Double contour integral formulas for the sum of GUE and one matrix model
|
|
- Clare Whitehead
- 6 years ago
- Views:
Transcription
1 Double contour integral formulas for the sum of GUE and one matrix model Based on arxiv: with Tom Claeys, Arno Kuijlaars, and Karl Liechty Dong Wang National University of Singapore Workshop on Random Product Matrices Bielefeld University, Germany, 3 August, 016
2 Gaussian Unitary Ensemble, everyone knows it Let M be an n n Hermitian matrix, with diagonal entries in i.i.d. normal distribution N(1, 0), and upper-triangular entries in i.i.d. complex normal distribution, with real and imaginary parts in N(0, 1 ). Then the random matrix M is in the Gaussian Unitary Ensemble. Its eigenvalues are a determinantal process, described by the correlation functions, which are R k (x 1,..., x k ) = det(k n (x i, x j )) k i,j=1, where K n is the correlation kernel K n (x, y) = 0 n 1 k=0 1 πk! H k (x)e x /4 H k (y)e y /4. It is well known (in [Abramowitz Stegun]) H k (x) = k! e xt t / t k 1 dt = 1 i e (s x)/ s k ds. πi πi i
3 Double contour integral formula for GUE Taking the sum with the help of the telescoping trick, we have K n (x, y) = 1 i (πi) ds dt e(s x) / ( s ) n 1 i 0 e (t y) / t s t. The local statistics of the GUE, namely the limiting Airy distribution and limiting Sine distribution, can be derived by the saddle point analysis of the double contour integral formula. Figure: The space between consecutive eigenvalues is O(n /3 ). Airy distribution. Figure: The space between consecutive eigenvalues is O(n 1 ). Sine distribution.
4 Deformation of contours: Airy Figure: The deformed contours for Airy limit. The integral converges to K Airy (x, y) = 1 (πi) ds Figure: The limiting local shape of the contours. dt es3 /3 xs 1 e t3 /3 yt s t.
5 Deformation of contours: Sine We need to deform the contour to a greater degree, and allow the vertical contour to cut through the circular one. plus Figure: The deformed contours for Sine limit. Figure: The contour for the residue integral (wrt s). The two intersecting points are the saddle points. It turns out that the integral on the right is bigger, and gives the Sine kernel.
6 Other models solved by double contour integrals 1. GUE/Wishart with external source (GUE + diag(a 1,..., a n )) [Brezin Hikami], [Zinn-Justin], [Tracy Widom], [Baki Peche Ben Arous], [El Karoui], [Bleher Kuijlaars]. GUE/Wishart minor process (all upper-left corners together) [Johansson Nordenstam], [Dieker Warren] 3. Upper-triangular ensemble (XX, upper-triangular entries of X are i.i.d. complex normal, diagonal ones are in independent gamma distribution) and the related Muttalib Borodin model [Adler van Moerbeke W], [Cheliotis], [Forrester W], [Zhang] 4. Determinantal particle systems (TASEP, polynuclear growth model, Schur process, etc) Johansson, Spohn, Borodin, Farrari, and too many others
7 Pros and cons Pro When a double contour integral formula is obtained, the computation of asymptotics is a straightforward application of saddle point analysis. Con All models are related to special functions which have their own contour integral representations. If the model is defined by functions that are not special, or not special enough, then there is little hope to find a double contour formula. Pro Thanks to the recent development of models related to the product random matrices, we have abundant of models associated to special functions, namely Meijer-G functions. [all participants here, and many more]
8 Proof of Airy and Sine universality for the product of Ginibre matrices and the Muttalib Borodin model Figure: The deformed contours for Airy limit. Figure: For Sine limit. (The residue integral is omitted). The schematic figures applies for both the two models [Liu W Zhang], [Forrester W]. The computation of the hard edge limit, which is more interesting, can be done in a technically easier way by double contour integral formula. [Kuijlaars Zhang]
9 Review of one matrix model Consider the n-dimensional random Hermitian matrix M with pdf 1 exp( n Tr V (M)), C V is a potential. The distribution of eigenvalues is a determinantal process. To express the kernels, we consider orthogonal polynomials with weight e nv (x) : p j (x)p k (x)e nv (x) dx = δ jk h k, p k (x) = x k +. and then we have two equivalent kernels: Kn alg (x, y) = 1 p n (x)p n 1 (y) p n 1 (x)p n (y) e nv (y), h n 1 x y K n (x, y) = 1 p n (x)p n 1 (y) p n 1 (x)p n (y) e n V (x) e n V (y). h n 1 x y
10 Note that at x = 0, the density vanishes like a square function, in contrast with the vanishing of density two edges that hase the square root behaviour. We say that x = 0 is an (interior) singular point of the potential V (x) = x 4 /4 x. Cases to be considered We are interested in a particular potential function V = x 4 /4 px, especially if p = 1 or very close to 1, or more precisely, p 1 = O(n /3 ). The density of eigenvalues is shown in the figure (from [Claeys Kuijlaars]).
