Index

A Bertola, M., 309, 346, 428 Ablowitz, M.J., 355 Bilinear identities Adler, M., XIII, XIV, 4, 35, 54, 88, integrable PDEs, 132 89, 113, 114, 118, 131, 141, τ-function PDEs 148–152, 157, 163, 167, 174, alternate proof, 175–176 175, 177, 189, 204, 207–213, fundamental identities, 175 219–221, 326, 372 Hirota form, 174, 175 Adler–van Moerbeke theorem 2-Toda, 174 central charge, 89 orthogonal polynomials and the KP contributions, 91–92 equation, 86–87 ε-derivative, 91 Bilodeau, M., 444 differential and multiplication Bleher, P.M., XIV, XV, 199, 202, 206, operators, 89, 90 314, 346, 372, 382, 383, 402, integration, 92–93 416 Virasoro generators, 90 Borel probability measures, 383 Airy function Borodin, A., 33, 39, 44, 54, 64, 134, 151 asymptotic behavior, 389 Borodin–Okounkov–Olshanski measure, branch point parametrix, 340 XIV geometrically distributed percolation Boundary value problem solver, 504 model, 74 Brenner, D., 444 limit theorem, 64 Br´ezin, E.,´ 199, 201, 206, 325, 326, 416 parametrix at edge points, 283, 332 Br´ezin–Hikami Pearcey kernel, 208 parametrix at end points, 389–390 Brock, P., 60 Airy kernel, XVII Brownian motion model, 326 Aldous, D., 15, 60 Brown, R., 183 Ambjorn, 416 Buckingham, R., 127 Anderson, T.W., 444 C Aptekarev, A., XIV, 199, 202, 206, 252 Asymptotic analysis, XVI Canonical correlation coefficients conditional distribution, 111, 112 B covariance , 109, 110 Baer, R.M., 60 density, 113 Baik, J., 64, 127, 346, 453 Gaussian populations, 109 Basor, E., 44 Jack polynomials, 113

J. Harnad (ed.), Random Matrices, Random Processes and Integrable Systems, 509 CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9514-8, c Springer Science+Business Media, LLC 2011 510 Index

Canonical correlation coefficients (cont.) D linear transformations, 110 David, F., 416 roots, 112 de Bruijn, J., 170, 455 sample , 111 Deift, P.A., 64, 96, 97, 104, 127, 267, Wishart distribution, 112 346, 355, 372, 416, 450 Case, K.M., 44 Deift–Zhou approach, 372 Cauchy transform, 287, 319 Deift–Zhou steepest descent method, Central limit theorem (CLT), 132 308, 326 Charlier polynomials, 54, 55 Del´epine, J., 4 Christoffel–Darboux formula, Deo, N., 416 256, 317 Diaconis, P., 15, 60, 134 Christoffel–Darboux kernel, XV, XVI, Dieng, M., XVI, XVII, 453–454 202, 203, 354 Differential equations Claeys, T., 4, 308 GOE Colomo, F., 346 B finite rank, 493 Cosgrove, C.M., 119 constant coefficient ordinary Costin, O., 346 differential equations, 498 Coupled random matrices and 2-Toda determinant limit, 496 lattice even and odd functions, 495 bilinear identities and τ-function finite dimensional perturbation, PDEs 492 alternate proof, 175–176 L2-inner product, 494 fundamental identities, 175 row operation, matrix, 497 Hirota form, 174, 175 Tracy–Widom distribution, 499 2-Toda, 174 GSE boundary operators, 167 constant coefficient ordinary final equations, 181–182 differential equations, 479 Lie algebra, 167 determinant limit, 479 link with 2-Toda hierarchy finite dimensional perturbation, HCIZ, 168–169 476 moment matrix, 170–171 Hermite polynomials, 478 2 τ-function, 169–170 L -inner product, 477 L-U decomposition Tracy–Widom distribution, 480 eigenfunctions identities, 173 limiting probability Hirota symbol, 174 Airy edge limit, 146–147 m∞, 171, 172 A derivatives in KdV, 148, 150 monic bi-orthogonal polynomials, Grassmannian invariance 172 condition, 147, 149 string orthogonal polynomials, hard edge limit, 148–149 171 Kontsevich integral, 147 2-Toda wave functions, 173 Laplace integral, 149 wave operators, 173–174 τ(t,E), condition, 147, 150 probability ensemble, 167 vacuum condition, 147, 149 third order nonlinear PDE, 167–168 Virasoro constraints, 148, 150 Virasoro constraints, 176–179 DiFranco, J., 127 Virasoro relations, 179–181 Discrete string equation, 354 wave functions, 167 Double scaling limit, XV Crnkovi´c, C.,ˇ 411 complex plane partition, 297 Index 511

