Bulk scaling limits, open questions
|
|
- Clemence Janel Stafford
- 5 years ago
- Views:
Transcription
1 Bulk scaling limits, open questions Based on: Continuum limits of random matrices and the Brownian carousel B. Valkó, B. Virág. Inventiones (2009). Eigenvalue statistics for CMV matrices: from Poisson to Clock via random matrix ensembles R. Killip, M. Stoiciu. Duke Math. Journal (2009).
2 Back to the β-hermite ensembles Bulk scaling P β (λ 1, λ 2,..., λ n ) = 1 Z n,β e β/4 n k=1 λ2 k For GUE we have the bulk scaling limit ( 1 (s t n) m ρm t + x 1,..., t + x ) m det s t n s t n ( j<k λ j λ k β. sin π(x i x j ) π(x i x j ) and similarly for GOE/GSE. Here, s t = 1 4 t 2 is the density of the limiting semi-circle law. 2π ) 1 i,j m,
3 Diffusion representation for general β > 0 Valkó and Virág prove the existence of a point process Sine β following holds. such that for Let Λ n be the point process corresponding to P β with Then, n 1/6 (2 n µ n ). 4n µ 2 n (Λ n µ n ) Sine β. and take any sequence µ n The Sine β process has a full description as a functional of Brownian motion on the hyperbolic plane, I ll just describe its marginals.
4 Counting points of Sine β Introduce the one-dimensional process t α t dα t = λf(t)dt + 2 sin(α t /2)db t for f(t) = (β/4) e βt/4 and α 0 = 0. As t, α t a (random) integer multiple of 2π. Then: Let law N(λ) be the number of points of Sine β in [0, λ]. There is the equality in N(λ) = α( )/2π. This is the n description for the asymptotic number of eigenvalues for the β-hermite ensemble in a window of length λ/ n.
5 Phase equations The idea is to study the limiting recursion equations (this is like doing the Riccati directly on the discrete model). Conjugate the β-hermite model to produce an equivalent random tridiaonal where H n = X n s n + Y n 1 s n 1 X n 1 s n 1 + Y n 1 s n 2 X n X j = g j, s j = j 1/2, Y j = χ2 β(j 1) s j. β βs j 1 Fix λ and let u l = u l,λ be a solution of s n l u l + X n l u l+1 + (Y n l + s n l )u l+2 = λu l+1, 0 l n 2 with u 0 = 0, u 1 = 1. Fact: λ is an eigenvalue only if u n+1 = 0.
6 Put r l = u l+1 /u l which solves r l+1 = ( 1 r l + Centered phase λ s n l X n l s n l ) ( 1 + Y n l s n l ) 1, starting at r 0 = (feels a bit more Riccati). Now, λ is an eigenvalue if r n = 0. What they show is that a mollified version of the phase θ l = arctan(r l ) converges to the process defined by the stochastic differential equation introduced above.
7 β-circular ensembles Turn to the measure on (θ 1, θ 2,..., θ n ) [0, 2π) n defined by the density P β (θ 1,..., θ n ) = 1 e iθ j e iθ k Z β j<k β These should be considered as the general β versions of the eigenvalue ensembles of the classical compact groups (O(n), U(n)...) introduced in P. Diaconis talks. For these objects, it is all bulk. Rather than tridiagonal matrix models, one has a family of five-diagonal matrices which produce these laws.
8 CMV Matrices Given a sequence α 0, α 1, D (the open unit disk in the complex plane), define the 2 2 matrices where ρ k = Ξ k = [ ] ᾱk ρ k ρ k α k 1 α k 2. From these build the block diagonals L = diag where Ξ 1 = [1] ( Ξ 0, Ξ 2, Ξ 4,... ) and M = diag Then, the CMV matrix associated to α 0, α 1,... is C(α) = LM. ( Ξ 1, Ξ 1, Ξ 3,... An important discovery of Killip and I. Nenciu (IMRN 2004) is that for a certain choice of random α s, the eigenvalues of C(α) are distributed according to circular beta ensemble. ),
9 Identifying the limit CMV matrices are connected to Orthogonal Polynomials on the unit circle in a way that Jacobi (tridiagonal) matrices are linked to OPs on the line. Following the phase Killip-Stoiciu prove: For any x consider dψ(x, t) = xdt + 2 Im βt { [e iψ(x,t) 1][db 1 (t) + idb 2 (t)] }. At t = 1 this is an increasing function of x. Then, with Ψ(x) = ψ 1 (x, t = 1) and any f C 0 : n ] [ lim E β [e 2π f(nθ k) k=1 = E exp n 0 ( m Z f Ψ(2πm + ω) ) ] dω.