11 Solution to 1MM We need to compute the limit of K n (x, y), which is reduced to the asymptotics of p n (x) and p n 1 (x). They trick is that they satisfy the following Riemann Hilbert problem for such that Y (z) = ( 1 1 ) h n p n (z) h n Cp n (z) πih n 1 p n 1 (z) πih n 1 Cp n 1 (z) where Cp n (z) = 1 πi Cp n 1 (z) = 1 πi R R p n (s)e nv (s) ds, s z p n (s)e nv (s) ds, s z 1. Y + (x) = Y (x) ( 1 e for x R. nv (x) 0 1 ). Y (z) = (1 + O(z 1 ) ( ) z n 0 0 z n as z.
12 The asymptotics of Y is obtained by the Deift Zhou nonlinear steepest-descent method [Bleher Its], [DKMVZ]. Around the singular point 0, K n (n 1/3 x, n 1/3 y) K PII (x, y) = Φ 1(x; σ)φ (y; σ) Φ (x; σ)φ(y; σ), π(x y) where σ is propotional to n /3 (p 1), and (ψ (1) and ψ () are defined in next slide) Φ 1 (x; σ) = ψ (1) 1 Φ (x; σ) = ψ (1) (x; σ) + ψ() 1 (x; σ), (x; σ) + ψ() (x; σ).
13 x Riemann Hilbert problem associated to the Hastings McLeod solution to Painlevé II Let Ψ be a x matrix valued function, such that 1. Ψ is analytic on C\ the four rays and continuous up to the boundary.. Ψ + = Ψ A j on each ray, where the jump matrix A j is given in the figure. 3. Ψ(ζ) = Ψ(ζ; σ) = (I + O(ζ 1 )e i( 4 3 ζ3 +σζ)σ 3 as ζ, where σ 3 = ( ). ( 1 ) ( ) ( 1 ) 1 ( 1 ) Now denote the Ψ(ζ; σ) in the left and right sectors by (ψ (1) (ζ; σ), ψ () (ζ; σ)), where ψ (1) and ψ () are -vectors. (Yes, Ψ in these two sectors are identical.)
14 Review of two matrix model In the general form, the two matrix model has two n n random Hermitian matrices M 1, M with pdf 1 C exp[ n Tr(V (M 1) + W (M ) τm 1 M )], where V, W are potentials and τ is the interaction factor. We are interested in the case V (x) = x / and W (y) = y 4 / + (α/)y, and are interested in the distribution of the eigenvalues of M 1. The distribution of eigenvalues of M is much easier, and we are going to explain it below. Then the eigenvalues of M 1 are a determinantal process, with the correlation kernel K n (x, y) = (0, w 0,n(y), w 1,n (y), w,n (y))y+ 1 (y)y +(x)(1, 0, 0, 0) T, πi(x y) where Y is defined by a RHP in next slide.
15 4 4 Riemann Hilbert problem Consider the following Riemann Hilbert problem 1. Y + (x) = Y (x) ( 1 w0,n (x) w 1,n (x) w,n (x) x R, where the exact formulas of w i,n (x) are omitted. ), for. As z, Y (z) = (1 + O(z 1 )) diag(z n, z n/3, z n/3, z n/3 ). The RHP seems not too bad, while how hard it is depends on the value of τ and α in the following phase diagram (Duits, Geudens, Kuijlaars, Mo, Delvaux, Zhang..., figure from [Duits]): 1390 DUITS and GEU q D 1 Case IV Case III b 1 a p 1 Case II Case I D p C Figure. The phase diagram in the -plane: the critical curves D p C and D p separate the four cases. The cases are distinguished by the fact whether 0 is in the support
16 MM with quadratic potential Consider the MM with pdf C 1 exp[ n Tr(M 1 / + W (M ) τm 1 M )], that is, the potential V is quadratic. Then let W (x) = W (x) τ Then the joint pdf for M 1 and M is x, and M 1 = M 1 τm. 1 [ ] C exp n Tr( M 1 / + W (M )), and then M 1 and M are independent. We can think M 1 as M 1 + τm. So the two matrix model with one quadratic potential is equivalent to the sum of a GUE matrix (i.e. a random matrix in 1MM with quadratic potential) and a random matrix in a 1MM [Duits].
