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WORKSHOP:
RANDOM PRODUCTMATRICES
BIELEFELD, AUGUST 22ND-26TH 2016
Organizers: Peter J. Forrester ([email protected])Mario Kieburg ([email protected])Roland Speicher ([email protected])
1
Monday (August 22nd 2016)
Time
8:30-9:00 Registration & Opening (8:50)
9:00-9:40Maciej NowakProducts of large random matrices: old laces and new pieces
9:40-10:20Lun ZhangLocal universality in biorthogonal Laguerre ensembles
10:20-11:00 Coffee Break
11:00-11:40Vladislav KarginLimit theorems for products of free variables
11:40-12:20Holger KostersProducts of Independent Bi-Invariant Random Matrices
12:20-15:30 Lunch & Discussion Time
15:30-16:10
Nanda Kishore ReedyLyapunov exponents and eigenvalues of products of randommatrices
16:10-16:40 Coffee Break
16:40-17:20
Yves TourignyLyapunov exponents for products of random matrices of SL(2,R):Impurity models and their continuum limit I.
17:20-18:00Christophe TexierLyapunov exponents for products of random matrices of SL(2,R):Impurity models and their continuum limit II.
Chairmen: Morning Sessions Mario KieburgAfternoon Sessions Arno Kuijlaars
Tuesday (August 23rd 2016)
Time
9:00-9:40Arul LakshminarayanReal eigenvalues of products of real random matrices
9:40-10:20Arno KuijlaarsTransformation results for sums and products of random matrices
10:20-11:00 Coffee Break
11:00-11:40
Dong WangDouble contour integral formulas for the sum of GUE and onematrix model
11:40-12:20
Thorsten NeuschelMoments and Spectral Densities of Singular Value Distributionsfor Products of Gaussian and Truncated Unitary Random Matrices
12:20-15:30 Lunch & Discussion Time
15:30-16:10
Karol ZyczkowskiDistinguishing two generic quantum states and SymmetrizedMarchenko–Pastur distribution
16:10-16:40 Coffee Break
16:40-17:20Florent Benaych-GeorgesSingle Ring Theorem and Spiked Models
17:20-18:00
Octavio ArizmendiOn first and second order free cumulants of products of freerandom variables
Chairmen: Morning Sessions Gernot AkemannAfternoon Sessions tba
Wednesday (August 24th 2016)
Time
9:00-9:40Aris MoustakasApplications of unitary matrices in fibre-optical communications
9:40-10:20Antonia M. Tulino
10:20-11:00 Coffee Break
11:00-11:40Karol A. Penson
11:40-12:20Alice GuionnetLocal statistics for words in several random matrices
12:20-15:30 Lunch & Discussion Time
15:30-16:10Yan FyodorovHow many stable equilibria will a large complex system have?
16:10-16:40 Coffee Break
16:40-17:20Mylene MaıdaLong time behavior of the free Fokker-Planck equation
17:20-18:00
Manon DefosseuxKirillov-Frenkel Character Formula for Loop Groups and RadialPart of Hermitian Brownian Sheet
20:00Conference Dinner atDas Wirtshaus 1802 im BultmannshofKurt-Schumacher-Str. 17a, 33615 Bielefeld
Chairmen: Morning Sessions Yan Fyodorov
Afternoon Sessions Karol Zyczkowski
Thursday (August 25th 2016)
Time
9:00-9:40
Gernot AkemannThe product of two coupled matrices and a new interpolatinghard-edge scaling limit
9:40-10:20Eugene StrahovDynamical correlation functions for products of random matrices
10:20-11:00 Coffee Break
11:00-11:40
Artur SwiechQuaternionic extension of R transform for non-Hermitian randommatrix models
11:40-12:20
Tobias MaiAsymptotic eigenvalue distributions of non-commutative polyno-mials and rational expressions in independent random matrices
12:20-15:30 Lunch & Discussion Time
15:30-16:10Tomasz ChecinskiSum of two complex correlated Wishart matrices
16:10-16:40 Coffee Break
16:40-17:20
Marco BertolaA model of coupled positive matrices; universality results andconjectures
17:20-18:00
Adrien HardyLarge complex correlated Wishart matrices: Local behavior of theeigenvalues
Chairmen: Morning Sessions Alice GuionnetAfternoon Sessions Eugene Strahov
Friday (August 26th 2016)
Time
9:00-9:40Friedrich Gotze
9:40-10:20Alexander N. TikhomirovOn the local laws for product of random matrices
10:20-11:00 Coffee Break
11:00-11:40
Dries StivignySmallest singular value distribution for products of Ginibre randommatrices
11:40-12:20Dang-Zheng LiuPhase transitions of singular values for products of random matrices
12:20-15:30 Lunch & Discussion Time
