Tracy-Widom GUE law and symplectic invariants

We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curve \Sigma_{TW} describing the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This explains directly how the Tracy-Widow law F_{GUE}, governing the distribution of the maximal eigenvalue in hermitian random matrices, can also be recovered from symplectic invariants.Comment: pdfLatex, 36 pages, 1 figure. v2: typos corrected, re-sectioning, a reference adde

The sine-law gap probability, Painlev\'e 5, and asymptotic expansion by the topological recursion

The goal of this article is to rederive the connection between the Painlev\'e $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, \textbf{to all orders}, that the WKB asymptotic expansions of the $\tau$-function as well as of determinantal formulas arising from the Painlev\'e $5$ Lax pair are identical to the large $N$ double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to $O(N^{-5})$.Comment: 37 pages, 1 figure, published in Random Matrices: Theory and Application

Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results

We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite union of discs of arbitrarily small but fixed radii. These exceptional discs are the neighborhoods of the so-called band edges of the associated equilibrium measure. As applications, we prove universality results for correlation functions of a general class of discrete orthogonal polynomial ensembles, and in particular we deduce asymptotic formulae with error bound for certain statistics relevant in the random tiling of a hexagon with rhombus-shaped tiles. The discrete orthogonal polynomials are characterized in terms of a a Riemann-Hilbert problem formulated for a meromorphic matrix with certain pole conditions. By extending the methods of [17, 22], we suggest a general and unifying approach to handle Riemann-Hilbert problems in the situation when poles of the unknown matrix are accumulating on some set in the asymptotic limit of interest.Comment: 28 pages, 7 figure

Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probablities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.Comment: 31 pages, 1 figur

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlev\'{e} IV System

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at $t_1,\cdots,t_m$. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second order partial differential equation which is reduced to the $\sigma$-form of a Painlev\'{e} IV equation when $m=1$. Moreover, under the assumption that $t_k-t_1$ is fixed for $k=2,\cdots,m$, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlev\'{e} IV system

Beta ensembles, quantum Painlev\'e equations and isomonodromy systems

This is a review of recent developments in the theory of beta ensembles of random matrices and their relations with conformal filed theory (CFT). There are (almost) no new results here. This article can serve as a guide on appearances and studies of quantum Painlev\'e and more general multidimensional linear equations of Belavin-Polyakov-Zamolodchikov (BPZ) type in literature. We demonstrate how BPZ equations of CFT arise from $\beta$-ensemble eigenvalue integrals. Quantum Painlev\'e equations are relatively simple instances of BPZ or confluent BPZ equations, they are PDEs in two independent variables ("time" and "space"). While CFT is known as quantum integrable theory, here we focus on the appearing links of $\beta$-ensembles and CFT with {\it classical} integrable structure and isomonodromy systems. The central point is to show on the example of quantum Painlev\'e II (QPII)~\cite{betaFP1} how classical integrable structure can be extended to general values of $\beta$ (or CFT central charge $c$), beyond the special cases $\beta=2$ ($c=1$) and $c\to\infty$ where its appearance is well-established. We also discuss an \'a priori very different important approach, the ODE/IM correspondence giving information about complex quantum integrable models, e.g.~CFT, from some stationary Schr\"odinger ODEs. Solution of the ODEs depends on (discrete) symmetries leading to functional equations for Stokes multipliers equivalent to discrete integrable Hirota-type equations. The separation of "time" and "space" variables, a consequence of our integrable structure, also leads to Schr\"odinger ODEs and thus may have a connection with ODE/IM methods.Comment: Submitted to Cont. Mat