17 Correlation functions for GUE + (fixed) external source Let H be a GUE and A a fixed Hermitian matrix with eigenvalues a 1,..., a n, then the correlation kernel of the eigenvalues of A + H is 1 i (πi) ds i dt e n (s x) e n (t y) n ( s ak k=1 t a k ) 1 s t, where Γ encloses a 1,..., a n. Here we can allow a 1,..., a n to be random, and need to integrate over the distribution of a 1,..., a n. How if they are eigenvalues of a matrix model too?
18 Correlation function for GUE + 1MM Theorem [Claeys Kuijlaars W] Let M be a random matrix in 1MM, with random eigenvalues a 1,..., a n, then the correlation kernel of the eigenvalues of M + H is K n,mm (x, y) = 1 i (πi) ds i = 1 i ds πi i R [ dt e n n (s x) E e n (t y) k=1 dt e n (s x) e n (t y) e nv (t) h n 1 ( s ak t a k = 1 i ds dt e n [(s x) +V (s)] πi i R e n [(t y) +V (t)] K n(s, t). ) ] 1 s t 1 (p n (s)p n 1 (t) p n 1 (s)p n (t)) s t }{{} Kn alg (s,t)
19 Result in figure Suppose the distribution of eigenvalues in for M is given in the upper-left Figure, then as τ becomes larger, the distribution of eigenvalues of M + τh evolves, shown in figures clockwise, into subcritical, critical), and then supercritical phases. Below we consider M + rh, whose kernel is given by Kn,MM r 1 i (x, y) = ds dt e n [(s x) /r+v (s)] πi e n [(t y) /r+v (t)] K n(s, t). i R
20 Derivation: subcritical Suppose the critical value for r is r cr (0, ). Then for r (0, t cr ) there are c r, c r depending on r in the way that as r runs from 0 to r cr, then c r, c r 0. such that for any ξ, η R, if x = c t n 1/3 ξ, y = c t n 1/3 η, we have that the the functions (s x) /r + V (s) and (t y) /r + V (t) have the saddle point approximation (s x) /r+v (s) c r n /3 (u ξ), (t y) /r+v (t) c r n /3 (v η), where u = n 1/3 s, v = n 1/3 t. Thus we have Kn,MM r 1 i (x, y) du πi i R 1 i du πi i R K PII (ξ, η). dv e n 1/3 c r [(u ξ) ] e n1/3 c r [(v η) ] 1/3 c r [(u ξ) ] dv e n e n1/3 c r [(v η) ] K n (n 1/3 u, n 1/3 v) K PII (u, v)
21 Derivation: critical When r = r cr, both c r and c r become 0, and the argument in last slide breaks down. However, we can still assume that as r = r cr + n 1/3 τ, x = n /3 ξ, y = n /3 η, and have that (s x) /r+v (s) n 1 (bu +ξu), (t y) /r+v (t) n 1 (bv ηv), where u = n 1/3 s, v = n 1/3 t and b depends on τ. So we have Kn,MM r 1 i (x, y) du dv ebu ξu πi i R e bv ηv K PII(u, v). The problem is that the integral may not be well defined, even if we consider it formally. The reason is that the sign of b depends on the sign of τ, and can be either positive or negative, while as v ±, K PII (u, v) does not vanish. (The correct form can be written down with the help of longer formulas, and we omit them.)
22 Result in formula Here we note that the PII singularity is quite robust. If M has the PII singularity, then M + rh has too, if r < r cr. The critical kernel, the most interesting one, has the kernel [Claeys Kuijlaars Liechty W] (formally) Kn,MM r 1 i (x, y) = du dv eau3 +bu +ξu πi i R e av 3 +bv +ηv K PII(u, v). If the potential is symmetric, then parameter a vanishes, as we discussed in previous slide. But our method allows us to consider asymmetric potentials, and generally there is a cubic term in the exponents. We can also deal with higher singularities, or singularities at the edge. The equivalence to the previous result by 4 4 RHP is obtained [Liechty-W].