15:30-16:10Jesper IpsenProducts of real asymmetric Gaussian matrices
16:10-16:40 Coffee Break
16:40-17:20Benoit CollinsOperator norm convergence for the unitary Brownian motion
17:20-18:00Zdzis law BurdaEigenvector statistics of the product of Ginibre matrices
18:00-18:10 Closing
Chairmen: Morning Sessions Benoit CollinsAfternoon Sessions Peter Forrester
Talks
(1) The product of two coupled matrices and anew interpolating hard-edge scaling limitGernot Akemann, Thursday 9:00-9:40Abstract: We consider the singular values of a family of products oftwo Gaussian matrices that are coupled through an Itzykson-Zubertype term. Our ensemble contains both the singular values of onematrix and of the product of two independent matrices as limitingcases. The corresponding point process is determinantal, going beyondthe class of polynomial ensembles. All its correlation functions canbe explicitly determined for finite matrix size N in terms of its kernelof bi-orthogonal functions. In the hard-edge limit at fixed couplingwe recover the limiting Meijer G-kernel valid for two independentmatrices and thus prove its universality for the entire family. In adouble scaling limit where the coupling scales with N we find a newparameter dependent hard-edge kernel that interpolates between theBessel-kernel for one and the aforementioned Meijer G-kernel for twomatrices. These results are based on joint work with Eugene Strahov.
(2) On first and second order free cumulantsof products of free random variablesOctavio Arizmendi, Tuesday 17:20-18:00Abstract: In this talk I describe results on explicit combinatorialformulas for moments and free cumulants of first and second order(in the sense of Speicher and Mingo) of product of free cumulants.Important examples come from products of Gaussian Matrices. Thistalk is based on joint works with Carlos Vargas and Jamie Mingo.
(3) Single Ring Theorem and Spiked ModelsFlorent Benaych-Georges, Tuesday 16:40-17:20Abstract: The Single Ring Theorem, proved by Guionnet, Krish-napur and Zeitouni in 2012 (but conjectured for long by physicists),gives the asymptotic relation between the singular values and theeigenvalues of large non-Hermitian isotropic random matrices (isotropicmatrices are random matrices which are invariant, in law, under theleft and right actions of the unitary group). In this talk, we will statethis result and then explain what happens when the matrix is subjectto a low rank perturbation: at the macroscopic level, the spectrumkeeps the same behavior, but certain eigenvalues (called outliers), can
go away from the bulk of the spectrum and give rise to surprisinggeometrical figures on the plane. Besides, correlations can appearbetween eigenvalues at a macroscopic distance from each other, evenwhen the perturbed matrix is Gaussian, which never happens withHermitian matrix models.
(4) A model of coupled positive matrices; univer-sality results and conjectures.Marco Bertola, Thursday 16:40-17:20Abstract: Universality results in the statistical behaviour of spectraof random matrices typically consider the fluctuations near a macro-scopic point of the spectrum. In an ensemble of positive definitematrices, there are two essentially different distinguished points; thelargest eigenvalues and the smallest. Since they are positive definitethe study of the smallest eigenvalue is the study of the statistics nearthe origin of the spectral axis. The simplest instance is the Laguerreensemble; in this case the fluctuations are described by a determinantrandom point field defined in terms of the celebrated Bessel kernel.This behaviour is also proven in the literature to be “universal”.We consider a model of several coupled matrices of Laguerre type(with a specific interaction) and address the corresponding study ofthe origin of the spectrum. We find a natural generalization of theBessel random point field to a multi-specie analog that involves specialfunctions (Meijer-G). It is then natural to formulate a universalityconjecture. Time permitting I will also discuss results concerning theexpectation of ratios of characteristic polynomials.