10 Problem #1: Equivalence of bulk limits Show that the limiting beta bulk statistics as described by Valko-Virág and Killip-Stoiciu are the same. Granted these start with different types of β-ensembles but for β = 1, 2, 4 the bulk limits are the same for G{O/U, S}E and C{O/U/S}E. Namely you see sine kernel.
11 Problem #2: Compute anything Does the general T W β Painlevé II? distribution function have description in terms of Maybe a hint: Want to compute lim lim P a (p(x, λ, β) does not explode for x L a L By the Cameron-Martin formula, can write this probability as in p(0)=a β 4 e L 0 [λ+x p 2 (x)]dp(x) β 8 L 0 [λ+x p 2 (x)] 2 dx e β 8 L 0 ) [p (x)] 2 dx d p. (2π0 + ) /2 By Itô, the first exponent only gives a boundary contribution. Thus we expect the path to reside (in a neighborhood) of where the functional p L 0 [ (λ + x p 2 (x)) 2 + (p (x)) 2] dx is minimized. The associated Euler-Lagrange equation is Painlevé II.
12 Problem #2: Compute anything, con t The Hard Edge is even more frustrating. Just from the density: one can see: P β,a (λ 1,..., λ n ) = 1 λ j λ k β Z β,a j<k n 1 k=0 λ β 2 (a+1) 1 k e β 2 λ k Whenever β 2 (a + 1) = 1 one has nλ min exp(β/2). Where is this in the limiting operator description? Recall there we have speed(dx) = exp [ ] (a + 1)x 2 b(x) β dx, scale(dx) = exp [ ] ax + 2 b(x) β dx. The only thing special I see when β 2 (a + 1) = 1 is that the speed measure (invariant measure) is a martingale.
13 Problem #3: Putting in time There is a dynamical version of GUE (Dyson s Brownian motion). replace each independent real component with a Brownian motion: M jk = m r jk + 1m c jk br jk (t) + 1b c jk (t) Namely, This gives reproduces GUE at t = 1 (alternatively could make a stationary version with Ornstein Uhlenbeck rather than Brownian entries.) More interestingly, the n-eigenvalues t (λ 1 (t),..., λ n (t)) performs a diffusion (is Markov) and a t n 1/6 ( λ max (t) b t n ) Airy(t), a continuous-time process with Tracy-Widom marginals. Should correspond to an evolution of Random Airy Operators. My conjecture is H β (t) = d2 dx + x + 2 B(x, t) 2 β where B(x, t) is White in space and Ornstein Uhlenbeck (!?) in time.
14 Problem #4: Proving the other half At the edges we get a limit operator and then a Riccati substitution gives a diffusion that counts eigenvalues. So... Can you get the edge diffusions directly from a limit theorem on the tridiagonal recursions? More interesting. In the bulk they went directly to the recursions (or phase function). Is there a continuum random operator description of the bulk?
15 Problem #5: Full hard to soft transition We proved a 2 Λ hard (2a, β) via the Riccati diffusions. a 4/3 Λ soft (β) Should be the case that the full hard-edge point processes converge to the soft edge process. Appears painful (and a bit besides the point) to prove this via the Riccati business. More satisfying would be a direct understanding of why (the rescaled) [ ] ) spec ( e x d 2 dx (a + 2 b (x)) d 2 β dx goes over to ( d 2 spec ) b (x) β dx + x as a. In particular, can we get convergence of the resolvents?
16 Our hard-edge operator Problem #6: Connections to RWRE [ G β,a = e x d 2 dx (a + 2 b (x)) d 2 β dx is intimately connected to the study of Random Walks in Random Environment it is practically Brox s example of a continuum version of Sinai s walk. It is a basic fact that: if X t is the process generated by G β,a Hard to soft transition implies 1 Prove it this way. lim a 1 lim t t log P(T 0 > t) = Λ 0 ( G β,a ). a lim 4/3 t ( ) 1 t log e a2t P(T 0 > t) = T W β. More interesting: Is there a qualitatively connected family of RWRE s that satisfy such a thing. ]
17 Problem #7: As a mechanism to prove Universality Wigner matrices: Can you understand enough about the tridiagonal process for general Wigner matrices to invoke our CLT spectral convergence result to get T W? Non-central Wisharts: The largest eigenvalues of sample covariance matrices of type XX T for i.i.d. X are known to have T W limits. If you take the non-central case XΣX T for even diagonal but non-iid Σ interesting things can happen. Let Σ = (r,..., r, 1, 1,..., 1): Then see T W if r < r c, a k fold GUE if r > r c and a new distribution at the critical r c. (Baik, Ben Arou, Peche.) Only can do this for β = 2. The conjecture is that what you get here is the Stochastic Airy Operator with different boundary conditions. TASEP, LPP, etc: Can you find Stochastic Airy or the attached diffusion in any of these other models??