23 Tacnode Riemann Hilbert problem Let M be a 4 4 matrix-valued function, and suppose it satisfies the following Riemann Hilbert problem: J = J 1 = J 3 = J 0 = J 4 = J 5 =
24 1. M is analytic in each of the sectors j, continuous up to the boundaries, and M(z) = O(1) as z 0.. On the boundaries of the sectors j, M = M (j) satisfies the jump conditions M (j) (z) = M (j 1) (z)j j, for j = 0,..., 5, M ( 1) M (5), for the jump matrices J 0,..., J 5 with constant entries specified in the figure in last page. 3. As z, M(z) satisfies the asymptotics M(z) = ( 1 + O(z 1 ) ) (v 1 (z), v (z), v 3 (z), v 4 (z)), where v 1, v, v 3, and v 4 are defined as v 1 (z) = 1 e θ 1 (( z) (z)+τz 4 1, 0, i( z) 1 T 4, 0), v (z) = 1 e θ (0, (z) τz z 4 1, 0, iz 1 ) T 4, v 3 (z) = 1 e θ 1 ( i( z) (z)+τz 4 1, 0, ( z) 1 T 4, 0), v 4 (z) = 1 e θ (0, (z) τz iz 4 1, 0, z 1 ) T 4, where θ 1 (z) = 3 r 1( z) 3 + s 1 ( z) 1, z C \ [0, ), θ (z) = 3 r z 3 + s z 1, z C \ (, 0],
25 Integral representation Then define the 4-vectors n (k) (z) = n (k) (z; r 1, r, s 1, s, τ) = Q Γ (k)(f (k), g (k) ), k = 0,..., 5, where Q Γ (f, g)(z) := Γ e izζ C 1 M Γ e izζ C 1 Γ e izζ C 1 Γ e izζ C 1 f 1 (ζ)g 1 (ζ)dζ + Γ e izζ C f (ζ)g (ζ)dζ + Γ e izζ C f 1 (ζ)g 3 (ζ)dζ + Γ e izζ C f (ζ)g 4 (ζ)dζ + Γ e izζ C g 1 (ζ)g 1 (ζ)dζ + Γ e izζ C 3 g (ζ)g (ζ)dζ + Γ e izζ C 3 g 1 (ζ)g 3 (ζ)dζ + Γ e izζ C 3 g (ζ)g 4 (ζ)dζ + Γ e izζ C 3 (f 1 (ζ) + g 1 (ζ))g 1 (ζ)dζ (f (ζ) + g (ζ))g (ζ)dζ, (f 1 (ζ) + g 1 (ζ))g 3 (ζ)dζ (f (ζ) + g (ζ))g 4 (ζ)dζ and r τz( 1 r r M = e 1 +r ) ( ) i τ r 1 r r 1 r 1 +r + τ s1 + C u i r q r 1 γ i 0 r 1 C r 1, i γ ( ) r 1 q i r r τ r 1 r C r r 1 +r τ + s C u i 0 r
26 such that (C is defined below) a = 4 ( r 1 r ) 3 r1 + r, b = ( γ 1 = exp 8τ C (r 1 + r ), 8r 4 1 τ 3 3(r 1 + r )3 + 4r 1s 1 τ r 1 + r ) c = 1 [ 4τ (r 1 r ) ( s1 C (r1 + r s ) ], ) r 1 r (, γ = exp 8r 4 τ 3 3(r1 + r + 4r ) s τ )3 r1 + r, and then the function ( G(ζ) = exp iaζ 3 + bζ ) + icζ, and the related functions G 1 (ζ) = π γ 1 C G(ζ), G (ζ) = r 1 π γ C r G(ζ), G 3 (ζ) = i C ζg 1(ζ), G 4 (ζ) = i C ζg (ζ). The entries of M are expressed in ( C = (r 1 + r ) 1/3 8 r1, γ = exp r 3 (r1 + r τ 3 4 r ) 1s 1 r s ) r1 + r τ, and q and u are functions of σ := C ( s1 r 1 + s r τ r 1 + r ).