(5) Eigenvector statistics of the product of GinibrematricesZdzis law Burda, Monday 17:20-18:00Abstract: We discuss correlations between left and right eigenvectorsof the product of Ginibre matrices. We show how to calculate the left-right eigenvector correlation function for small matrix size N=2,3,...and in the large N limit. The limiting law has a surprisingly simpleform.
(6) Sum of two complex correlated WishartmatricesTomasz Checinski, Thursday 15:30-16:10Abstract: We study the sum of two statistically independent complex
Wishart matrices which are separately correlated by empirical covari-ance matrices. They model the simplest set of time series with timedependent spatial correlations. Thereby we also consider a certainlimit where one of the empirical covariance matrices degenerates. Thisparticular case exhibits a determinantal point process. Our results aregiven in closed form in terms of a supermatrix integral for the k-pointresolvent and in terms of an explicit kernel of the corresponding deter-minantal point process in the case of half degeneracy. Additionally,I present the analytical result of the spectral density and illustrate itwith Monte Carlo simulations.
(7) Operator norm convergence for the unitaryBrownian motionBenoit Collins, Friday 16:40-17:20Abstract: We consider the unitary Brownian motion U
(n)t on the
n-dimensional unitary group. We fix k times t1, . . . , tk and consider a
non-commutative polynomial in the non-commuting variables U(n)t1 , . . . ,
U(n)tk
. We show that the operator norm of P (U(n)t1 , . . . , U
(n)tk
) convergealmost surely in the limit of large dimension, and we explain how tocompute it with free probability. We will also provide motivation forthe study of this problem, and applications. This is joint work withA. Dahlqvist and T. Kemp.
(8) Kirillov-Frenkel Character Formula for LoopGroups and Radial Part of Hermitian BrownianSheetManon Defosseux, Wednesday 17:20-18:00Abstract: We will present in the framework of affine Lie algebraan analogue of the Kirillov/Harish-Chandra/itzykson-Zuber formulaproved by Igor Frenkel and will give an analogue of the closely relatedfamous result which states that a real Brownian motion conditionedin Doob’s sense to remain positive, is distributed as a Bessel 3 process.For this, we will consider the coadjoint action of a Loop group of acompact group on the dual of the corresponding centrally extendedLoop algebra and show that a Brownian motion in a Cartan subal-gebra conditioned to remain in an affine Weyl chamber is distributedas the radial part process of a Brownian sheet on the underlying Liealgebra.
(9) How many stable equilibria will a largecomplex system have?Yan Fyodorov, Wednesday 15:30-16:10Abstract: We aim to provide the quantitative answer to the classicalquestion posed by Robert May (1972) “Will a Large Complex Systembe Stable?”. To this end we analyse a generic autonomous nonlinearsystem of N � 1 randomly coupled degrees of freedom relaxing withthe common rate µ > 0. We show that with decreasing rate µ suchsystems experience an abrupt transition at some critical value µ = µCfrom a trivial phase portrait with a single stable equilibrium into atopologically non-trivial ‘absolute instability’ regime for µB < µ <µC where equilibria are exponentially abundant, but typically all ofthem are unstable. Finally, at even smaller relaxation rate µ < µBstable equilibria become exponentially abundant, but their fractionto totality of all equilibria remains exponentially small. The revealedpicture goes much beyond the May’s linear analysis and is expectedto be of relevance in the applications of complex systems to ecology,population biology, neural network theory and other areas. This is ajoint work with Gerard Ben Arous and Boris Khoruzhenko.
(10) tbaFriedrich Gotze, Friday 9:00-9:40Abstract: tba
(11) Local statistics for words in several randommatricesAlice Guionnet, Wednesday 11:40-12:20Abstract: I will discuss the use of approximate transport techniquesto study the local fluctuations of perturbative words in several matrices.
(12) Large complex correlated Wishart matrices:Local behavior of the eigenvaluesAdrien Hardy, Thursday 17:20-18:00Abstract: The aim of this talk is to provide an overview on the localbehavior of the eigenvalues of complex Wishart matrices at differentregimes of interest. In a few words, the Tracy-Widom law appears atthe soft edges of the limiting spectrum, and processes involving Besseland Pearcey kernels arise at the hard edge and cusp points respec-tively. The fluctuations for the condition number are also considered,
and open problems as well. Joint works with W. Hachem and J.Najim.
