This page intentionally left blank LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor:Professor M.Reid,Mathematics Institute, University ofWarwick,Coventry CV4 7AL,United Kingdom The titles below are available from booksellers,or from Cambridge University Press at www.cambridge.org/mathematics 247 Analytic number theory, Y.MOTOHASHI(ed) 248 Tame topology and O-minimalstructures, L.VAN DEN DRIES 249 The atlas offinite groups -ten years on, R.T.CURTIS & R.A.WILSON (eds) 250 Characters and blocks offinite groups, G.NAVARRO 251 Gro¨bner bases and applications, B.BUCHBERGER & F.WINKLER (eds) 252 Geometry and cohomology in group theory, P.H.KROPHOLLER,G.A.NIBLO & R. STO¨HR (eds) 253 The q-Schur algebra, S.DONKIN 254 Galois representations in arithmetic algebraic geometry, A.J.SCHOLL & R.L.TAYLOR (eds) 255 Symmetries and integrability ofdifference equations, P.A.CLARKSON & F.W. NIJHOFF (eds) 256 Aspects ofGalois theory, H.VO¨LKLEIN,J.G.THOMPSON,D.HARBATER & P. MU¨LLER (eds) 257 An introduction to noncommutative differentialgeometry and its physicalapplications (2nd Edition), J.MADORE 258 Sets and proofs, S.B.COOPER & J.K.TRUSS (eds) 259 Models and computability, S.B.COOPER & J.TRUSS (eds) 260 Groups St Andrews 1997 in Bath I, C.M.CAMPBELL etal (eds) 261 Groups St Andrews 1997 in Bath II, C.M.CAMPBELL etal (eds) 262 Analysis and logic, C.W.HENSON,J.IOVINO,A.S.KECHRIS & E.ODELL 263 Singularity theory, W.BRUCE & D.MOND (eds) 264 New trends in algebraic geometry, K.HULEK,F.CATANESE,C.PETERS & M.REID (eds) 265 Elliptic curves in cryptography, I.BLAKE,G.SEROUSSI& N.SMART 267 Surveys in combinatorics,1999, J.D.LAMB & D.A.PREECE (eds) 268 Spectralasymptotics in the semi-classicallimit, M.DIMASSI& J.SJO¨STRAND 269 Ergodic theory and topologicaldynamics ofgroup actions on homogeneous spaces, M.B. BEKKA & M.MAYER 271 Singular perturbations ofdifferentialoperators, S.ALBEVERIO & P.KURASOV 272 Character theory for the odd order theorem, T.PETERFALVI.Translated by R. SANDLING 273 Spectraltheory and geometry, E.B.DAVIES & Y.SAFAROV (eds) 274 The Mandelbrot set,theme and variations, T.LEI(ed) 275 Descriptive set theory and dynamicalsystems, M.FOREMAN,A.S.KECHRIS,A. LOUVEAU & B.WEISS (eds) 276 Singularities ofplane curves, E.CASAS-ALVERO 277 Computationaland geometric aspects ofmodern algebra, M.ATKINSON etal (eds) 278 Globalattractors in abstract parabolic problems, J.W.CHOLEWA & T.DLOTKO 279 Topics in symbolic dynamics and applications, F.BLANCHARD,A.MAASS & A. NOGUEIRA (eds) 280 Characters and automorphism groups ofcompact Riemann surfaces, T.BREUER 281 Explicit birationalgeometry of3-folds, A.CORTI& M.REID (eds) 282 Auslander-Buchweitz approximations ofequivariant modules, M.HASHIMOTO 283 Nonlinear elasticity, Y.B.FU & R.W.OGDEN (eds) 284 Foundations ofcomputationalmathematics, R.DEVORE,A.ISERLES & E.SU¨LI(eds) 285 Rationalpoints on curves over finite fields, H.NIEDERREITER & C.XING 286 Clifford algebras and spinors (2nd Edition), P.LOUNESTO 287 Topics on Riemann surfaces and Fuchsian groups, E.BUJALANCE,A.F.COSTA & E. MART´INEZ (eds) 288 Surveys in combinatorics,2001, J.W.P.HIRSCHFELD (ed) 289 Aspects ofSobolev-type inequalities, L.SALOFF-COSTE 290 Quantum groups and Lie theory, A.PRESSLEY (ed) 291 Tits buildings and the modeltheory ofgroups, K.TENT (ed) 292 A quantum groups primer, S.MAJID 293 Second order partialdifferentialequations in Hilbert spaces, G.DA PRATO & J. ZABCZYK 294 Introduction to operator space theory, G.PISIER 295 Geometry and integrability, L.MASON & Y.NUTKU (eds) 296 Lectures on invariant theory, I.DOLGACHEV 297 The homotopy category ofsimply connected 4-manifolds, H.