27 Furthermore, q = q(σ) satisfies the Painlevé II equation with Hastings McLeod initial condition (Ai is the Airy function) q (σ) = σq + q 3, q(σ) Ai(σ) as σ +, = q (σ) is the derivative with respect to σ, and u is the PII Hamiltonian q ψ () 0 n (0) ψ () ψ (1) + ψ () n (1) ψ () ψ (1) u(σ) := q (σ) q(σ) q(σ) 4. ψ () At last, we can specify the contours Γ (k) j and functions f (k) and g (k) in the integrands as in the figure. ψ () ψ (1) ψ (1) + ψ () n () ψ (1) + ψ () ψ (1) ψ (1) ψ (1) 0 n (3) ψ (1) + ψ () ψ () ψ (1) n (4) n (5)
28 Theorem [Liechty-W] The 4x4 RHP M can be expressed by n (k), the integrals involving entries of Ψ. ( M (0) = n (5) n (0), n (0), n (1), n ()), ( M (1) = n (3), n (0), n (1), n ()), ( M () = n (3), n (4), n (1) + n (), n ()), ( M (3) = n (3), n () n (3), n (5), n (4)), ( M (4) = n (3), n (0), n (5), n (4)), ( M (5) = n (1), n (0), n (5), n (4) + n (5)).
Universality 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 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 informationPainlevé 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 informationNumerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations
Numerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations Tamara Grava (SISSA) joint work with Christian Klein (MPI Leipzig) Integrable Systems in Applied Mathematics Colmenarejo,
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 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 informationCentral Limit Theorems for linear statistics for Biorthogonal Ensembles
Central Limit Theorems for linear statistics for Biorthogonal Ensembles Maurice Duits, Stockholm University Based on joint work with Jonathan Breuer (HUJI) Princeton, April 2, 2014 M. Duits (SU) CLT s
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 informationDomino tilings, non-intersecting Random Motions and Critical Processes
Domino tilings, non-intersecting Random Motions and Critical Processes Pierre van Moerbeke Université de Louvain, Belgium & Brandeis University, Massachusetts Institute of Physics, University of Edinburgh
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 informationOPSF, Random Matrices and Riemann-Hilbert problems
OPSF, Random Matrices and Riemann-Hilbert problems School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, ICMAT 23 27 October, 207 Plan of the course lecture : Orthogonal Polynomials
More informationTwo-periodic Aztec diamond
Two-periodic Aztec diamond Arno Kuijlaars (KU Leuven) joint work with Maurice Duits (KTH Stockholm) Optimal and Random Point Configurations ICERM, Providence, RI, U.S.A., 27 February 2018 Outline 1. Aztec
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 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 informationJINHO BAIK Department of Mathematics University of Michigan Ann Arbor, MI USA
LIMITING DISTRIBUTION OF LAST PASSAGE PERCOLATION MODELS JINHO BAIK Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA E-mail: baik@umich.edu We survey some results and applications
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 informationThe Hermitian two matrix model with an even quartic potential. Maurice Duits Arno B.J. Kuijlaars Man Yue Mo
The Hermitian two matrix model with an even quartic potential Maurice Duits Arno B.J. Kuijlaars Man Yue Mo First published in Memoirs of the American Mathematical Society in vol. 217, no. 1022, 2012, published
More informationBulk scaling limits, open questions
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
More informationOPSF, Random Matrices and Riemann-Hilbert problems
OPSF, Random Matrices and Riemann-Hilbert problems School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, ICMAT 23 27 October, 2017 Plan of the course lecture 1: Orthogonal
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 informationIntegrable probability: Beyond the Gaussian universality class
SPA Page 1 Integrable probability: Beyond the Gaussian universality class Ivan Corwin (Columbia University, Clay Mathematics Institute, Institute Henri Poincare) SPA Page 2 Integrable probability An integrable
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 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 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 informationUniformly Random Lozenge Tilings of Polygons on the Triangular Lattice
Interacting Particle Systems, Growth Models and Random Matrices Workshop Uniformly Random Lozenge Tilings of Polygons on the Triangular Lattice Leonid Petrov Department of Mathematics, Northeastern University,
More informationFrom the Pearcey to the Airy process
From the Pearcey to the Airy process arxiv:19.683v1 [math.pr] 3 Sep 1 Mark Adler 1, Mattia Cafasso, Pierre van Moerbeke 3 Abstract Putting dynamics into random matrix models leads to finitely many nonintersecting