(13) Products of real asymmetric Gaussian matricesJesper Ipsen, Friday 15:30-16:10Abstract: A study of the eigenvalues of a real asymmetric Gaussianmatrix (also known as a real Ginibre matrix) has been a famouslychallenging problem in random matrix theory. One of the difficultieswith these matrices is that their eigenvalues come as a mixture ofboth complex conjugate pairs and unpaired real eigenvalues. In thistalk, we provide a summary of recent results for the eigenvalues ofproducts of such real asymmetric matrices. Our focus will be onintegrable properties available for an arbitrary number factors witharbitrary matrix dimensions. In particular, we will see how the jointdensity may be found and (skew-)orthogonalised. This allows us tolook at correlation functions as well as the probability of finding acertain number of real eigenvalues. Different known and conjecturedasymptotic properties will be discussed.
(14) Limit theorems for products of free variablesVladislav Kargin, Monday 11:00-11:40Abstract: I will talk about recent results on the limit laws forproducts of free random variables. In particular, I will talk aboutthe growth of the largest exponents and about the limit laws for thedistribution of singular values. I will review known results and suggestsome open questions.
(15) Lyapunov exponents and eigenvalues of productsof random matricesNanda Kishore Reedy, Monday 15:30-16:10Abstract: We will discuss the equality of asymptotic distributionsof singular values and eigenvalues of product random matrices. Weshall briefly discuss the existing knowledge of Lyapunov exponents forproduct random matrices.
(16) Products of Independent Bi-Invariant RandomMatricesHolger Kosters, Monday 11:40-12:20
Abstract: The investigation of products of independent randommatrices has a long history, dating back to work by Furstenberg inthe early 1960s. Interest in this area has resurged in recent years,sparked by explicit results on the joint spectral densities of certainrandom product matrices.
My talk will be concerned with products of independent bi-unitarilyinvariant complex random matrices as well as the associated jointsingular value and eigenvalue densities. In particular, I will discussseveral transformation formulas for the class of polynomial ensemblesof derivative type. These transformation formulas make it possible tore-derive and generalize a number of results on products of independentGinibre or truncated unitary random matrices from the last few years.
This is based on joint work with Mario Kieburg.
(17) Transformation results for sums and productsof random matricesArno Kuijlaars, Tuesday 9:40-10:20Abstract: Since the remarkable works of Akemann, Ipsen, Kieburgand Wei [1,2], it is known that singular values of products of complexGinibre matrices have a determinantal structure. This fact was inter-preted in [8] as a transformation result for polynomial ensembles,see also [6,7]. Recently, Kieburg and Kosters [4,5] gave a profoundgroup theoretical approach in terms of the spherical functions for theGelfand pair (GL(n,C), U(n)).
I will give an overview of these results for products of randommatrices, as well as a similar constructions for sums of random matrices[3] with their group theoretical interpretation as spherical functionsfor a Gelfand pair.[1] G. Akemann, J. Ipsen and M. Kieburg, Products of rectangular
random matrices: singular values and progressive scattering, Phys.Rev. E 88 (2013), 052118.
[2] G. Akemann, M. Kieburg, and L. Wei, Singular value correlationfunctions for products of Wishart random matrices, J. Phys. A46 (2013), 275205.
[3] T. Claeys, A.B.J. Kuijlaars, and D. Wang, Correlation kernelsfor sums and products of random matrices, Random MatricesTheory Appl. 4 (2015), article nr. 1550017, 31 pp.
[4] M. Kieburg and H. Kosters, Exact relation between the singularvalue and eigenvalue statistics, arXiv:1601.02586.
[5] M. Kieburg and H. Kosters, Products of random matrices frompolynomial ensembles, arXiv:1601.03724.
[6] M. Kieburg, A.B.J. Kuijlaars, and D. Stivigny, Singular valuestatistics of matrix products with truncated unitary matrices,Int. Math. Res. Not. 2016 (2016), no.11, pp. 3392-3424.