Painlevé Representations for Distribution Functions for Next-Largest, Next-Next-Largest, etc., Eigenvalues of GOE, GUE and GSE
Painlevé Representations for Distribution Functions for Next-Largest, Next-Next-Largest, etc., Eigenvalues of GOE, GUE and GSE Craig A. Tracy UC Davis RHPIA 2005 SISSA, Trieste 1 Figure 1: Paul Painlevé,
More informationUniversality of distribution functions in random matrix theory Arno Kuijlaars Katholieke Universiteit Leuven, Belgium
Universality of distribution functions in random matrix theory Arno Kuijlaars Katholieke Universiteit Leuven, Belgium SEA 06@MIT, Workshop on Stochastic Eigen-Analysis and its Applications, MIT, Cambridge,
More informationDifferential Equations for Dyson Processes
Differential Equations for Dyson Processes Joint work with Harold Widom I. Overview We call Dyson process any invariant process on ensembles of matrices in which the entries undergo diffusion. Dyson Brownian
More informationExponential tail inequalities for eigenvalues of random matrices
Exponential tail inequalities for eigenvalues of random matrices M. Ledoux Institut de Mathématiques de Toulouse, France exponential tail inequalities classical theme in probability and statistics quantify
More informationUpdate on the beta ensembles
Update on the beta ensembles Brian Rider Temple University with M. Krishnapur IISC, J. Ramírez Universidad Costa Rica, B. Virág University of Toronto The Tracy-Widom laws Consider a random Hermitian n
More information1 Tridiagonal matrices
Lecture Notes: β-ensembles Bálint Virág Notes with Diane Holcomb 1 Tridiagonal matrices Definition 1. Suppose you have a symmetric matrix A, we can define its spectral measure (at the first coordinate
More informationRANDOM MATRIX THEORY AND TOEPLITZ DETERMINANTS
RANDOM MATRIX THEORY AND TOEPLITZ DETERMINANTS David García-García May 13, 2016 Faculdade de Ciências da Universidade de Lisboa OVERVIEW Random Matrix Theory Introduction Matrix ensembles A sample computation:
More informationMarkov operators, classical orthogonal polynomial ensembles, and random matrices
Markov operators, classical orthogonal polynomial ensembles, and random matrices M. Ledoux, Institut de Mathématiques de Toulouse, France 5ecm Amsterdam, July 2008 recent study of random matrix and random
More informationUniversality for random matrices and log-gases
Universality for random matrices and log-gases László Erdős IST, Austria Ludwig-Maximilians-Universität, Munich, Germany Encounters Between Discrete and Continuous Mathematics Eötvös Loránd University,
More informationUniversality of local spectral statistics of random matrices
Universality of local spectral statistics of random matrices László Erdős Ludwig-Maximilians-Universität, Munich, Germany CRM, Montreal, Mar 19, 2012 Joint with P. Bourgade, B. Schlein, H.T. Yau, and J.