double-well quartic matrix model, random walk, 182 294 scale invariant, 183 final transformation, 345–346 Dyson determinantal formula, 255 free energy, 315 Dyson diffusion process, XIV large N asymptotics, random Dyson express, 354 matrices, 394, 402 Dyson, F.J., 3, 127, 131, 135, 182, 185, modified Pastur equation, 341–343 186, 353, 453 monic orthogonal polynomials, 294 Painlev´e II equation, 295 E parametrix construction (see El Karoui, N., 450 Parametrix construction) Ercolani, N.M., 311 Pauli matrix, 297 Erdos, P., 60 Psi-matrix, 296 Euler characteristics, 421–423 recurrent coefficient, 295 Euler–Lagrange equations, 257 scaling functions, 294–295 Eynard, B., XVI, 186, 252, 309, 314, Schr¨odinger equation, 296 346, 415 semiclassical approximation, 297 Double-well quartic matrix model, F 294 Fa´a di Bruno’s formula, 500 Douglas, M.R., 411 Fay identity, 142 Duits, M., 4, 308 Feynman diagrams, 254 Dyson Brownian motion and Airy Flaschka–Newell Lax pair, 405 process Fokas, A.S., 308–310 Dyson diffusion, 185 Fokin, V., 346 Fredholm determinant, 187 Formal matrix integrals full force, 185 ABAB models, 439 heat diffusion equation, 183 ADE models, 438 Hermite kernel, 186, 187 bi-orthogonal polynomials, 440 joint probability distribution, 184 chain of matrices, 434 Markovian process, 183, 184 closed chain of matrices, 435 Ornstein–Uhlenbeck process, 185, 3-color model, 438 186 Feynman’s graphical representation, PDEs and asymptotics 421 Dyson process, 189 formal power series, 418 large time, 191–192 free energy, 419 Sine process, 190–191 Gaussian integral, 417, 422 probability density, diffusion Hermitian matrices, 418 equation, 183, 184 invariant monomial, 419, 420 proof of results invariant noncommutative monomial asymptotic expansion, 196 and polynomials, 417 boundary operators, 192 invariant polynomial, 420–421 boundary terms, 199 Laurent polynomial, 418 conditional probability, 197 loop equations constants, 198 integral, Hermitian matrices, 192 427–428 nonexplosion condition, 197 formal expectation value, 425–426 plausible conjecture, 196 Gaussian integrals, 426 , 192, 193 Leibnitz rule, 424 , 194, 195 linear combination, 429 512 Index

Formal matrix integrals (cont.) ordered space, 481 merge rule, 425 Vandermonde determinant, 481 “multicut” solutions, 427 GSE no Mk rule, 425 integral operator, 466–467 random matrices, 423 matrix kernel, 466, 467 Schwinger–Dyson equations, 2N × 2 matrix, 465 423–424 2 × 2n matrix, 466 split rule, 424 Pfaffian square, 464 Taylor series expansion, 428 GUE, 458–459, 463 maps, 420 KP-vertex operator, 141 matrix element, 422 link with kernels, 142 1-matrix model Meixner kernel, 71, 72 F (2) expression, 430 partition probability potential V, 429 integrable, Borodin–Okounkov, 44 spectral curve, 429 kernel, 47–48 square maps, 431 wedging-contracting operation, triangulated maps, 430–431 45–47 2-matrix model, 431–433 soliton Fredholm determinant, 144 multi-cut formal matrix models, 439 style soliton formula, 143–144 −2 N Fl polynomial, 422–423 vertex identities, 142–143 in nuclear physics, solid state Virasoro relations, 144–146 physics, quantum chaos, 416 Freilikher, V., 416 O(n) model, 435–437 Freud equation, 354, 372, 411 oriented discrete surfaces, 420 Potts model, 437–438 G Q-gon, 419 Gauden, 353 in quantum chromodynamics, Gaudin, M., 131 quantum gravity, string theory, Gaussian Hermitian ensemble, 68 conformal field theory, 416 classical ensembles, Riemann–Hilbert problem, 440 108 Temperly–Lieb algebra, 440 Painlev´e generation topological expansion, 423 differential operators, 114–116 6-vertex model, 438 disjoint union of intervals, 113, Wick’s theorem, 421, 422 114 Forrester, F.J., 453 KP equation, 114 Forrester, P.J., 19 PDE and ODE, 114, 117 Fredholm determinants, XIV, XV, 187 t-differentiation, 115 correlations Gaussian matrix model, 321–322 1- and 2-point correlations, 105 Gaussian model computation, 106 b ∈ R, 325 differentiation, 107 bulk of spectrum, 324 multinomial coefficient, 106 Deift–Zhou steepest descent method, probability distribution, 104 326 edge distribution functions, 448–449 Fredholm determinant, 326 GOE limiting mean eigenvalue density, arbitrary polynomials, 482 322–324 column vector, 484 non-intersecting Brownian bridges, matrix kernel, 485, 486 321–322 , 483–484 Pastur equation, 322 Index 513