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 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 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 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 informationNUMERICAL CALCULATION OF RANDOM MATRIX DISTRIBUTIONS AND ORTHOGONAL POLYNOMIALS. Sheehan Olver NA Group, Oxford
NUMERICAL CALCULATION OF RANDOM MATRIX DISTRIBUTIONS AND ORTHOGONAL POLYNOMIALS Sheehan Olver NA Group, Oxford We are interested in numerically computing eigenvalue statistics of the GUE ensembles, i.e.,
More informationPartition function of one matrix-models in the multi-cut case and Painlevé equations. Tamara Grava and Christian Klein
Partition function of one matrix-models in the multi-cut case and Painlevé equations Tamara Grava and Christian Klein SISSA IMS, Singapore, March 2006 Main topics Large N limit beyond the one-cut case
More informationDirected random polymers and Macdonald processes. Alexei Borodin and Ivan Corwin
Directed random polymers and Macdonald processes Alexei Borodin and Ivan Corwin Partition function for a semi-discrete directed random polymer are independent Brownian motions [O'Connell-Yor 2001] satisfies
More informationProducts of Rectangular Gaussian Matrices
Products of Rectangular Gaussian Matrices Jesper R. Ipsen Department of Physics Bielefeld University in collaboration with Gernot Akemann Mario Kieburg Summer School on Randomness in Physics & Mathematics
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 informationIntegrable Probability. Alexei Borodin
Integrable Probability Alexei Borodin What is integrable probability? Imagine you are building a tower out of standard square blocks that fall down at random time moments. How tall will it be after a large
More informationLECTURE III Asymmetric Simple Exclusion Process: Bethe Ansatz Approach to the Kolmogorov Equations. Craig A. Tracy UC Davis
LECTURE III Asymmetric Simple Exclusion Process: Bethe Ansatz Approach to the Kolmogorov Equations Craig A. Tracy UC Davis Bielefeld, August 2013 ASEP on Integer Lattice q p suppressed both suppressed
More informationMultiple orthogonal polynomials. Bessel weights
for modified Bessel weights KU Leuven, Belgium Madison WI, December 7, 2013 Classical orthogonal polynomials The (very) classical orthogonal polynomials are those of Jacobi, Laguerre and Hermite. Classical
More informationThe Density Matrix for the Ground State of 1-d Impenetrable Bosons in a Harmonic Trap
The Density Matrix for the Ground State of 1-d Impenetrable Bosons in a Harmonic Trap Institute of Fundamental Sciences Massey University New Zealand 29 August 2017 A. A. Kapaev Memorial Workshop Michigan
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 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 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 informationAppearance of determinants for stochastic growth models
Appearance of determinants for stochastic growth models T. Sasamoto 19 Jun 2015 @GGI 1 0. Free fermion and non free fermion models From discussions yesterday after the talk by Sanjay Ramassamy Non free
More informationarxiv:hep-th/ v1 14 Oct 1992
ITD 92/93 11 Level-Spacing Distributions and the Airy Kernel Craig A. Tracy Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, CA 95616, USA arxiv:hep-th/9210074v1
More informationNumerical solution of Riemann Hilbert problems: random matrix theory and orthogonal polynomials
Numerical solution of Riemann Hilbert problems: random matrix theory and orthogonal polynomials Sheehan Olver and Thomas Trogdon February, 013 Abstract Recently, a general approach for solving Riemann
More informationDuality, Statistical Mechanics and Random Matrices. Bielefeld Lectures
Duality, Statistical Mechanics and Random Matrices Bielefeld Lectures Tom Spencer Institute for Advanced Study Princeton, NJ August 16, 2016 Overview Statistical mechanics motivated by Random Matrix theory
More informationarxiv: v3 [math-ph] 23 May 2017
arxiv:161.8561v3 [math-ph] 3 May 17 On the probability of positive-definiteness in the ggue via semi-classical Laguerre polynomials Alfredo Deaño and icholas J. Simm May 5, 17 Abstract In this paper, we
More informationDiscrete Toeplitz Determinants and their Applications
Discrete Toeplitz Determinants and their Applications by Zhipeng Liu A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Mathematics in The University
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 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 informationNONINTERSECTING BROWNIAN MOTIONS ON THE UNIT CIRCLE
The Annals of Probability 206, Vol. 44, No. 2, 34 2 DOI: 0.24/4-AOP998 Institute of Mathematical Statistics, 206 NONINTERSECTING BROWNIAN MOTIONS ON THE UNIT CIRCLE BY KARL LIECHTY AND DONG WANG DePaul
More information1 Assignment 1: Nonlinear dynamics (due September
Assignment : Nonlinear dynamics (due September 4, 28). Consider the ordinary differential equation du/dt = cos(u). Sketch the equilibria and indicate by arrows the increase or decrease of the solutions.