[7] A.B.J. Kuijlaars, Transformations of polynomial ensembles, in”Modern Trends in Constructive Function Theory” (D.P. Hardin,D.S. Lubinsky and B. Simanek, eds.) Contemp. Math. 661,2016, pp. 253-268.
[8] A.B.J. Kuijlaars and D. Stivigny, Singular values of products ofrandom matrices and polynomial ensembles, Random MatricesTheory Appl. 3 (2014), article nr. 1450011, 22 pp.
(18) Real eigenvalues of products of real randommatricesArul Lakshminarayan, Tuesday 9:00-9:40Abstract: Talk reviews an initial motivation from quantum mechanicsfor studying the number of real eigenvalues of products of two randommatrices, and lays out possible generalizations of interest. It is nowunderstood that products of Gaussian random matrices tend to havemore real eigenvalues as the number of terms in the product increases.Numerical experiments and some analytical results are presented formore general measures, which indicates, at the same time, the specialnature of the Gaussian ensemble and the generality of the phenomenonthat all the eigenvalues in products tend to be real.
(19) Phase transitions of singular values for productsof random matricesDang-Zheng Liu, Friday 11:40-12:20Abstract: In this talk we are devoted to describe phase transitionsof squared singular values appearing recently in products of randommatrices. For products of independent Gaussian random matrices,but only one of which has a non-zero mean matrix, a phase transitionphenomenon at the origin is observed from Meijer G-kernels to a newfamily of critical Meijer G-kernels. However, for products of twocoupled Gaussian random matrices, the limiting correlation kernelis proved to admit a four-term phase transition phenomenon at theorigin, which are in turn Meijer G-kernel associated with products oftwo independent Gaussian matrices, a new interpolating kernel, theperturbed Bessel kernel and finite coupled product kernel. This isbased on joint work with P.J. Forrester.
(20) Asymptotic eigenvalue distributions of non-commutative polynomials and rational expres-sions in independent random matricesTobias Mai, Thursday 11:40-12:20
Abstract: Free probability theory, which was invented around 1985by D. Voiculescu, can be seen as a highly non-commutative analogueof classical probability theory, that comes with its own notion ofindependence, the so-called free independence. It was originally intendedto serve as a tool for the theory of operator algebras, where it openeda completely new point of view on some very intricate problems,since it allowed to treat them in probabilistic terms. But later on,deep and fascinating connections also to random matrix theory werefound. More precisely, it turned out that free independence governsthe asymptotic behavior of independent random matrices of manytypes as their dimension tends to infinity. This provided in particularan effective way to understand the asymptotic eigenvalue distributionof sums and products of independent random matrices of many types.However, there was for a long time no machinery for dealing in auniform way with more general non-commutative polynomials (not tomention rational expressions) in independent random matrices. Fornon-commutative polynomials, this problem was solved in 2013 injoint work with S. Belinschi and R. Speicher, and the more generalcase of rational expressions was treated in 2015 in joint work withJ. W. Helton and R. Speicher. These solutions build both on a verynice interplay between the analytic machinery of (operator-valued)free probability on one side and the purely algebraic theory of linearrepresentations (aka realizations or linearizations) on the other side.In my talk, I will explain these results and I will show by someconcrete examples that the obtained algorithms are easily accessiblefor numerical computations.
(21) Long time behavior of the free Fokker-Planck equationMylene Maıda, Wednesday 16:40-17:20Abstract: In this talk, we will explain how ideas and techniquesfrom free probability or orthogonal polynomials helped us to tacklethe convergence to equilibrium of the granular media equation withlogarithmic interaction (a.k.a. Fokker-Planck or non linear McKean-Vlasov equation), for some non-convex potentials. This is joint workwith Catherine Donati-Martin and Benjamin Groux.
(22) Applications of unitary matrices in fiber-optical communicationsAris Moustakas, Wednesday 9:00-9:40Abstract: A promising new technology in fibre-optical communi-cations entails the simultaneous transmission of information through
multiple modes or cores of the fibre. Due to imperfections the trans-mitted signal is mixed, as a result of which the channel can be describedby a random unitary matrix. In this work we apply random matrixtheory to obtain analytic estimates of the statistics of the informationthroughput and the associated error probability.