More informationProgress in the method of Ghosts and Shadows for Beta Ensembles
Progress in the method of Ghosts and Shadows for Beta Ensembles Alan Edelman (MIT) Alex Dubbs (MIT) and Plamen Koev (SJS) Oct 8, 2012 1/47 Wishart Matrices (arbitrary covariance) G=mxn matrix of Gaussians
More informationDeterminantal point processes and random matrix theory in a nutshell
Determinantal point processes and random matrix theory in a nutshell part II Manuela Girotti based on M. Girotti s PhD thesis, A. Kuijlaars notes from Les Houches Winter School 202 and B. Eynard s notes
More informationPOINT PROCESS LIMITS OF RANDOM MATRICES
POINT PROCESS LIMITS OF RANDOM MATRICES By Diane Holcomb A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Mathematics at the UNIVERSITY OF WISCONSIN
More informationStochastic Differential Equations Related to Soft-Edge Scaling Limit
Stochastic Differential Equations Related to Soft-Edge Scaling Limit Hideki Tanemura Chiba univ. (Japan) joint work with Hirofumi Osada (Kyushu Unv.) 2012 March 29 Hideki Tanemura (Chiba univ.) () SDEs
More informationDeterminantal point processes and random matrix theory in a nutshell
Determinantal point processes and random matrix theory in a nutshell part I Manuela Girotti based on M. Girotti s PhD thesis and A. Kuijlaars notes from Les Houches Winter School 202 Contents Point Processes
More informationDeterminantal Processes And The IID Gaussian Power Series
Determinantal Processes And The IID Gaussian Power Series Yuval Peres U.C. Berkeley Talk based on work joint with: J. Ben Hough Manjunath Krishnapur Bálint Virág 1 Samples of translation invariant point
More informationAiry and Pearcey Processes
Airy and Pearcey Processes Craig A. Tracy UC Davis Probability, Geometry and Integrable Systems MSRI December 2005 1 Probability Space: (Ω, Pr, F): Random Matrix Models Gaussian Orthogonal Ensemble (GOE,
More informationUniversal phenomena in random systems
Tuesday talk 1 Page 1 Universal phenomena in random systems Ivan Corwin (Clay Mathematics Institute, Columbia University, Institute Henri Poincare) Tuesday talk 1 Page 2 Integrable probabilistic systems
More informationNear extreme eigenvalues and the first gap of Hermitian random matrices
Near extreme eigenvalues and the first gap of Hermitian random matrices Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI Sydney Random Matrix Theory Workshop 13-16 January 2014 Anthony Perret, G. S.,
More informationLectures 6 7 : Marchenko-Pastur Law
Fall 2009 MATH 833 Random Matrices B. Valkó Lectures 6 7 : Marchenko-Pastur Law Notes prepared by: A. Ganguly We will now turn our attention to rectangular matrices. Let X = (X 1, X 2,..., X n ) R p n
More informationFluctuations from the Semicircle Law Lecture 4
Fluctuations from the Semicircle Law Lecture 4 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 23, 2014 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, 2014
More informationRandom matrices: Distribution of the least singular value (via Property Testing)
Random matrices: Distribution of the least singular value (via Property Testing) Van H. Vu Department of Mathematics Rutgers vanvu@math.rutgers.edu (joint work with T. Tao, UCLA) 1 Let ξ be a real or complex-valued
More informationFredholm determinant with the confluent hypergeometric kernel
Fredholm determinant with the confluent hypergeometric kernel J. Vasylevska joint work with I. Krasovsky Brunel University Dec 19, 2008 Problem Statement Let K s be the integral operator acting on L 2
More informationFrom the mesoscopic to microscopic scale in random matrix theory
From the mesoscopic to microscopic scale in random matrix theory (fixed energy universality for random spectra) With L. Erdős, H.-T. Yau, J. Yin Introduction A spacially confined quantum mechanical system
More informationA Remark on Hypercontractivity and Tail Inequalities for the Largest Eigenvalues of Random Matrices
A Remark on Hypercontractivity and Tail Inequalities for the Largest Eigenvalues of Random Matrices Michel Ledoux Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse, France E-mail: ledoux@math.ups-tlse.fr
More informationThe Matrix Dyson Equation in random matrix theory
The Matrix Dyson Equation in random matrix theory László Erdős IST, Austria Mathematical Physics seminar University of Bristol, Feb 3, 207 Joint work with O. Ajanki, T. Krüger Partially supported by ERC
More informationConvergence of spectral measures and eigenvalue rigidity
Convergence of spectral measures and eigenvalue rigidity Elizabeth Meckes Case Western Reserve University ICERM, March 1, 2018 Macroscopic scale: the empirical spectral measure Macroscopic scale: the empirical
More informationGaussian Free Field in beta ensembles and random surfaces. Alexei Borodin
Gaussian Free Field in beta ensembles and random surfaces Alexei Borodin Corners of random matrices and GFF spectra height function liquid region Theorem As Unscaled fluctuations Gaussian (massless) Free
More informationGaussian Free Field in (self-adjoint) random matrices and random surfaces. Alexei Borodin
Gaussian Free Field in (self-adjoint) random matrices and random surfaces Alexei Borodin Corners of random matrices and GFF spectra height function liquid region Theorem As Unscaled fluctuations Gaussian
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationMaximal height of non-intersecting Brownian motions
Maximal height of non-intersecting Brownian motions G. Schehr Laboratoire de Physique Théorique et Modèles Statistiques CNRS-Université Paris Sud-XI, Orsay Collaborators: A. Comtet (LPTMS, Orsay) P. J.