Pearcey kernel, 326 Gaussian weight function, 468 recurrence equation, 321 Hermite polynomials, 469, 470 v ∈ R, 324–325 infinite antisymmetric tridiagonal Gaussian orthogonal ensemble (GOE) matrix, 471 edge scaling integrand, 470 determinant reduction, 490–492 matrix kernel, 473–474 differential equations (see skew-symmetry, 468 Differential equations) Geometrically distributed percolation Fredholm determinant model arbitrary polynomials, 482 Airy function, 74 column vector, 484 Fp(z) function, 73, 74 matrix kernel, 485, 486 Fredholm determinant, Meixner matrix multiplication, 483–484 kernel, 71, 72 ordered space, 481 Johansson’s percolation model, 67 Vandermonde determinant, 481 Johansson theorem, 68–71 Gaussian specialization mean and standard deviation, 67 Gaussian weight function, 486 random variable, 72 Hermite polynomials, 487 Tracy–Widom distribution, 72 matrix kernel, 489–490 variables, 68 odd and even functions, 488 Geronimo, J.S., 44 orthogonality, 487, 488 Gessel, I.M., 4, 35, 60, 154 Painlev´e function, 486 g-function, DKMVZ method skew-symmetry, 487 Abelian, 382 Painlev´e representations, 453, 454 constant jump matrix, 376 Gaussian symplectic ensemble (GSE) diagonal jump matrix, 377 edge scaling and differential equilibrium measure, 383 equations ∞ Φ (z) function, 377 δ function, 461 Φ-jump matrix, 375, 378 first-order differential equations, inequalities, 376, 381 462 lens-shaped region Ω and contour Γ , Hermite polynomials, 459 374 kernel, 460 one-cut condition, 381 Painlev´e representation, 463 “one interval” g-function, 380 Plancherel–Rotach asymptotics, Plemelj–Sokhotskii formulae, 459 378–379 s →−∞, 463 properties, 373–374 Fredholm determinant, 458–459, 463 quartic potential, 379, 380 Gaussian unitary ensemble (GUE), 253 Riemann theta-functions, 383 edge scaling determinant reduction, 474–476 Gioev, D., 346, 450 differential equations (see Gross, D., 314 Differential equations) Gr¨unbaum, A., 4 Fredholm determinant integral operator, 466–467 H matrix kernel, 466, 467 Haine, L., 4, 118 2N × 2 matrix, 465 Hammersley, J.M., 60 2 × 2n matrix, 466 Hankel determinants, 354 Pfaffian square, 464 Harish-Chandra-Itzykson-Zuber Gaussian specialization, 472 formula, 168–169 514 Index

Harish-Chandra-Itzykson-Zuber disjoint cycles, 236 integral, 446 finite-dimensional identity, 232, 233 Harnad, J., XIII, 4, 114, 252, 309, 346 finite-rank operators, 231 Hastings–Mcleaod Painlev´e transcen- inequality, 236 dents, 406 kernel, 234, 235 Hastings–McLeod solution, 295, 300, Lebesgue measure, 235 406 Lidski˘ı theorem, 233–234 Hastings, S.P., 406 operator with rank, 231, 236 Heine–Borel formula, 354 polar decomposition, 230 Heine’s formula singular numbers, 230 g-function, 268–269 triangle inequality, 231 heuristic semiclassical approxima- Hirota symbol residue identity, 221–223 tion, 268 Hotelling, H., 109, 110 matrix integral, 267–268 Huq, A., 4 Heisenberg, W., 142 Hermite polynomials I GOE, 487 Infinite wedge spaces GSE, 469, 470, 478 bi-infinite matrix, 42 GUE, 459 Cauchy–Binet formula, 43 Hermitian matrices Cauchy identity, 40–41 Chazy and Painlev´eequations, definition, 39 119–120 shift, 40 classical Hermitian matrix ensembles wedge products, 43 canonical correlation coefficients, wedging and contracting operator, 109–113 39 complex Gaussian population, 109 Integral operators GUE, 108 correlation functions and kernels Laguerre Hermitian ensemble, 109 Gaussian unitary ensemble, 243 diagonalization, 98 matrix, 241, 242 formal matrix integrals, 418 n-point correlation function, 238, Haar measure, 96 240 inner product, 97 orthogonal function, 240 model, XIV probability density of eigenvalues, n2 × n2 , 97 238 Painlev´e generation sine and Airy kernel, 243 GUE, 113–117 skew-orthogonal polynomials, 242 Jacobi ensemble, 118 two Hilbert spaces, 239 Laguerre ensemble, 117–118 Hilbert–Schmidt and trace class tangent space, 99 operators vectors, 97, 98 classical Fredholm determinant Hermitian , 315 formula, 236 Hikami, S., 199, 201, 206, 325, 326 continuous linear function, 232 Hilbert–Schmidt and trace class definition, 229, 230 operators determinant, 232, 237 classical Fredholm determinant disjoint cycles, 236 formula, 236 finite-dimensional identity, 232, continuous linear function, 232 233 definition, 229, 230 finite-rank operators, 231 determinant, 232, 237 inequality, 236 Index 515