-J.BAUES 298 Higher operads,higher categories, T.LEINSTER (ed) 299 Kleinian groups and hyperbolic 3-manifolds, Y.KOMORI,V.MARKOVIC & C.SERIES (eds) 300 Introduction to Mo¨bius differentialgeometry, U.HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F.E.A.JOHNSON 302 Discrete and continuous nonlinear Schro¨dinger systems, M.J.ABLOWITZ,B.PRINARI & A.D.TRUBATCH 303 Number theory and algebraic geometry, M.REID & A.SKOROBOGATOV (eds) 304 Groups St Andrews 2001 in Oxford I, C.M.CAMPBELL,E.F.ROBERTSON & G.C. SMITH (eds) 305 Groups St Andrews 2001 in Oxford II, C.M.CAMPBELL,E.F.ROBERTSON & G.C. SMITH (eds) 306 Geometric mechanics and symmetry, J.MONTALDI& T.RATIU (eds) 307 Surveys in combinatorics 2003, C.D.WENSLEY (ed.) 308 Topology,geometry and quantum field theory, U.L.TILLMANN (ed) 309 Corings and comodules, T.BRZEZINSKI& R.WISBAUER 310 Topics in dynamics and ergodic theory, S.BEZUGLYI& S.KOLYADA (eds) 311 Groups:topological,combinatorialand arithmetic aspects, T.W.MU¨LLER (ed) 312 Foundations ofcomputationalmathematics,Minneapolis 2002, F.CUCKER etal (eds) 313 Transcendentalaspects ofalgebraic cycles, S.MU¨LLER-STACH & C.PETERS (eds) 314 Spectralgeneralizations ofline graphs, D.CVETKOVIC´,P.ROWLINSON & S.SIMIC´ 315 Structured ring spectra, A.BAKER & B.RICHTER (eds) 316 Linear logic in computer science, T.EHRHARD,P.RUET,J.-Y.GIRARD & P.SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F.BLAKE,G.SEROUSSI& N.P.SMART (eds) 318 Perturbation ofthe boundary in boundary-value problems ofpartialdifferentialequations, D.HENRY 319 Double affine Hecke algebras, I.CHEREDNIK 320 L-functions and Galois representations, D.BURNS,K.BUZZARD & J.NEKOVA´Rˇ (eds) 321 Surveys in modern mathematics, V.PRASOLOV & Y.ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F.MEZZADRI& N.C. SNAITH (eds) 323 Poisson geometry,deformation quantisation and group representations, S.GUTT etal (eds) 324 Singularities and computer algebra, C.LOSSEN & G.PFISTER (eds) 325 Lectures on the Ricciflow, P.TOPPING 326 Modular representations offinite groups ofLie type, J.E.HUMPHREYS 327 Surveys in combinatorics 2005, B.S.WEBB (ed) 328 Fundamentals ofhyperbolic manifolds, R.CANARY,D.EPSTEIN & A.MARDEN (eds) 329 Spaces ofKleinian groups, Y.MINSKY,M.SAKUMA & C.SERIES (eds) 330 Noncommutative localization in algebra and topology, A.RANICKI(ed) 331 Foundations ofcomputationalmathematics,Santander 2005, L.M PARDO,A.PINKUS, E.SU¨LI& M.J.TODD (eds) 332 Handbook oftilting theory, L.ANGELERIHU¨GEL,D.HAPPEL & H.KRAUSE (eds) 333 Synthetic differentialgeometry (2nd Edition), A.KOCK 334 The Navier–Stokes equations, N.RILEY & P.DRAZIN 335 Lectures on the combinatorics offree probability, A.NICA & R.SPEICHER 336 Integralclosure ofideals,rings,and modules, I.SWANSON & C.HUNEKE 337 Methods in Banach space theory, J.M.F.CASTILLO & W.B.JOHNSON (eds) 338 Surveys in geometry and number theory, N.YOUNG (ed) 339 Groups St Andrews 2005 I, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON & G.C.SMITH (eds) 341 Ranks ofelliptic curves and random matrix theory, J.B.CONREY,D.W.FARMER,F. MEZZADRI& N.C.SNAITH (eds) 342 Elliptic cohomology, H.R.MILLER & D.C.RAVENEL (eds) 343 Algebraic cycles and motives I, J.NAGEL & C.PETERS (eds) 344 Algebraic cycles and motives II, J.NAGEL & C.PETERS (eds) 345 Algebraic and analytic geometry, A.NEEMAN 346 Surveys in combinatorics 2007, A.HILTON & J.TALBOT (eds) 347 Surveys in contemporary mathematics, N.YOUNG & Y.CHOI(eds) 348 Transcendentaldynamics and complex analysis, P.J.RIPPON & G.M.STALLARD (eds) 349 Modeltheory with applications to algebra and analysis I, Z.CHATZIDAKIS,D. MACPHERSON,A.PILLAY & A.WILKIE (eds) 350 Modeltheory with applications to algebra and analysis II, Z.CHATZIDAKIS,D. MACPHERSON,A.PILLAY & A.WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M.SINCLAIR & R.R.SMITH 352 Number theory and polynomials, J.MCKEE & C.SMYTH (eds) 353 Trends in stochastic analysis, J.BLATH,P.MO¨RTERS & M.SCHEUTZOW (eds) 354 Groups and analysis, K.TENT (ed) 355 Non-equilibrium statisticalmechanics and turbulence, J.CARDY,G.FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D.DELBOURGO 357 Algebraic theory ofdifferentialequations, M.A.H.MACCALLUM & A.V.MIKHAILOV (eds) 358 Geometric and cohomologicalmethods in group theory, M.R.BRIDSON,P.H. KROPHOLLER & I.J.LEARY (eds) 359 Modulispaces and vector bundles, L.BRAMBILA-PAZ,S.B.BRADLOW,O. GARC´IA-PRADA & S.RAMANAN (eds) 360 Zariskigeometries, B.ZILBER 361 Words:Notes on verbalwidth in groups, D.SEGAL 362 Differentialtensor algebras and their module categories, R.BAUTISTA,L.SALMERO´N & R.ZUAZUA 363 Foundations ofcomputationalmathematics,Hong Kong 2008, F.CUCKER,A.PINKUS & M.J.TODD (eds) 364 Partialdifferentialequations and fluid mechanics, J.C.ROBINSON & J.L.RODRIGO (eds) 365 Surveys in combinatorics 2009, S.HUCZYNSKA,J.D.MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B.ENGQUIST,A.FOKAS,E.HAIRER & A.ISERLES (eds) London Mathematical Society Lecture Note Series: 367 Random Matrices: High Dimensional Phenomena GORDON BLOWER Lancaster University cambridge university press Cambridge,New York,Melbourne,Madrid,Cape Town,Singapore, S˜ao Paulo,Delhi Cambridge University Press The Edinburgh Building,Cambridge CB2 8RU,UK Published in the United States ofAmerica by Cambridge University Press,New York www.cambridge.org Information on this title:www.cambridge.org/9780521133128 (cid:1)C G.Blower 2009 This publication is in copyright.Subject to statutory exception and to the provisions ofrelevant collective licensing agreements, no reproduction ofany part may take place without the written permission ofCambridge University Press. First published 2009 Printed in the United Kingdom at the University Press,Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-13312-8 Paperback Cambridge University Press has no responsibility for the persistence or accuracy ofURLs forexternalor third-party Internet websites referred to in this publication,and does not guarantee that any content on such websites is,or willremain,accurate orappropriate. To the memory of my father Ronald Frederick Blower Contents Introduction page 1 1 Metric measure spaces 4 1.1 Weak convergence on compact metric spaces 4 1.2 Invariant measure on a compact metric group 10 1.3 Measures on non-compact Polish spaces 16 1.4 The Brunn–Minkowski inequality 22 1.5 Gaussian measures 25 1.6 Surface area measure on the spheres 27 1.7 Lipschitz functions and the Hausdorff metric 31 1.8 Characteristic functions and Cauchy transforms 33 2 Lie groups and matrix ensembles 42 2.1 The classical groups, their eigenvalues and norms 42 2.2 Determinants and functional calculus 49 2.3 Linear Lie groups 56 2.4 Connections and curvature 63 2.5 Generalized ensembles 66 2.6 The Weyl integration formula 72 2.7 Dyson’s circular ensembles 78 2.8 Circular orthogonal ensemble 81 2.9 Circular symplectic ensemble 83 3 Entropy and concentration of measure 84 3.1 Relative entropy 84 3.2 Concentration of measure 93 3.3 Transportation 99 3.4 Transportation inequalities 103 3.5 Transportation inequalities for uniformly convex potentials 106 3.6 Concentration of measure in matrix ensembles 109 vii viii Contents 3.7 Concentration for rectangular Gaussian matrices 114 3.8 Concentration on the sphere 123 3.9 Concentration for compact Lie groups 126 4 Free entropy and equilibrium 132 4.1 Logarithmic energy and equilibrium measure 132 4.2 Energy spaces on the disc 134 4.3 Free versus classical entropy on the spheres 142 4.4 Equilibrium measures for potentials on the real line 147 4.5 Equilibrium densities for convex potentials 154 4.6 The quartic model with positive leading term 159 4.7 Quartic models with negative leading term 164 4.8 Displacement convexity and relative free entropy 169 4.9 Toeplitz determinants 172 5 Convergence to equilibrium 177 5.1 Convergence to arclength 177 5.2 Convergence of ensembles 179 5.3 Mean field convergence 183 5.4 Almost sure weak convergence for uniformly convex potentials 189 5.5 Convergence for the singular numbers from the Wishart distribution 193 6 Gradient flows and functional inequalities 196 6.1 Variation of functionals and gradient flows 196 6.2 Logarithmic Sobolev inequalities 203 6.3 Logarithmic Sobolev inequalities for uniformly convex potentials 206 6.4 Fisher’s information and Shannon’s entropy 210 6.5 Free information and entropy 213 6.6 Free logarithmic Sobolev inequality 218 6.7 Logarithmic Sobolev and spectral gap inequalities 221 6.8 Inequalities for Gibbs measures on Riemannian manifolds 223 7 Young tableaux 227 7.1 Group representations 227 7.2 Young diagrams 229 7.3 The Vershik Ω distribution 237 7.4 Distribution of the longest increasing subsequence 243 7.5 Inclusion-exclusion principle 250