BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Martin Vogel (Université de Strasbourg) DTSTART:20200515T120000Z DTEND:20200515T130000Z DTSTAMP:20260422T212705Z 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:20200515T133000Z DTEND:20200515T143000Z DTSTAMP:20260422T212705Z 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:20200605T120000Z DTEND:20200605T130000Z DTSTAMP:20260422T212705Z 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:20200605T133000Z DTEND:20200605T143000Z DTSTAMP:20260422T212705Z 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:20201113T130000Z DTEND:20201113T140000Z DTSTAMP:20260422T212705Z 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:20201113T143000Z DTEND:20201113T153000Z DTSTAMP:20260422T212705Z 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:20201113T093000Z DTEND:20201113T110000Z DTSTAMP:20260422T212705Z 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:20201211T093000Z DTEND:20201211T110000Z DTSTAMP:20260422T212705Z 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:20201211T130000Z DTEND:20201211T140000Z DTSTAMP:20260422T212705Z 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:20201211T143000Z DTEND:20201211T153000Z DTSTAMP:20260422T212705Z 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:20210115T093000Z DTEND:20210115T110000Z DTSTAMP:20260422T212705Z 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:20210115T140000Z DTEND:20210115T150000Z DTSTAMP:20260422T212705Z 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:20210115T153000Z DTEND:20210115T163000Z DTSTAMP:20260422T212705Z 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:20210205T093000Z DTEND:20210205T110000Z DTSTAMP:20260422T212705Z 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:20210312T130000Z DTEND:20210312T140000Z DTSTAMP:20260422T212705Z 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:20210312T143000Z DTEND:20210312T153000Z DTSTAMP:20260422T212705Z 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:20210205T130000Z DTEND:20210205T140000Z DTSTAMP:20260422T212705Z 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:20210205T143000Z DTEND:20210205T153000Z DTSTAMP:20260422T212705Z 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:20210611T120000Z DTEND:20210611T130000Z DTSTAMP:20260422T212705Z 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:20210312T093000Z DTEND:20210312T110000Z DTSTAMP:20260422T212705Z 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:20210409T083000Z DTEND:20210409T100000Z DTSTAMP:20260422T212705Z 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:20210409T133000Z DTEND:20210409T143000Z DTSTAMP:20260422T212705Z 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:20210409T120000Z DTEND:20210409T130000Z DTSTAMP:20260422T212705Z 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:20210507T083000Z DTEND:20210507T100000Z DTSTAMP:20260422T212705Z 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:20210507T123000Z DTEND:20210507T133000Z DTSTAMP:20260422T212705Z 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:20210507T140000Z DTEND:20210507T150000Z DTSTAMP:20260422T212705Z 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:20210611T083000Z DTEND:20210611T100000Z DTSTAMP:20260422T212705Z 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:20210611T133000Z DTEND:20210611T143000Z DTSTAMP:20260422T212705Z 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