raney distributions and random matrix theory recent works have shown that the family of probability distributions with moments given by the fuss catalan numbers permit a simple parameterized form for their density we extend this result to the raney distribution which by definition has its moments given by a generalization of the fuss catalan numbers such computations begin with an algebraic equation satisfied by the stieltjes transform which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the raney distribution for the fuss catalan distribution an equilibrium problem characterizing the density is identified the stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard gaussian matrices and s standard gaussian matrices is shown to satisfy a variant of the algebraic equation relating to the raney distribution supported on 0 infty we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization as an application the leading asymptotic form of the density as the endpoints of the support are approached is computed and is shown to have some universal features

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

universality in several matrix models via approximate transport maps we construct approximate transport maps for perturbative several matrix models as a consequence we deduce that local statistics have the same asymptotic as in the case of independent gue or goe matrices i e they are given by the sine kernel in the bulk and the tracy widom distribution at the edge and we show averaged energy universality i e universality for averages of m points correlation functions around some energy level e in the bulk as a corollary these results yield universality for self adjoint polynomials in several independent gue or goe matrices which are close to the identity

second order asymptotics for matrix models we study several matrix models and show that when the potential is convex and a small perturbation of the gaussian potential the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one in order to do that we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

transport maps for beta matrix models and universality we construct approximate transport maps for non critical beta matrix models that is maps so that the push forward of a non critical beta matrix model with a given potential is a non critical beta matrix model with another potential up to a small error in the total variation distance one of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension in addition we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

sum rules via large deviations in this paper a sum rule means a relationship between a functional defined on a subset of all probability measures on r involving the reverse kullback leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction the first sum rule is the szeg o verblunsky theorem see simon 2011 theorem 1 8 6 p 29 and concerns the case of trigonometrical polynomials on the torus more recently killip and simon killip and simon 2003 have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution indeed this sum rule includes a contribution of the atomic part of the measure away from the support of the circular law in this paper we recover this sum rules by using only large deviations tools on random matrices furthermore we obtain new up to our knowledge magic sum rules for the reverse kullback leibler divergence with respect to the pastur marchenko or kesten mckay distributions as in the semicircular case these formulas generally include a contribution of the atomic part appearing away from the support of the reference measure from the point of view of large deviations we obtain these formulas by showing a large deviation principle for the point mass measure concentrated on random eigenvalues of classical ensembles and combine it with a large deviation principle due to gamboa and rouault 2011 for the recursion coefficients

concentration for coulomb gases and coulomb transport inequalities we study the non asymptotic behavior of coulomb gases in dimension two and more such gases are modeled by an exchangeable boltzmann gibbs measure with a singular two body interaction we obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure with respect to bounded lipschitz and wasserstein distances this implies macroscopic as well as mesoscopic convergence in such distances in particular we improve the concentration inequalities known for the empirical spectral distribution of ginibre random matrices our approach is remarkably simple and bypasses the use of renormalized energy it crucially relies on new inequalities between probability metrics including coulomb transport inequalities which can be of independent interest our work is inspired by the one of maida and maurel segala itself inspired by large deviations techniques our approach allows to recover extend and simplify previous results by rougerie and serfaty

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

free monotone transport by solving a free analog of the monge ampere equation we prove a non commutative analog of brenier s monotone transport theorem if an n tuple of self adjoint non commutative random variables z 1 z n satisfies a regularity condition its conjugate variables 1 n should be analytic in z 1 z n and j should be close to z j in a certain analytic norm then there exist invertible non commutative functions f j of an n tuple of semicircular variables s 1 s n so that z j f j s 1 s n moreover f j can be chosen to be monotone in the sense that and g is a non commutative function with a positive definite hessian in particular we can deduce that c z 1 z n c s 1 s n and w z 1 dots z n cong l mathbb f n thus our condition is a useful way to recognize when an n tuple of operators generate a free group factor we obtain as a consequence that the q deformed free group factors vargamma q mathbb r n are isomorphic for sufficiently small q with bound depending on n to free group factors we also partially prove a conjecture of voiculescu by showing that free gibbs states which are small perturbations of a semicircle law generate free group factors lastly we show that entrywise monotone transport maps for certain gibbs measure on matrices are well approximated by the matricial transport maps given by free monotone transport