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 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 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 informationDomino tilings, non-intersecting Random Motions and Integrable Systems
Domino tilings, non-intersecting Random Motions and Integrable Systems Pierre van Moerbeke Université de Louvain, Belgium & Brandeis University, Massachusetts Conference in honour of Franco Magri s 65th
More informationRiemann-Hilbert problems from Donaldson-Thomas theory
Riemann-Hilbert problems from Donaldson-Thomas theory Tom Bridgeland University of Sheffield Preprints: 1611.03697 and 1703.02776. 1 / 25 Motivation 2 / 25 Motivation Two types of parameters in string
More informationProducts and ratios of characteristic polynomials of random Hermitian matrices
JOURAL OF ATHEATICAL PHYSICS VOLUE 44, UBER 8 AUGUST 2003 Products and ratios of characteristic polynomials of random Hermitian matrices Jinho Baik a) Department of athematics, Princeton University, Princeton,
More informationSolutions to practice problems for the final
Solutions to practice problems for the final Holomorphicity, Cauchy-Riemann equations, and Cauchy-Goursat theorem 1. (a) Show that there is a holomorphic function on Ω = {z z > 2} whose derivative is z
More informationFourier-like bases and Integrable Probability. Alexei Borodin
Fourier-like bases and Integrable Probability Alexei Borodin Over the last two decades, a number of exactly solvable, integrable probabilistic systems have been analyzed (random matrices, random interface
More informationRandom matrix theory and log-correlated Gaussian fields
Random matrix theory and log-correlated Gaussian fields Nick Simm Collaborators : Yan Fyodorov and Boris Khoruzhenko (Queen Mary, London) Mathematics Institute, Warwick University, Coventry, UK XI Brunel-Bielefeld
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 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 informationarxiv: v2 [math.pr] 9 Aug 2008
August 9, 2008 Asymptotics in ASEP with Step Initial Condition arxiv:0807.73v2 [math.pr] 9 Aug 2008 Craig A. Tracy Department of Mathematics University of California Davis, CA 9566, USA email: tracy@math.ucdavis.edu
More informationφ 3 theory on the lattice
φ 3 theory on the lattice Michael Kroyter The Open University of Israel SFT 2015 Chengdu 15-May-2015 Work in progress w. Francis Bursa Michael Kroyter (The Open University) φ 3 theory on the lattice SFT
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 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 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 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 informationRandom Growth and Random Matrices
Random Growth and Random Matrices Kurt Johansson Abstract. We give a survey of some of the recent results on certain two-dimensional random growth models and their relation to random matrix theory, in
More informationarxiv: v1 [math-ph] 30 Jun 2011
arxiv:1106.6168v1 [math-ph] 30 Jun 2011 Orthogonal polynomials in the normal matrix model with a cubic potential Pavel M. Bleher and Arno B.J. Kuijlaars July 1, 2011 Abstract We consider the normal matrix
More informationGeometric Dyson Brownian motion and May Wigner stability
Geometric Dyson Brownian motion and May Wigner stability Jesper R. Ipsen University of Melbourne Summer school: Randomness in Physics and Mathematics 2 Bielefeld 2016 joint work with Henning Schomerus
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 informationThe Pearcey Process. Mathematical Physics. Communications in. Craig A. Tracy 1, Harold Widom 2
Commun. Math. Phys. 263, 381 400 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1506-3 Communications in Mathematical Physics The Pearcey Process Craig A. Tracy 1, Harold Widom 2 1 Department
More informationInsights into Large Complex Systems via Random Matrix Theory
Insights into Large Complex Systems via Random Matrix Theory Lu Wei SEAS, Harvard 06/23/2016 @ UTC Institute for Advanced Systems Engineering University of Connecticut Lu Wei Random Matrix Theory and Large
More informationSPA07 University of Illinois at Urbana-Champaign August 6 10, 2007
Integral Formulas for the Asymmetric Simple Exclusion Process a Craig A. Tracy & Harold Widom SPA07 University of Illinois at Urbana-Champaign August 6 10, 2007 a arxiv:0704.2633, supported in part by
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 informationRandom Matrices and Determinantal Process
Random Matrices and Determinantal Process February 27 - March 3, 2017 Gernot Akemann: Universality in products of two coupled random matrices: Finite rank perturbations The singular values of products
More informationarxiv: v2 [math.pr] 29 Nov 2009
Some Examples of Dynamics for Gelfand-Tsetlin Patterns Jon Warren Peter Windridge warren@stats.warwick.ac.uk p.windridge@warwick.ac.uk Department of Statistics, Department of Statistics, University of
More informationDEVIATION INEQUALITIES ON LARGEST EIGENVALUES
DEVIATION INEQUALITIES ON LARGEST EIGENVALUES M. Ledoux University of Toulouse, France In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and
More informationFluctuations of interacting particle systems
Stat Phys Page 1 Fluctuations of interacting particle systems Ivan Corwin (Columbia University) Stat Phys Page 2 Interacting particle systems & ASEP In one dimension these model mass transport, traffic,
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 informationKPZ growth equation and directed polymers universality and integrability
KPZ growth equation and directed polymers universality and integrability P. Le Doussal (LPTENS) with : Pasquale Calabrese (Univ. Pise, SISSA) Alberto Rosso (LPTMS Orsay) Thomas Gueudre (LPTENS,Torino)
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 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 information4.3 Classical model at low temperature: an example Continuum limit and path integral The two-point function: perturbative
Contents Gaussian integrals.......................... Generating function.......................2 Gaussian expectation values. Wick s theorem........... 2.3 Perturbed gaussian measure. Connected contributions.......