kernel, 234, 235 asymptotic analysis, 399–402 Lebesgue measure, 235 double scaling limit, 394, 402 Lidskii theorem, 233–234 Painlev´e asymptotics, recurrence operator with rank, 231, 236 coefficients (see Painlev´e polar decomposition, 230 asymptotics) singular numbers, 230 parametrix at z =0(see triangle inequality, 231 Parametrix) Painlev´e functions second Painlev´eequation(see chain rule, 248 Second Painlev´eequation) commutator relations, 246–248 DKMVZ method differential equation, 247, 249 g-function (see g-function, distribution functions, 243 DKMVZ method) Ky(x)=K(x, y) condition, Naive approach, 373 244–245 eigenvalue statistics, 352–353 Painlev´e II equation, 243 elements of RH theory Sobolev space, 244, 246 Abelian case, 364 Its, A.R., XVI, 127, 134, 252, 308–310, Cauchy integral, 363 416 Cauchy theorem, 365, 368 Itzykson, C., 416 contour integral, 371 cyclic/trivial monodromy J condition, 362 Jacobi matrix, 309 Γ contours, 360, 361 James, A.T., 445, 446 Γ R contours, 366, 367 Jimbo, M., 114, 131, 138 jump matrix, 362, 372 Johansson, K., 24, 54, 60, 64, 65, 68, 72, Lax pair, 372 75, 120, 134, 311 L2-boundedness of C± and Johnstone, I.M., XVI, 449, 450 small-norm theorem, 369–371 non-Abelian case, 364–365 K non-tangential limit, 368 Kac, V.G., 39, 141 normalization condition, 368 Kanzieper, E., 416 piece-wise smooth contours, 363 Karlin, S., 200 Plemelj–Sokhotskii formulae, 362 Kazakov, V.A., 416, 431, 438 p × p invertible matrices, 360 Kerov, S.V., 60–62 Soliton theory, 372 Kitaev, A.V., 308–310 unbounded piece-wise smooth Koekoek, R., 58 contour, 366 Korteweg-de Vries equation, 84 Hermitian matrix model, 351–352 Kostov, I.K., 435 orthogonal polynomials Krasovsky, I., 127 arbitrary monic polynomial, 357 Kriecherbauer, T., 267, 346, 372 Cauchy integrals, 356 Kroenecker, 418 collection of polynomials, 353 Kuijlaars, A.B.J., XIV, 4, 199, 202, 206, domain ΩΨ 0, 358 252, 308, 311, 314, 346 Γ ± and C contours, 357 Kuznetsov, V., 4 Heine–Borel formula, 354 Liouville’s theorem, 359 L matrix valued function, 356 Laguerre polynomials, 76 partition function, 354 Large N asymptotics, random matrices Pn(z) orthogonality, 359 critical case positive constant Cq, 358 516 Index

Large N asymptotics, random matrices hard edge limit, 148–149 (cont.) Kontsevich integral, 147 residue theorem, 357 Laplace integral, 149 RH representation, 360 τ(t,E), condition, 147, 150 parametrix at end points vacuum condition, 147, 149 Airy equation, 388 Virasoro constraints, 148, 150 Airy functions, 389–390 KP hierarchy asymptotic analysis, 391–394 bilinear identity, τ-function, asymptotic relation, 387–388 140–141 differential equation, 388 isospectral deformation of L, final formula, 390–391 139 jump and asymptotic condition, ψ parametrization, 140 386 large n-limits “logarithmic derivative,” matrix bulk scaling limit, 137–138 valued function, 387 edge scaling limit, 138 model problem near z = z0, Gaussian case, 137 383–386 hard edge scaling limit, 139 monodromy theory, 387 Wigner’s semi-circle law, 137 Riemann–Hilbert method, 354–355 random matrix ensembles Lax pair equations, 296 crucial property, 135–136 Lebesgue measure, 351 Fredholm determinant, 136 Letendre, L., 134 Haar measure, n × n Hermitian Limit theorems, XIV matrices, 134–135 exponentially distributed percolation probability, 134 model, 75–76 vertex operators, soliton formulas geometrically distributed percolation and Fredholm determinants model (see Geometrically Fredholm determinant style distributed percolation model) soliton formula, 143–144 longest increasing sequences, 64–67 KP-vertex operator, 141 Plancherel measure, 60–64 link with kernels, 142 Liouville theorem, 262, 405 vertex identities, 142–143 Logan, B.F., 60, 61 Virasoro relations, Fredholm Longest increasing sequences determinant, 144–146 Bessel kernel, 64 McGregor, J., 200 probability of length, 64 McLaughlin, K.D.T.-R., 134, 267, 311, random permutation, 64–67 314, 346, 372, 373, 383, 416 McLeod, J.B., 406 M McMillan, E.M., 155 MacDonald, I.G., 6, 446 Mehta Hermite kernel, 186 Manakov, S.V., 355, 372 Mehta, M.L., 3, 96, 97, 104, 131, 186, Marcenko, V.A., 120 353, 452, 453, 480 Markovian process, 183, 184 Miller, P.D., 346 Matrix integrals and solitons Miwa, T., 114, 131, 138 differential equations, limiting Modified Pastur equation, 341–343 probability Mo, M.Y., 346, 428 Airy edge limit, 146–147 Moody, R., 141 A derivatives in KdV, 148, 150 Moor, G., 411 Grassmannian invariance Mori, Y., 114, 131, 138 condition, 147, 149 m-point correlation function, 352 Index 517