More informationOperator limits of random matrices
Operator limits of random matrices arxiv:804.06953v [math.pr] 9 Apr 208 Bálint Virág May 25, 204 Abstract We present a brief introduction to the theory of operator limits of random matrices to non-experts.
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationComparison Method in Random Matrix Theory
Comparison Method in Random Matrix Theory Jun Yin UW-Madison Valparaíso, Chile, July - 2015 Joint work with A. Knowles. 1 Some random matrices Wigner Matrix: H is N N square matrix, H : H ij = H ji, EH
More informationConcentration Inequalities for Random Matrices
Concentration Inequalities for Random Matrices M. Ledoux Institut de Mathématiques de Toulouse, France exponential tail inequalities classical theme in probability and statistics quantify the asymptotic
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationRandom Matrix: From Wigner to Quantum Chaos
Random Matrix: From Wigner to Quantum Chaos Horng-Tzer Yau Harvard University Joint work with P. Bourgade, L. Erdős, B. Schlein and J. Yin 1 Perhaps I am now too courageous when I try to guess the distribution
More informationSTATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, = (2n 1)(2n 3) 3 1.
STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, 26 Problem Normal Moments (A) Use the Itô formula and Brownian scaling to check that the even moments of the normal distribution
More informationLocal semicircle law, Wegner estimate and level repulsion for Wigner random matrices
Local semicircle law, Wegner estimate and level repulsion for Wigner random matrices László Erdős University of Munich Oberwolfach, 2008 Dec Joint work with H.T. Yau (Harvard), B. Schlein (Cambrigde) Goal:
More informationarxiv: v3 [math-ph] 21 Jun 2012
LOCAL MARCHKO-PASTUR LAW AT TH HARD DG OF SAMPL COVARIAC MATRICS CLAUDIO CACCIAPUOTI, AA MALTSV, AD BJAMI SCHLI arxiv:206.730v3 [math-ph] 2 Jun 202 Abstract. Let X be a matrix whose entries are i.i.d.
More informationNumerical Methods for Random Matrices
Numerical Methods for Random Matrices MIT 18.95 IAP Lecture Series Per-Olof Persson (persson@mit.edu) January 23, 26 1.9.8.7 Random Matrix Eigenvalue Distribution.7.6.5 β=1 β=2 β=4 Probability.6.5.4.4
More informationDLR equations for the Sineβ process and applications
DLR equations for the Sineβ process and applications Thomas Leble (Courant Institute - NYU) Columbia University - 09/28/2018 Joint work with D. Dereudre, A. Hardy, M. Maı da (Universite Lille 1) Log-gases
More informationLarge deviations of the top eigenvalue of random matrices and applications in statistical physics
Large deviations of the top eigenvalue of random matrices and applications in statistical physics Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI Journées de Physique Statistique Paris, January 29-30,
More informationBrown University Analysis Seminar
Brown University Analysis Seminar Eigenvalue Statistics for Ensembles of Random Matrices (especially Toeplitz and Palindromic Toeplitz) Steven J. Miller Brown University Providence, RI, September 15 th,
More informationCOMPLEX HERMITE POLYNOMIALS: FROM THE SEMI-CIRCULAR LAW TO THE CIRCULAR LAW
Serials Publications www.serialspublications.com OMPLEX HERMITE POLYOMIALS: FROM THE SEMI-IRULAR LAW TO THE IRULAR LAW MIHEL LEDOUX Abstract. We study asymptotics of orthogonal polynomial measures of the
More informationThe one-dimensional KPZ equation and its universality
The one-dimensional KPZ equation and its universality T. Sasamoto Based on collaborations with A. Borodin, I. Corwin, P. Ferrari, T. Imamura, H. Spohn 28 Jul 2014 @ SPA Buenos Aires 1 Plan of the talk
More informationStrong Markov property of determinatal processes
Strong Markov property of determinatal processes Hideki Tanemura Chiba university (Chiba, Japan) (August 2, 2013) Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 1 / 27 Introduction The
More informationEigenvalue PDFs. Peter Forrester, M&S, University of Melbourne
Outline Eigenvalue PDFs Peter Forrester, M&S, University of Melbourne Hermitian matrices with real, complex or real quaternion elements Circular ensembles and classical groups Products of random matrices
More informationBeyond the Gaussian universality class
Beyond the Gaussian universality class MSRI/Evans Talk Ivan Corwin (Courant Institute, NYU) September 13, 2010 Outline Part 1: Random growth models Random deposition, ballistic deposition, corner growth
More informationRectangular Young tableaux and the Jacobi ensemble
Rectangular Young tableaux and the Jacobi ensemble Philippe Marchal October 20, 2015 Abstract It has been shown by Pittel and Romik that the random surface associated with a large rectangular Young tableau
More informationTriangular matrices and biorthogonal ensembles
/26 Triangular matrices and biorthogonal ensembles Dimitris Cheliotis Department of Mathematics University of Athens UK Easter Probability Meeting April 8, 206 2/26 Special densities on R n Example. n
More informationA determinantal formula for the GOE Tracy-Widom distribution
A determinantal formula for the GOE Tracy-Widom distribution Patrik L. Ferrari and Herbert Spohn Technische Universität München Zentrum Mathematik and Physik Department e-mails: ferrari@ma.tum.de, spohn@ma.tum.de
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationThe Tracy-Widom distribution is not infinitely divisible.