(23) Moments and Spectral Densities of SingularValue Distributions for Products of Gaussianand Truncated Unitary Random MatricesThorsten Neuschel, Tuesday 11:40-12:20Abstract: We study moments and densities of limiting distributionsof singular values of large dimensional matrix products composedof independent complex Gaussian (complex Ginibre) and truncatedunitary matrices which are taken from Haar distributed unitary matriceswith appropriate dimensional growth. It turns out that the momentscan be expressed by Jacobi polynomials with varying parameterswhereas the densities admit explicit integral representations with elem-entary integrands. The derivation is based on an approach to obtaincomplex integral representations for densities of measures whose Stieltjestransforms satisfy algebraic equations of a certain type.
(24) Products of large random matrices: oldlaces and new piecesMaciej Nowak, Monday 9:00-9:40Abstract: We recall how to extend the formalism of random matrixtheory to evaluate spectral properties of products of large complex,hermitian or unitary matrices. We comment on the emergence of a“topological phase transition” in the limit when number of matricestends to infinity and on the link between this transition and so-calledinviscid “Burgulence” in the spectral flow of eigenvalues. Then, wepoint at the crucial role of eigenvector correlations for non-hermitianproblems. In particular, we extend the so-called “single ring theorem”,also known as Haagerup-Larsen theorem, providing a universal formulafor the mean eigenvalue condition number. We give several examplesand we present cross-checks of analytic prediction by large scale numerics.
(25) tbaKarol A. Penson, Wednesday 11:00-11:40
Abstract: tba
(26) Smallest singular value distribution for productsof Ginibre random matricesDries Stivigny, Friday 11:00-11:40Abstract: We study the distribution of the smallest (squared) singularvalue for products of Ginibre random matrices, in the limit where thesize of the matrices becomes large. The limiting distributions thatwe will study can be expressed as Fredholm determinants of certainintegral operators, and generalize in a natural way the extensivelystudied hard edge Bessel kernel determinant. We will express thelogarithmic derivative of those Fredholm determinants identically interms of a 2x2 Riemann-Hilbert problem, and use this representationto obtain the so-called large gap asymptotics. This is joint work withTom Claeys and Manuela Girotti.
(27) Dynamical correlation functions for productsof random matricesEugene Strahov, Thursday 9:40-10:20Abstract: I will talk about a family of random processes with adiscrete time related to products of random matrices (product matrixprocesses). Such processes are formed by singular values of randommatrix products, and the number of factors in a random matrixproduct plays a role of a discrete time. The correlation functions ofthese processes can be understood as dynamical correlation functionsfor products of random matrices.
I will explain that in certain cases product matrix processes arediscrete-time determinantal point processes, whose correlation kernelscan be expressed in terms of double contour integrals. In particular,this enables to compute the hard edge scaling limits of the dynamicalcorrelation functions for products of random matrices.
(28) Quaternionic extension of R transform fornon-Hermitian random matrix modelsArtur Swiech, Thursday 11:00-11:40Abstract: We rephrase the non-Hermitian planar diagrammatic form-alism using the Cayley-Dickson construction to generalize the freeprobability calculus to asymptotically large non-Hermitian randommatrices. The main object in this generalization is a quaternionic
extension of the R transform which is a generating function for noncross-ing cumulants. We demonstrate that the quaternionic R transformgenerates all connected averages of all distinct powers of a randommatrix and its Hermitian conjugate in the infinite matrix size limit.We show that the R transform for Gaussian elliptic laws, the equiv-alent of semicircle law for Hermitian matrices, is given by a linearquaternionic map described by a total of 5 real parameters. Wedemonstrate how to use the quaternionic R transform to calculatethe limiting eigenvalue densities of sums and products of determin-istic and Gaussian random matrices in a few examples.
(29) Lyapunov exponents for products of randommatrices of SL(2,R): Impurity models andtheir continuum limit II.Christophe Texier, joint talk with Yves Tourigny,Monday 17:20-18:00Abstract: Product of random matrices arise in many models ofstatistical physics; for example in random spin chains or in quantumlocalisation problem. An important quantity associated with theproduct is the Lyapunov exponent, which measures its growth, andoften has a physical interpretation: for instance, in a spin chain, it isthe free energy per spin; in quantum localisation problems, it is thereciprocal of the localisation length.