Muirhead, R.J., 109, 113, 444 N Multiple Hermite polynomials, 326 Naive approach, 373 Multivariate Gamma function, 444 Newell, A.C., 355 Multivariate statistics Nikiforov, A.F., 58 algorithms, 503–504 Normalized free energy, 254 determinant, 454–455 edge distribution functions O Fredholm determinant representa- Odlyzko, A., 61 tions, 448–449 Okounkov, A., 33, 39, 44, 64 universality theorems, 449–450 Olshanski, G., 33, 39, 64 F1 vs. empirical distributions, Ornstein–Uhlenbeck process, 185, 186 505 Orthogonal polynomials interlacing property, 499–502 equilibrium measure mean, standard deviation, skewness end-points, 266–267 and kurtosis data, 504, 505 equilibrium density, 265 mth largest eigenvalue distribution minimization problem, 264 GOE (see Gaussian orthogonal Wigner semicircle law, 265 ensemble) Euler–Lagrange variational condi- GSE (see Gaussian unitary tions, 265 ensemble) heuristics, 263–264 GUE (see Gaussian symplectic large N asymptotics, random ensemble) matrices arbitrary monic polynomial, 357 Painlev´erepresentations Cauchy integrals, 356 Airy equation, 452 collection of polynomials, 353 edge scaling limit, 453 domain ΩΨ 0, 358 Gaussian β-ensembles, 451 Γ ± and C contours, 357 GOE matrices, 453, 454 Heine–Borel formula, 354 interlacing property, 453 R Liouville’s theorem, 359 MATLAB , 454 matrix valued function, 356 Tracy–Widom distributions, 451 partition function, 354 partial derivatives of q(x, λ), 503 Pn(z) orthogonality, 359 recursion formula, eigenvalue positive constant Cq, 358 distributions residue theorem, 357 β-ensemble, 455, 457 RH representation, 360 binomial coefficient, 456 m∞ factorization, 77 integrand, 456, 457 moment matrix, 76 joint density function, 455 monic, 78–80 symmetric function, 456 n degree polynomial and Stieltjes Σ =c Ip, 446–447 transform, 77 Wishart distribution off-diagonal elements, 81 covariance matrix, 444, 445 orthonormal polynomials, 78 2 χ test, 443 RHP joint density function, 445 adjoint functions, 260–261 multivariate Gamma function, eigenvalue correlation functions, 444 263 principle component analysis, 445 with external source, 318–319 Schur polynomials, 446 first transformation, 269–270 zonal polynomials, 446 Heine’s formula, 267–269 518 Index

Orthogonal polynomials (cont.) Painlev´e functions Liouville theorem, 262 chain rule, 248 matrix-valued function, 261, 262 commutator relations, multicut case, 274–280 246–248 normalizing constant, 262–263 differential equation, 247, 249 one-cut case, 272–274 distribution functions, 243 parametrix construction, edge Ky(x)=K(x, y) condition, points, 280–285 244–245 recurrence coefficients, 262 Painlev´e II equation, 243 recurrent coefficients, asymp- Sobolev space, 244, 246 totics, 288–291 Painlev´e generation RN (z), 287–288 GUE second transformation, 271–272 differential operators, 114–116 third and final transformation, disjoint union of intervals, 113, 286 114 universality, 291–294 KP equation, 114 semi-infinite column, 77 PDE and ODE, 114, 117 semi-infinite , 77 t-differentiation, 115 tridiagonal , 80 Jacobi ensemble, 118 unitary ensembles Laguerre ensemble, 117–118 Christoffel–Darboux formula, 256 Painlev´e II equation, 295, 300 differential equations, ψ-functions, Painlev´e, P., 132, 148, 151, 191 259 Painlev´e type relations, 132 discrete string equations, 257 Parametrix eigenvalues, 254–255 at end points GUE, 253 Airy equation, 388 infinite Hamiltonian, 257–259 Airy functions, 389–390 Lax pair, discrete string equations, asymptotic analysis, 391–394 260 asymptotic relation, 387–388 polynomial interaction, 252 differential equation, 388 symmetric tridiagonal Jacobi final formula, 390–391 matrix, 256 jump and asymptotic condition, three-term recurrence relation, 386 255–256 “logarithmic derivative,” matrix topological large N expansion, valued function, 387 253–254 model problem near z = z0, Ostensson,¨ J., 308 383–386 monodromy theory, 387 P at z =0 Painlev´e asymptotics contour Γ c, 395 asymptotic series, 407, 409 Φˆ-RH problem, 398–399 Hastings–McLeod Painlev´e function, h(z) function analysis, 394–395 410 holomorphic and invertible holomorphicity, 408 functions, 396 jump matrix, 408 “interior” and “exterior” function, matrix coefficient, 407 398 Rn, 410–411 matrix function, 397 uniqueness theorem, 409 normalization condition, 398 z →∞, 407 z → w, 397 Index 519