The Tracy-Widom distribution is not infinitely divisible. arxiv:1601.02898v1 [math.pr] 12 Jan 2016 J. Armando Domínguez-Molina Facultad de Ciencias Físico-Matemáticas Universidad Autónoma de Sinaloa, México
More informationDouble contour integral formulas for the sum of GUE and one matrix model
Double contour integral formulas for the sum of GUE and one matrix model Based on arxiv:1608.05870 with Tom Claeys, Arno Kuijlaars, and Karl Liechty Dong Wang National University of Singapore Workshop
More informationLarge Deviations for Random Matrices and a Conjecture of Lukic
Large Deviations for Random Matrices and a Conjecture of Lukic Jonathan Breuer Hebrew University of Jerusalem Joint work with B. Simon (Caltech) and O. Zeitouni (The Weizmann Institute) Western States
More informationFluctuation results in some positive temperature directed polymer models
Singapore, May 4-8th 2015 Fluctuation results in some positive temperature directed polymer models Patrik L. Ferrari jointly with A. Borodin, I. Corwin, and B. Vető arxiv:1204.1024 & arxiv:1407.6977 http://wt.iam.uni-bonn.de/
More informationFluctuations of random tilings and discrete Beta-ensembles
Fluctuations of random tilings and discrete Beta-ensembles Alice Guionnet CRS (E S Lyon) Advances in Mathematics and Theoretical Physics, Roma, 2017 Joint work with A. Borodin, G. Borot, V. Gorin, J.Huang
More informationSpectral Theory of Orthogonal Polynomials
Spectral Theory of Orthogonal Polynomials Barry Simon IBM Professor of Mathematics and Theoretical Physics California Institute of Technology Pasadena, CA, U.S.A. Lectures 11 & 12: Selected Additional,
More informationRandom Correlation Matrices, Top Eigenvalue with Heavy Tails and Financial Applications
Random Correlation Matrices, Top Eigenvalue with Heavy Tails and Financial Applications J.P Bouchaud with: M. Potters, G. Biroli, L. Laloux, M. A. Miceli http://www.cfm.fr Portfolio theory: Basics Portfolio
More informationGeometric RSK, Whittaker functions and random polymers
Geometric RSK, Whittaker functions and random polymers Neil O Connell University of Warwick Advances in Probability: Integrability, Universality and Beyond Oxford, September 29, 2014 Collaborators: I.
More informationLow-temperature random matrix theory at the soft edge. Abstract
Low-temperature random matrix theory at the soft edge Alan Edelman Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 239, USA Per-Olof Persson Department of Mathematics, University
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),
More informationNonparametric Drift Estimation for Stochastic Differential Equations
Nonparametric Drift Estimation for Stochastic Differential Equations Gareth Roberts 1 Department of Statistics University of Warwick Brazilian Bayesian meeting, March 2010 Joint work with O. Papaspiliopoulos,
More informationInteger-Valued Polynomials
Integer-Valued Polynomials LA Math Circle High School II Dillon Zhi October 11, 2015 1 Introduction Some polynomials take integer values p(x) for all integers x. The obvious examples are the ones where
More informationBurgers equation in the complex plane. Govind Menon Division of Applied Mathematics Brown University
Burgers equation in the complex plane Govind Menon Division of Applied Mathematics Brown University What this talk contains Interesting instances of the appearance of Burgers equation in the complex plane
More informationCharacterizations of free Meixner distributions
Characterizations of free Meixner distributions Texas A&M University March 26, 2010 Jacobi parameters. Matrix. β 0 γ 0 0 0... 1 β 1 γ 1 0.. m n J =. 0 1 β 2 γ.. 2 ; J n =. 0 0 1 β.. 3............... A
More informationEigenvalue variance bounds for Wigner and covariance random matrices
Eigenvalue variance bounds for Wigner and covariance random matrices S. Dallaporta University of Toulouse, France Abstract. This work is concerned with finite range bounds on the variance of individual
More informationQuasi-Diffusion in a SUSY Hyperbolic Sigma Model
Quasi-Diffusion in a SUSY Hyperbolic Sigma Model Joint work with: M. Disertori and M. Zirnbauer December 15, 2008 Outline of Talk A) Motivation: Study time evolution of quantum particle in a random environment
More informationQuantum Chaos and Nonunitary Dynamics
Quantum Chaos and Nonunitary Dynamics Karol Życzkowski in collaboration with W. Bruzda, V. Cappellini, H.-J. Sommers, M. Smaczyński Phys. Lett. A 373, 320 (2009) Institute of Physics, Jagiellonian University,
More informationNext is material on matrix rank. Please see the handout
B90.330 / C.005 NOTES for Wednesday 0.APR.7 Suppose that the model is β + ε, but ε does not have the desired variance matrix. Say that ε is normal, but Var(ε) σ W. The form of W is W w 0 0 0 0 0 0 w 0
More informationStatistical Inference and Random Matrices
Statistical Inference and Random Matrices N.S. Witte Institute of Fundamental Sciences Massey University New Zealand 5-12-2017 Joint work with Peter Forrester 6 th Wellington Workshop in Probability and
More informationL n = l n (π n ) = length of a longest increasing subsequence of π n.
Longest increasing subsequences π n : permutation of 1,2,...,n. L n = l n (π n ) = length of a longest increasing subsequence of π n. Example: π n = (π n (1),..., π n (n)) = (7, 2, 8, 1, 3, 4, 10, 6, 9,
More informationRandom Toeplitz Matrices
Arnab Sen University of Minnesota Conference on Limits Theorems in Probability, IISc January 11, 2013 Joint work with Bálint Virág What are Toeplitz matrices? a0 a 1 a 2... a1 a0 a 1... a2 a1 a0... a (n
More informationMicroscopic behavior for β-ensembles: an energy approach
Microscopic behavior for β-ensembles: an energy approach Thomas Leblé (joint with/under the supervision of) Sylvia Serfaty Université Paris 6 BIRS workshop, 14 April 2016 Thomas Leblé (Université Paris
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationSTA 294: Stochastic Processes & Bayesian Nonparametrics
MARKOV CHAINS AND CONVERGENCE CONCEPTS Markov chains are among the simplest stochastic processes, just one step beyond iid sequences of random variables. Traditionally they ve been used in modelling a
More informationStrong Markov property of determinantal processes associated with extended kernels
Strong Markov property of determinantal processes associated with extended kernels Hideki Tanemura Chiba university (Chiba, Japan) (November 22, 2013) Hideki Tanemura (Chiba univ.) () Markov process (November
More informationSpectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues László Erdős Antti Knowles 2 Horng-Tzer Yau 2 Jun Yin 2 Institute of Mathematics, University of Munich, Theresienstrasse
More informationFinite Rank Perturbations of Random Matrices and Their Continuum Limits. Alexander Bloemendal
Finite Rank Perturbations of Random Matrices and Their Continuum Limits by Alexander Bloemendal A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department
More informationarxiv: v2 [cond-mat.dis-nn] 9 Feb 2011
Level density and level-spacing distributions of random, self-adjoint, non-hermitian matrices Yogesh N. Joglekar and William A. Karr Department of Physics, Indiana University Purdue University Indianapolis
More informationDifferential Geometry qualifying exam 562 January 2019 Show all your work for full credit
Differential Geometry qualifying exam 562 January 2019 Show all your work for full credit 1. (a) Show that the set M R 3 defined by the equation (1 z 2 )(x 2 + y 2 ) = 1 is a smooth submanifold of R 3.