We shall describe a class of quantum disordered systems called“impurity models” that lead naturally to products of 2x2 matrices.Such products may be viewed as random walks in the Lie groupSL(2,R). We shall use these impurity models to highlight some ofthe ideas and concepts that are useful in the study of random walkson Lie groups: decompositions, invariant measures, transforms and”continuum” limits.
(30) On the local laws for product of randommatricesAlexander N. Tikhomirov, Friday 9:40-10:20Abstract: This talk will review the local limit theorems for thedistribution of eigenvalues and singular values of products of randommatrices.
(31) Lyapunov exponents for products of randommatrices of SL(2,R): Impurity models andtheir continuum limit I.Yves Tourigny, joint talk with Christophe Texier,Monday 16:40-17:20Abstract: See abstract of Christophe Texier.
We show that, in the continuum limit, the Lyapunov exponentof the product may be systematically obtained by considering theHilbert transform of the invariant measure of the matrix product.The Lyapunov exponent is shown to present a scaling form. Thisgeneral analysis allows us, among other things, to recover in a unifiedframework few results known previously from exactly solvable modelsof one-dimensional disordered systems and find several new ones,providing a classification of possible solutions for such 1D models.Finally the fluctuations of random matrix product are discussed: fortwo particular choices of random matrix products, one gets explicitintegral representations of the variance of the logarithm of the matrixproduct. The application for quantum localisation problem is discussed.
(32) tbaAntonia M. Tulino, Wednesday 9:40-10:20Abstract: tba
(33) Double contour integral formulas for thesum of GUE and one matrix modelDong Wang, Tuesday 11:00-11:40Abstract: In the recent development of random matrix theory,especially the product of random matrices, double contour integralformulas are often economical ways to express the exact formulasof the eigenvalue correlation kernels. They can also be handy toolsto derive the limiting correlation kernels. We introduce the doublecontour integral formula for the eigenvalue correlation kernel of arandom Hermitian matrix that is the sum of a GUE random matrixand an independent random matrix in a so-called one matrix model.Then we show the computation of the limiting eigenvalue correlationkernel. This is joint work with Tom Claeys, Arno Kuijlaars and KarlLiechty.
(34) Local universality in biorthogonal LaguerreensemblesLun Zhang, Monday 9:40-10:20Abstract: In this talk, we consider n particles x1, . . . , xn ∈ [0,∞),distributed according to a probability measure of the form
1
Zn
∏1≤i<j≤n
(xj − xi)∏
1≤i<j≤n
(xθj − xθi )n∏j=1
xαj e−xjdxj, α > −1, θ > 0,
where Zn is the normalization constant. This distribution arises inthe context of modeling disordered conductors in the metallic regime,and can also be realized as the distribution for squared singular valuesof certain triangular random matrices. We give a double contourintegral formula for the correlation kernel, which allows us to establishuniversality for the local statistics of the particles, namely, the bulkuniversality and the soft edge universality via the sine kernel andthe Airy kernel, respectively. In particular, our analysis also leadsto new double contour integral representations of scaling limits atthe origin (hard edge), which are different from those found in theclassical work of Borodin. We next show that, if θ is rational, i.e.,θ = m
nwith m,n ∈ N, gcd(m,n) = 1, and α > m − 1 − m
n, the hard
edge scaling limit is integrable in the sense of Its-Izergin-Korepin-Slavnov. We then come to the Fredholm determinant of this limitover the union of several scaled intervals, which can also be inter-preted as the gap probability (the probability of finding no particles)on these intervals. From the integrable structure, we obtain a systemof coupled partial differential equations associated with the corre-sponding Fredholm determinant as well as a Hamiltonian interpre-tation. As a consequence, we are able to represent the gap proba-bility over a single interval (0, s) in terms of a solution of a systemof nonlinear ordinary differential equations and to derive its small sasymptotics.