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

criticality and the λ property for elliptic equations abstract let l denote a nonnegative second order elliptic expression defined on c 0 and let l be an associated self adjoint operator l is said to have the property in iff l for some fixed positive function and any c 0 l is termed subcritical iff a generalized green s function may be defined for l in various applications of these and closely related concepts may be found in the literature it is the main purpose of this article to show that l is subcritical iff l has the property several corollaries and an example are given as consequences of this equivalence

on the 1 n expansion for some unitary invariant ensembles of random matrices we present a version of the 1 n expansion for random matrix ensembles known as matrix models the case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated in these cases we construct the expansion scheme for the jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices prove the asymptotic character of the scheme and give an explicit form of the first two terms this allows us in particular to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices we also find the asymptotic form of several related objects such as smoothed squares of certain orthogonal polynomials the normalized trace and the matrix elements of the resolvent of the jacobi matrices etc

beta ensembles stochastic airy spectrum and a diffusion we prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low lying eigenvalues of the random schrodinger operator d2 dx2 x 2 bx restricted to the positive half line where b x is white noise in doing so we extend the definition of the tracy widom distributions to all 0 and also analyze their tails last in a parallel development we provide a second characterization of these laws in terms of a one dimensional diffusion the proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit in particular we show how tracy widom laws arise from a functional central limit theorem

limit theorems for biorthogonal ensembles and related combinatorial identities abstract we study the fluctuations of certain biorthogonal ensembles for which the underlying family p q satisfies a finite term recurrence relation of the form x p x j p x for polynomial linear statistics of such ensembles we reformulate the cumulant method introduced in 53 in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix j in the spirit of 12 we show that the asymptotic fluctuations are described by the right limits of the matrix j moreover whenever the right limit is a laurent matrix we show that the clt is equivalent to soshnikov s main combinatorial lemma we discuss several applications to unitary invariant hermitian random matrices in particular we provide a general central limit theorem clt and a law of large numbers in the one cut regime we also prove a clt for the square singular values of the product of independent complex rectangular ginibre matrices as well as for the laguerre and jacobi biorthogonal ensembles introduced in 7 and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients finally we discuss the connection with the strong szeg limit theorem where this combinatorial method originates

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

a unified fluctuation formula for one cut β ensembles of random matrices using a coulomb gas approach we compute the generating function of the covariances of power traces for one cut ensembles of random matrices in the limit of large matrix size this formula depends only on the support of the spectral density and is therefore universal for a large class of models this allows us to derive a closed form expression for the limiting covariances of an arbitrary one cut ensemble as particular cases of the main result we consider the classical gaussian wishart and jacobi ensembles for which we derive previously available results as well as new ones within a unified simple framework we also discuss the connections between the problem of trace fluctuations for the gaussian unitary ensemble and the enumeration of planar maps

large deviation principle for empirical fields of log and riesz gases we study a system of n particles with logarithmic coulomb or riesz pairwise interactions confined by an external potential we examine a microscopic quantity the tagged empirical field for which we prove a large deviation principle at speed n the rate function is the sum of an entropy term the specific relative entropy and an energy term the renormalized energy introduced in previous works coupled by the temperature we deduce a variational property of the sine beta processes which arise in random matrix theory we also give a next to leading order expansion of the free energy of the system proving the existence of the thermodynamic limit

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

asymptotic expansion of β matrix models in the one cut regime we prove the existence of a 1 n expansion to all orders in matrix models with a confining offcritical potential corresponding to an equilibrium measure with a connected support thus the coefficients of the expansion can be obtained recursively by the topological recursion derived in chekhov and eynard jhep 0612 026 2006 our method relies on the combination of a priori bounds on the correlators and the study of schwinger dyson equations thanks to the uses of classical complex analysis techniques these a priori bounds can be derived following boutet de monvel et al in j stat phys 79 3 4 585 611 1995 johansson in duke math j 91 1 151 204 1998 kriecherbauer and shcherbina in fluctuations of eigenvalues of matrix models and their applications 2010 or for strictly convex potentials by using concentration of measure anderson et al in an introduction to random matrices sect 2 3 cambridge university press cambridge 2010 doing so we extend the strategy of guionnet and maurel segala ann probab 35 2160 2212 2007 from the hermitian models 2 and perturbative potentials to general models the existence of the first correction in 1 n was considered in johansson 1998 and more recently in kriecherbauer and shcherbina 2010 here by taking similar hypotheses we extend the result to all orders in 1 n

incomplete determinantal processes from random matrix to poisson statistics we study linear statistics of a class of determinantal processes which interpolate between poisson and gue ginibre statistics in dimension 1 or 2 these processes are obtained by performing an independent bernoulli percolation on the particle configuration of a log gas confined in a general potential we show that depending on the expected number of deleted particles there is a universal transition for mesoscopic linear statistics namely at small scales the point process behave according to random matrix theory while at large scales it needs to be renormalized because the variance of any linear statistic diverges the crossover is explicitly characterized as the superposition of a h 1 or h 1 2 correlated gaussian noise depending on the dimension and an independent poisson process the proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents