEND:20210507T133000Z
DTSTAMP:20260404T094534Z
UID:MEGA/26
DESCRIPTION:Title: Spectral properties of structured random matrices\nby Olivier Gir
aud (Université Paris Saclay) as part of Séminaire MEGA\n\n\nAbstract\nM
otivated by the problem of metal-insulator transition in the Anderson mode
l of condensed matter physics\, I will discuss some spectral properties of
Hermitian Toeplitz\, Hankel\, and Toeplitz-plus-Hankel random matrices wi
th independent identically distributed entries. Spectral statistics of all
these random matrices turns out to be of intermediate type\, as found for
instance at the metal-insulator transition\, or in certain pseudo-integra
ble billiards. Namely\, nearest-neighbor spacing distributions are charact
erized by level repulsion at small distances and an exponential decrease a
t large distances\, while the spectral compressibility has a non-trivial v
alue. Such statistics are usually associated with multifractal eigenstates
\, and I will show that it is also the case here.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tatyana Shcherbina (University of Wisconsin - Madison)
DTSTART:20210507T140000Z
DTEND:20210507T150000Z
DTSTAMP:20260404T094534Z
UID:MEGA/27
DESCRIPTION:Title: Universality for random band matrices\nby Tatyana Shcherbina (Uni
versity of Wisconsin - Madison) as part of Séminaire MEGA\n\n\nAbstract\n
Random band matrices (RBM) are natural intermediate models to study eigenv
alue statistics and quantum propagation in disordered systems\, since they
interpolate between mean-field type Wigner matrices and random Schrodinge
r operators. In particular\, RBM can be used to model the Anderson metal-i
nsulator phase transition (crossover) even in 1d. In this talk we will dis
cuss some recent progress in application of the supersymmetric method (SUS
Y) and transfer matrix approach to the analysis of local spectral characte
ristics of some specific types of 1d RBM.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cécilia Lancien (Université de Toulouse)
DTSTART:20210611T083000Z
DTEND:20210611T100000Z
DTSTAMP:20260404T094534Z
UID:MEGA/28
DESCRIPTION:Title: Mini-course: Quantum expander graphs\nby Cécilia Lancien (Univer
sité de Toulouse) as part of Séminaire MEGA\n\n\nAbstract\nThe goal of t
his lecture is to understand what quantum expander graphs are\, what they
are useful for\, and how they can be constructed. We will first recall the
definition of classical expander graphs\, and explain how quantum analogu
es of these objects can be defined. We will then show that\, both classica
lly and quantumly\, random constructions provide with high probability exa
mples of expander graphs. In the quantum case\, such result is derived fro
m a spectral analysis for random matrix models with a tensor product struc
ture. The presentation will be based\, among others\, on:\n\n-Random unita
ries give quantum expanders. M.B.Hastings. 2007.\n\n- Quantum expanders an
d geometry of operator spaces. G.Pisier. 2014\n\n- Correlation length in r
andom MPS and PEPS. C.Lancien and D.Peréz-García. 2019.\n\n- Characteriz
ing expansion\, classicaly and quantumly. C.Lancien. 2020.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Pastur (B.Verkin Institute for Low Temperature Physics and
Engineering\, Kharkiv\, Ukraine)
DTSTART:20210611T133000Z
DTEND:20210611T143000Z
DTSTAMP:20260404T094534Z
UID:MEGA/29
DESCRIPTION:Title: On Random Matrices Arising in Deep Neural Networks\nby Leonid Pas
tur (B.Verkin Institute for Low Temperature Physics and Engineering\, Khar
kiv\, Ukraine) as part of Séminaire MEGA\n\n\nAbstract\nWe study the dist
ribution of singular values of product of random matrices pertinent to the
analysis of deep neural networks. The matrices resemble the product of th
e sample covariance matrices. However\, an important dierence is that the
analog the of the population covariance matrices\, assumed to be non-rand
om or random but independent of the random data matrix in statistics and r
andom matrix theory\, are now certain functions of random data matrices (s
ynaptic weight matrices in the deep neural network terminology). For the G
aussian synaptic weight matrices the problem has been treated in recent wo
rk [1] and certain subsequent works by using the techniques of free probab
ility theory. Since\, however\, free probability theory deals with populat
ion covariance matrices which are independent of the data matrices\, its a
pplicability to this case has to be justi\ned. We use a version of the tec
hniques of random matrix theory to justify and generalize the results of [
1] to the case where the entries of the synaptic weight matrices are just
independent identically distributed random variables with zero mean and \n
nite fourth moment [2]. This\, in particular\, extends the property of the
so-called macroscopic universality to the considered random matrices.\n\n
[1] J. Pennington\, S. Schoenholz\, and S. Ganguli\, The emergence of spec
tral universality In: Proc. Mach. Learn. Res. (PMLR 70) 84 (2018) 1924-193
2\, http://arxiv.org/abs/1802.09979\n\n[2] L. Pastur and V. Slavin\, On Ra
ndom Matrices Arising in Deep Neural Networks: General I.I.D. Case\, http:
//arxiv.org/abs/2011.11439.\n
LOCATION:https://stable.researchseminars.org/talk/MEGA/29/
END:VEVENT
END:VCALENDAR