(35) Distinguishing two generic quantum statesand Symmetrized Marchenko–Pastur distri-butionKarol Zyczkowski, Tuesday 15:30-16:10Abstract: Properties of random density matrices of sizeN distributeduniformly with respect to the Hilbert-Schmidt measure are investi-gated. We show that for large N , due to the concentration of measurephenomenon, the trace distance between two random states tendsto a fixed number 1/4 + 1/π, which yields the Helstrom bound ontheir distinguishability. To arrive at this result we apply free randomcalculus and derive the symmetrized Marchenko–Pastur distribution.
Asymptotic value for the root fidelity between two random states,√F = 3/4, can serve as a universal reference value for further theoretical
and experimental studies. Analogous results for quantum relativeentropy and Chernoff quantity provide other bounds on the distin-guishablity of both states in a multiple measurement setup due to thequantum Sanov theorem. Entanglement of a generic mixed state of abi–partite system is estimated.
List of Participants
Name Affiliation E-Mail Address
Gernot Akemann Bielefeld University [email protected]
Giusi Alfano Politec. di Torino [email protected]
Octavio Arizmendi CIMAT (Guanajuato) [email protected]
Theodoros Assiotis University of Warwick [email protected]
Florent Benaych-Georges Universite Paris Descartes [email protected]
Tristan Benoist Universite Paul Sabatier [email protected](Toulouse)
Marco Bertola Concordia University [email protected](Montreal/Canada) and
SISSA/ISAS (Italy)
Zdzis law Burda AGH (Krakow) [email protected]
Tomasz Checinski Bielefeld University [email protected]
Benoit Collins Kyoto University [email protected]
Manon Defosseux Universite Paris Descartes [email protected]
Peter J. Forrester The University of Melbourne [email protected]
Yan Fyodorov Kings College London [email protected]
Friedrich Gotze Bielefeld University [email protected]
Aurelien Grabsch Universite Paris-Saclay [email protected]
Ewa Gudowska-Nowak Jagiellonian University [email protected]
Alice Guionnet Ecole Normale [email protected] de Lyon
Adrien Hardy Universite des Sciences [email protected] Technologies de Lille
Jesper Ipsen The University of Melbourne [email protected]
Christopher H. Joyner Queen Mary University [email protected] London
Vladislav Kargin Palo Alto [email protected]
Mario Kieburg Bielefeld University [email protected]
Nanda Kishore Reedy Indian Institute of Science) [email protected](Bangalore
Holger Kosters Bielefeld University [email protected]
Valerie Kovaleva Moscow Institute of [email protected] and Technology
Arno Kuijlaars KU Leuven [email protected]
Sushma Kumari Kyoto University [email protected]
Arul Lakshminarayan IIT Madras [email protected]
Dang-Zheng Liu University of Science and [email protected] of China (Hefei)
Tobias Mai Saarland University [email protected]
Mylene Maıda Universite des Sciences et [email protected] de Lille
Name Affiliation E-Mail Address
Leslie Molag KU Leuven [email protected]
Aris Moustakas National & Capodistrian [email protected] of Athens
Alexey Naumov Lomonosov Moscow State [email protected]
Thorsten Neuschel Universite catholique [email protected] Louvain
Maciej A. Nowak Jagiellonian University [email protected]
Karol A. Penson Universite Pierre et [email protected] Curie
Miguel Angel Pluma Saarland University [email protected]
Tulasi Ram Reddy Indian Statistical tulasi [email protected] Institute (Bangalore)
Koushik Saha Indian Institute of [email protected] Bombay
Roland Speicher Saarland University [email protected]
Dries Stivigny KU Leuven [email protected]
Eugene Strahov The Hebrew University [email protected] Jerusalem
Artur Swiech University of Cologne [email protected]
Wojciech Tarnowski Jagiellonian University [email protected]
Christophe Texier Universite Paris-Sud [email protected]
Alexander N. Komi Research Center of [email protected] Ural Division of RAS
Yves Tourigny University of Bristol [email protected]
Petri Tuisku University of Helsinki [email protected]
Antonia M. Tulino Nokia Bell Labs (New Jersey/USA) [email protected] Universita di Napoli Federico II (Italy)
Dong Wang National University of Singapore [email protected]
Sheng Yin Saarland University [email protected]
Lun Zhang Fudan University [email protected]
Karol Zyczkowski Jagiellonian University [email protected]