Parametrix construction Cauchy transform, 211 a > 1 identity, 209 at edge points, 329–332 probability, 209 first transformation, RHP, 327 probability distribution, 208, 209 integrals of ξ-functions, 326–327 self-evident formal residue model RHP, 328–329 identity, 211 Riemann surface, 326 wave functions, 210 second transformation, RHP, 328 wave matrices, 209–210 third and final transformation, MOPS and Riemann–Hilbert 333 problem, 202–204 a < 1 universal behavior at branch points, 338–340 Brownian kernel, 206–207 first transformation Y → U, Brownian motion, 205 334–335 density of eigenvalues, 205 fourth transformation S → R, 340 extended Pearcey kernel, 208 λ−functions, 333–334 mean density of states, 206 second transformation U → T, PDE, 204 335, 336 random matrices, 205 third transformation T → S, Virasoro constraints, integrable 335–338 deformations, 213–218 near critical point Pearcey kernels, 325–326 double scaling asymptotics, Pearson, K., 443 305–308 Percolation and growth models model solution, 300–301 discrete polynuclear growth models, V (z) construction, 303–304 32–34 ζ(z) construction, 301–305 exponentially distributed entries, near turning points, 299–300 28–30 at origin, 343–345 geometrically distributed entries, in ΩWKB, 297–299 26–28 Parisi, G., 416 probability, 24, 25 Partition probability queuing problem, 30–32 Cauchy’s identity, 33 Periwal, V., 411 Fredholm determinant, 44–48 Permutations infinite wedge spaces (see Infinite Cauchy identity, 11–13 wedge spaces) percolation and growth models Toeplitz determinants, 35–39 discrete polynuclear growth U(n) integrals, 48–50 models, 32–34 Pastur equation, 322 exponentially distributed entries, Pastur, L.A., 120 28–30 Pauli matrix, 297 geometrically distributed entries, Pearcey distribution 26–28 Gaussian ensemble, 218–221 probability, 24, 25 GUE, external source and Brownian queuing problem, 30–32 motion probability measure on words, 22–24 non-intersecting Brownian paths, Robinson–Schensted–Knuth corre- 200–201 spondence, 9–11 random matrix, 201–202 subsequences, 4–6 3-KP deformation uniform probability bilinear identity, 210, 212–213 anticipation, n asymptotics, 15–16 520 Index

Permutations (cont.) R Plancherel measure, 13–15 Raina, A., 39 random walks, 17–22 Rains, E.M., 61, 155, 453 solitaire game, 15 Random and integrable models transition probability, 16–17 correlations and Fredholm determi- Young diagrams and Schur nants polynomials 1- and 2-point correlations, 105 hook length, 7 computation, 106 number of semi-standard Young differentiation, 107 tableaux, 8 multinomial coefficient, 106 number of standard Young probability distribution, 104 tableaux, 8 Hermitian matrices, 96–99 Schur function, 7 Chazy and Painlev´eequations, semi-standard Young tableau, 6 119–120 skew-Schur polynomial, 7 classical Hermitian matrix standard Young tableau, 6 ensembles, 108–113 weight, 6 Painlev´e generation, 113–118 Pieri’s formula, 8 kernel reproduction Plancherel, M., XIII, XIV determinant, 102, 103 Plancherel measure m induction, 104 and Gessel’s theorem n × n Hermitian matrix, 102 generating function, 51 symmetric matrix, 103 poissonized Plancherel measure, large Hermitian matrix ensembles 51, 52 GUE and Wigner’s semi-circle, probability restriction, 50, 51 120–122 recurrence relation, Bessel soft edge scaling limit (see Soft functions, 53 edge scaling limit) limit theorems limit theorems function, 62 exponentially distributed hook length, 61 percolation model, 75–76 Monte Carlo simulation, 60, 61 geometrically distributed percola- numerical computation, 60 tion model (see Geometrically Vershik–Kerov and Logan–Shepp distributed percolation model) theorem, 62–64 longest increasing sequences, Plancherel–Rotach asymptotics, 459 64–67 Plemelj–Sokhotskii formulae, 378–379 Plancherel measure, 60–64 Poincar´e, J.H., 136 partition probability Pr¨ahofer, M.M., 188, 191 Cauchy’s identity, 33 Principle component analysis, 445 Fredholm determinant, 44–48 Pronko, A.G., 346 infinite wedge spaces (see Infinite Psi-matrix, 296 wedge spaces) Toeplitz determinants, 35–39 Q U(n) integrals, 48–50 Queuing problem percolation departure time, 31 Gauss’ hypergeometric function, first-in first-out service rule, 30 58 random matrix, 31 Laurent series expansion, 58 servers, 30, 31 Meixner polynomials, 57, straightforward calculation, 32 59–60 Index 521 permutations Random matrix models. See also Cauchy identity, 11–13 Random and integrable models models (see Percolation and double scaling limit growth models) complex plane partition, 297 probability measure on words, double-well quartic matrix model, 22–24 294 Robinson–Schensted–Knuth final transformation, 345–346 correspondence, 9–11 free energy, 315 subsequences, 4–6 modified Pastur equation, uniform probability (see Uniform 341–343 probability) monic orthogonal polynomials, Young diagrams and Schur 294 polynomials, 6–8 Painlev´e II equation, 295 Plancherel measure and Gessel’s parametrix construction (see theorem Parametrix construction) generating function, 51 Pauli matrix, 297 poissonized Plancherel measure, Psi-matrix, 296 51, 52 recurrent coefficient, 295 probability restriction, 50, 51 scaling functions, 294–295 recurrence relation, Bessel Schr¨odinger equation, 296 functions, 53 semiclassical approximation, 297 probability, random words, 54–56 with external source random Hermitian ensemble Christoffel–Darboux formula, Christoffel–Darboux identity, 101 317 eigenvalue correlation functions, orthonormal polynomials, 100 317–318 probability distribution, 99 Gaussian model (see Gaussian symmetric kernel, 100, 101 model) Zn normalization, 101 Hermitian random matrix, 315 statistical distributions, 4 monic polynomial, 316 time-dependent weight and KP parametrix construction (see equation, 81–88 Parametrix construction) Hirota symbol, 81 recurrence and differential holomorphic functions, 82 equations, 319–321 Lie algebra splitting, 81 RHP, 318–319 m∞ factorization, 81 free energy analyticity, regular V, moment matrix determinants, 310–311 82–88 one-sided analyticity, critical point, orthogonal polynomials (see 313–314 Orthogonal polynomials) orthogonal polynomials (see Taylor’s formula, 82 Orthogonal polynomials) Virasoro equations, 82 partition function, 308–310 weight, parameters, 81 topological expansion, 311–313 Virasoro constraints Random matrix theory (RMT), 131, differential operators, 93 352 β-integrals, 88–93 Random walks Jacobi ensemble, 95–96 analogous argument, 22 Laguerre ensemble, 94–95 canonical way, 19 linear combinations, 94 configurations, 19, 20 522 Index

Random walks (cont.) first transformation, 269–270 Dyson Brownian motion and Airy formal matrix integrals, 440 process, 182 Heine’s formula horizontal and vertical segments, 21 g-function, 268–269 2m possibilities, 17 heuristic semiclassical approxima- one-to-one map, 20 tion, 268 probability, 17 matrix integral, 267–268 six nonintersecting walkers, 18 Liouville theorem, 262 Recursion relations, unitary integrals matrix-valued function, 261, 262 basic involution, 152 multicut case basic recursion variables, 151–152 arbitrary constant vectors, 277, bi-orthogonal polynomials 278 dynamical variables, 158 auxiliary RHP, 274 inner product, 157 basis of cycles, 275 Ladik–Ablowitz lattice, 159 combining formulas, 278–280 Toda equations, 158–159 holomorphic one-forms, 277 2-Toda τ functions, 157–158 linear space, 275–276 Toeplitz lattice, 157 matrix-valued function, 278 combinatorics, 154–156 normalization, 276 rational relations, (x, y), 153–154 periodicity properties, 276–277 recursion matrices, 152 Riemann theta function, 276 self-dual case, 152 two-sheeted Riemannian surface, singularity confinement 274–275 analogous property, 163 normalizing constant, 262–263 differential equations, 166 one-cut case, 272–274 formal Laurent solutions, 163, parametrix construction, edge points 165–166 Airy function, 283 general case, 164 analyticity of EN (z), 284–285 self-dual case, 163, 165 analytic prefactor, 284 Toeplitz matrices, 152 Virasoro constraints and difference convex neighborhood, 281–282 relations discontinuity rays, 283 explicit Virasoro formulas, 161 jump condition, 282–283 integrated version, 159 local parametrix, 280–281 main facts, 160–161 singularity, 285 Toda vertex operator, 159 upper half-plane, 281 Toeplitz flow, 162 recurrence coefficients, 262 Virasoro operator, 159 RN (z), 287–288 Virasoro relations, 161–162 second transformation, 271–272 weight r(z), 151 third and final transformation, 286 Riemann–Hilbert method, XV, universality 354–355 N →∞, 293–294 Riemann–Hilbert problem (RHP) N → 1, 292–293 adjoint functions, 260–261 reproducing kernel, 291–292 asymptotics, recurrent coefficients, Riemann theta-functions, 276, 383 288–291 Robinson–Schensted–Knuth (RSK) eigenvalue correlation functions, correspondence, XIII, 9–11 263 Runge–Kutta–based ODE solver, with external source, 318–319 503 Index 523