More informationSeptember 29, Gaussian integrals. Happy birthday Costas
September 29, 202 Gaussian integrals Happy birthday Costas Of course only non-gaussian problems are of interest! Matrix models of 2D-gravity Z = dme NTrV (M) in which M = M is an N N matrix and V (M) =
More informationA Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices
Journal of Statistical Physics, Vol. 18, Nos. 5/6, September ( ) A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices Alexander Soshnikov 1 Received
More informationEcon 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines
Econ 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines Maximilian Kasy Department of Economics, Harvard University 1 / 37 Agenda 6 equivalent representations of the
More informationA Riemann Hilbert approach to Jacobi operators and Gaussian quadrature
A Riemann Hilbert approach to Jacobi operators and Gaussian quadrature Thomas Trogdon trogdon@cims.nyu.edu Courant Institute of Mathematical Sciences New York University 251 Mercer St. New York, NY, 112,
More informationModification of the Porter-Thomas distribution by rank-one interaction. Eugene Bogomolny
Modification of the Porter-Thomas distribution by rank-one interaction Eugene Bogomolny University Paris-Sud, CNRS Laboratoire de Physique Théorique et Modèles Statistiques, Orsay France XII Brunel-Bielefeld
More informationRandom Matrices Numerical Methods for Random Matrices. Per-Olof Persson
18.325 Random Matrices Numerical Methods for Random Matrices Per-Olof Persson (persson@mit.edu) December 19, 22 1 Largest Eigenvalue Distributions In this section, the distributions of the largest eigenvalue
More information30 August Jeffrey Kuan. Harvard University. Interlacing Particle Systems and the. Gaussian Free Field. Particle. System. Gaussian Free Field
s Interlacing s Harvard University 30 August 2011 s The[ particles ] live on a lattice in the quarter plane. There are j+1 2 particles on the jth level. The particles must satisfy an interlacing property.
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 informationFluctuations of TASEP on a ring in relaxation time scale
Fluctuations of TASEP on a ring in relaxation time scale Jinho Baik and Zhipeng Liu March 1, 2017 Abstract We consider the totally asymmetric simple exclusion process on a ring with flat and step initial
More informationON THE SECOND-ORDER CORRELATION FUNCTION OF THE CHARACTERISTIC POLYNOMIAL OF A HERMITIAN WIGNER MATRIX
ON THE SECOND-ORDER CORRELATION FUNCTION OF THE CHARACTERISTIC POLYNOMIAL OF A HERMITIAN WIGNER MATRIX F. GÖTZE AND H. KÖSTERS Abstract. We consider the asymptotics of the second-order correlation function
More informationNumerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann Hilbert problems
Numerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann Hilbert problems Thomas Trogdon 1 and Bernard Deconinck Department of Applied Mathematics University of
More informationSecond Midterm Exam Name: Practice Problems March 10, 2015
Math 160 1. Treibergs Second Midterm Exam Name: Practice Problems March 10, 015 1. Determine the singular points of the function and state why the function is analytic everywhere else: z 1 fz) = z + 1)z
More informationPrimes, queues and random matrices
Primes, queues and random matrices Peter Forrester, M&S, University of Melbourne Outline Counting primes Counting Riemann zeros Random matrices and their predictive powers Queues 1 / 25 INI Programme RMA
More informationThe polyanalytic Ginibre ensembles
The polyanalytic Ginibre ensembles Haakan Hedenmalm (joint with Antti Haimi) Royal Institute of Technology, Stockholm haakanh@kth.se June 30, 2011 Haakan Hedenmalm (joint with Antti Haimi) (U of X) Short
More information