Open string theory and planar algebras

In this note we show that abstract planar algebras are algebras over the topological operad of moduli spaces of stable maps with Lagrangian boundary conditions, which in the case of the projective line are described in terms of real rational functions. These moduli spaces appear naturally in the formulation of open string theory on the projective line. We also show two geometric ways to obtain planar algebras from real algebraic geometry, one based on string topology and one on Gromov-Witten theory. In particular, through the well known relation between planar algebras and subfactors, these results establish a connection between open string theory, real algebraic geometry, and subfactors of von Neumann algebras.Comment: 13 pages, LaTeX, 7 eps figure

Order in glassy systems

A directly measurable correlation length may be defined for systems having a two-step relaxation, based on the geometric properties of density profile that remains after averaging out the fast motion. We argue that the length diverges if and when the slow timescale diverges, whatever the microscopic mechanism at the origin of the slowing down. Measuring the length amounts to determining explicitly the complexity from the observed particle configurations. One may compute in the same way the Renyi complexities K_q, their relative behavior for different q characterizes the mechanism underlying the transition. In particular, the 'Random First Order' scenario predicts that in the glass phase K_q=0 for q>x, and K_q>0 for q<x, with x the Parisi parameter. The hypothesis of a nonequilibrium effective temperature may also be directly tested directly from configurations.Comment: Typos corrected, clarifications adde

Spectral density asymptotics for Gaussian and Laguerre β\beta-ensembles in the exponentially small region

The first two terms in the large $N$ asymptotic expansion of the $\beta$ moment of the characteristic polynomial for the Gaussian and Laguerre $\beta$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.Comment: 19 page

Genus expansion for real Wishart matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central limit-type theorem for the asymptotic distribution around the expected value

Rigorous Inequalities between Length and Time Scales in Glassy Systems

Glassy systems are characterized by an extremely sluggish dynamics without any simple sign of long range order. It is a debated question whether a correct description of such phenomenon requires the emergence of a large correlation length. We prove rigorous bounds between length and time scales implying the growth of a properly defined length when the relaxation time increases. Our results are valid in a rather general setting, which covers finite-dimensional and mean field systems. As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin glass models on random regular graphs. We present the first proof that a model of this type undergoes a purely dynamical phase transition not accompanied by any thermodynamic singularity.Comment: 24 pages, 3 figures; published versio

Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law.Comment: minor corrections; the manuscript will appear in the Journal of Statistical Physic

Large deviations of the maximal eigenvalue of random matrices

We present detailed computations of the 'at least finite' terms (three dominant orders) of the free energy in a one-cut matrix model with a hard edge a, in beta-ensembles, with any polynomial potential. beta is a positive number, so not restricted to the standard values beta = 1 (hermitian matrices), beta = 1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This model allows to study the statistic of the maximum eigenvalue of random matrices. We compute the large deviation function to the left of the expected maximum. We specialize our results to the gaussian beta-ensembles and check them numerically. Our method is based on general results and procedures already developed in the literature to solve the Pastur equations (also called "loop equations"). It allows to compute the left tail of the analog of Tracy-Widom laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos corrected and preprint added ; v4 few more numbers adde

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

On the large N limit of matrix integrals over the orthogonal group

We reexamine the large N limit of matrix integrals over the orthogonal group O(N) and their relation with those pertaining to the unitary group U(N). We prove that lim_{N to infty} N^{-2} \int DO exp N tr JO is half the corresponding function in U(N), and a similar relation for lim_{N to infty} \int DO exp N tr(A O B O^t), for A and B both symmetric or both skew symmetric.Comment: 12 page