S Stokes matrices, 405, 406 Sagan, B.E., 6 Suslov, S.K., 58 Sato, M., 114, 131, 138 Swarttouw, R.F., 58 Schlesinger-type systems, 372 Szekeres, G., 60 Schr¨odinger equation, 296 T Schur polynomials, 12, 13, 446 Schwinger–Dyson equations, 423–424 Takasaki, K., 133, 158 Scoufis, G., 119 Taylor, B., 213 Second Painlev´eequation Temperly–Lieb algebra, 440 Airy-solvable problem, 404 t’Hooft, G., 416, 420–423 compatibility condition, 405 Toeplitz determinants, XIII–XIV contour Γ p, 403, 404 differential equations, 36 equilibrium measure, 403 Fourier series, 37 Flaschka–Newell Lax pair, 405 initial value problem, 36 × Hastings–McLeod solution, 406 n n upper-left blocks, 37 jump and asymptotic condition, 404 probability measure, 35 linear differential equations, 405 semi-infinite , 36 Liouville theorem, 405 sequence of integers, 38 Stokes matrices, 405, 406 Young diagram, 37, 38 trivial gauge transformation, 403 Toeplitz Lattice, 133 Seiberg, N., 411 Tracy, C.A., XVI, 35, 54, 104, 114, 117, Selberg integral approach, 354 127, 131, 134, 138, 139, 155, Semengue, J.-P., 118 156, 188, 206–208, 321, 325, Sepp¨al¨ainen, T., 60 326, 452, 474, 490, 499 Shabat, A.B., 355 Tracy–Widom distributions Shenker, S.H., 411 GOE, 499 Shepp, L.A., 60 GSE, 480 Shevitz, D., 411 Painlev´e representations, 451 Shiota, T., 114, 134, 141, 148–151 Tutte, W.T., 426 Sine process, 188, 190–191 Two-sheeted Riemannian surface, Soft edge scaling limit 274–275 Airy function, 127 U Fredholm determinant, 124, 126 Ueno, K., 133, 158 Hastings–McLeod solution, 123 Ulam problem, XIII × n GUE, 122 Uniform probability PDE, 125 anticipation, n asymptotics, 15–16 Plancherel–Rotach asymptotic Plancherel measure, 13–15 formula, 123 random walks, 17–22 3rd-order non-linear PDE, 122 solitaire game, 15 Tracy–Widom distribution, 123 transition probability, 16–17 Soliton-like tau function, XIV Uvarov, V.B., 58 Soliton theory, 383 Soshnikov, A., 450 V Spectral theory, XV Van Assche, W., 4 Spohn, H., 188, 191 Vandermonde, A.-T., 134, 185, 238 Stanley, R.P., 6, 456 Vanderstichelen, D., 4 Stanton, D., 6 Vanhaecke, P., 4, 88, 134, 163, 174, 175, Stirling’s formula, 68 209–213 Stojanovic, A., 346 Vanlessen, M., 4, 308, 346 524 Index van Moerbeke, P., XIII, 35, 88, 99, 113, Wigner, E., 3, 131 114, 118, 134, 141, 148–152, Wigner semicircle law, 265 157, 163, 167, 174, 175, 177, Wishart distribution, XVI 189, 204, 207–213, 219–221, Wishart, J., 443–446, 450, 454, 505, 326, 372 XVI Venakides, S., 267, 372, 416 Witte, N., 134 Verbaarschot, J., 102 Witten, E., 314 Vershik, A.M., 60–62 Vertex Lax equation, 142 Y Virasoro constraints, XIV, 372 Young diagrams differential operators, 93 and Schur polynomials β-integrals hook length, 7 Adler–van Moerbeke theorem (see number of semi-standard Young Adler–van Moerbeke theorem) tableaux, 8 weights, 88 number of standard Young Jacobi ensemble, 95–96 tableaux, 8 Laguerre ensemble, 94–95 Schur function, 7 linear combinations, 94 semi-standard Young tableau, 6 Virasoro identities, 132 skew-Schur polynomial, 7 Virasoro Lie algebra, 132 standard Young tableau, 6 Virasoro, M.A., 132, 142 weight, 6 W Toeplitz determinants, 37, 38 Wang, D., 4 White, D., 6 Z Wick’s theorem, 421, 422 Zakharov, V.E., 372 Widom, H., XV, 35, 44, 54, 104, 114, Zhou, X., 127, 267, 355, 372 117, 127, 131, 134, 138, 139, Zhou, Z., 416 155, 156, 188, 206–208, 229, Zinn-Justin, P., 317, 346, 449 321, 325, 326, 452, 474, 490, Zonal polynomials, 446 499 Zuber, J., 416