More informationMath 306 Topics in Algebra, Spring 2013 Homework 7 Solutions
Math 306 Topics in Algebra, Spring 203 Homework 7 Solutions () (5 pts) Let G be a finite group. Show that the function defines an inner product on C[G]. We have Also Lastly, we have C[G] C[G] C c f + c
More informationMath 108b: Notes on the Spectral Theorem
Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator
More informationOn the concentration of eigenvalues of random symmetric matrices
On the concentration of eigenvalues of random symmetric matrices Noga Alon Michael Krivelevich Van H. Vu April 23, 2012 Abstract It is shown that for every 1 s n, the probability that the s-th largest
More informationNon white sample covariance matrices.
Non white sample covariance matrices. S. Péché, Université Grenoble 1, joint work with O. Ledoit, Uni. Zurich 17-21/05/2010, Université Marne la Vallée Workshop Probability and Geometry in High Dimensions
More informationInfinitely divisible distributions and the Lévy-Khintchine formula
Infinitely divisible distributions and the Cornell University May 1, 2015 Some definitions Let X be a real-valued random variable with law µ X. Recall that X is said to be infinitely divisible if for every
More informationRandom matrix pencils and level crossings
Albeverio Fest October 1, 2018 Topics to discuss Basic level crossing problem 1 Basic level crossing problem 2 3 Main references Basic level crossing problem (i) B. Shapiro, M. Tater, On spectral asymptotics
More informationRandom matrices and determinantal processes
Random matrices and determinantal processes Patrik L. Ferrari Zentrum Mathematik Technische Universität München D-85747 Garching 1 Introduction The aim of this work is to explain some connections between
More informationUniversality in Numerical Computations with Random Data. Case Studies.
Universality in Numerical Computations with Random Data. Case Studies. Percy Deift and Thomas Trogdon Courant Institute of Mathematical Sciences Govind Menon Brown University Sheehan Olver The University
More informationNumerical analysis and random matrix theory. Tom Trogdon UC Irvine
Numerical analysis and random matrix theory Tom Trogdon ttrogdon@math.uci.edu UC Irvine Acknowledgements This is joint work with: Percy Deift Govind Menon Sheehan Olver Raj Rao Numerical analysis and random
More informationNumerical Analysis Comprehensive Exam Questions
Numerical Analysis Comprehensive Exam Questions 1. Let f(x) = (x α) m g(x) where m is an integer and g(x) C (R), g(α). Write down the Newton s method for finding the root α of f(x), and study the order
More informationFluctuations of random tilings and discrete Beta-ensembles
Fluctuations of random tilings and discrete Beta-ensembles Alice Guionnet CRS (E S Lyon) Workshop in geometric functional analysis, MSRI, nov. 13 2017 Joint work with A. Borodin, G. Borot, V. Gorin, J.Huang
More informationFree Probability and Random Matrices: from isomorphisms to universality
Free Probability and Random Matrices: from isomorphisms to universality Alice Guionnet MIT TexAMP, November 21, 2014 Joint works with F. Bekerman, Y. Dabrowski, A.Figalli, E. Maurel-Segala, J. Novak, D.
More informationWigner s semicircle law
CHAPTER 2 Wigner s semicircle law 1. Wigner matrices Definition 12. A Wigner matrix is a random matrix X =(X i, j ) i, j n where (1) X i, j, i < j are i.i.d (real or complex valued). (2) X i,i, i n are
More informationAdvances in Random Matrix Theory: Let there be tools
Advances in Random Matrix Theory: Let there be tools Alan Edelman Brian Sutton, Plamen Koev, Ioana Dumitriu, Raj Rao and others MIT: Dept of Mathematics, Computer Science AI Laboratories World Congress,
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationHow long does it take to compute the eigenvalues of a random symmetric matrix?
How long does it take to compute the eigenvalues of a random symmetric matrix? Govind Menon (Brown University) Joint work with: Christian Pfrang (Ph.D, Brown 2011) Percy Deift, Tom Trogdon (Courant Institute)
More informationOrthogonal Polynomial Ensembles
Chater 11 Orthogonal Polynomial Ensembles 11.1 Orthogonal Polynomials of Scalar rgument Let wx) be a weight function on a real interval, or the unit circle, or generally on some curve in the comlex lane.
More informationBefore you begin read these instructions carefully.
MATHEMATICAL TRIPOS Part IB Tuesday, 5 June, 2012 9:00 am to 12:00 pm PAPER 1 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each question
More information