on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

large deviations upper bounds for the laws of matrix valued processes and non communicative entropies using ito s calculus we study the large deviations properties of the law of the spectral measure of the hermitian brownian motion we extend this strategy to the symmetric unitary and wishart processes this dynamical approach is generalized to the study of the large deviations of the non commutative laws of several independent hermitian brownian motions as a consequence we can bound from above entropies defined in the spirit of the microstates entropy introduced by voiculescu

asymptotics of unitary and orthogonal matrix integrals abstract in this paper we prove that in small parameter regions arbitrary unitary matrix integrals converge in the large n limit and match their formal expansion secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the hciz integral our convergence result also leads us to new results of smoothness of microstates we finally generalize our approach to integrals over the orthogonal group

asymptotics of unitary multimatrix models the schwinger dyson lattice and topological recursion abstract we prove the existence of a 1 n 2 expansion in unitary multimatrix models which are gibbs perturbations of the haar measure and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem the recursion obtained is closely related to the topological recursion which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems our approach consists of two main ingredients an asymptotic study of the schwinger dyson lattice over noncommutative laurent polynomials and uniform control on the cumulants of gibbs measures on product unitary groups the required cumulant bounds are obtained by concentration of measure arguments and change of variables techniques

hydrodynamics and large deviation for simple exclusion processes we prove the hydrodynamical limit for weakly asymmetric simple exclusion processes a large deviation property with respect to this limit is established for the symmetric case we treat also the situation where a slow reaction creation and annihilation of particles is present

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

stochastic calculus with respect to free brownian motion and analysis on wigner space we define stochastic integrals with respect to free brownian motion and show that they satisfy burkholder gundy type inequalities in operator norm we prove also a version of ito s predictable representation theorem as well as product form and functional form of ito s formula finally we develop stochastic analysis on the free fock space in analogy with stochastic analysis on the wiener space

the dilute potts model on random surfaces we present a new solution of the asymmetric two matrix model in the large n limit which only involves a saddle point analysis the model can be interpreted as ising in the presence of a magnetic field on random dynamical lattices with the topology of the sphere resp the disk for closed resp open surfaces we elaborate on the resulting phase diagram the method can be equally well applied to a more general q 1 matrix model which represents the dilute potts model on random dynamical lattices we discuss in particular duality of boundary conditions for open random surfaces

global spectrum fluctuations for the β hermite and β laguerre ensembles via matrix models we study the global spectrum fluctuations for hermite and laguerre ensembles via the tridiagonal matrix models introduced previously by the present authors j math phys 43 5830 2002 and prove that the fluctuations describe a gaussian process on polynomials we extend our results to slightly larger classes of random matrices

a method of integration over matrix variables ii the integral over twon n hermitan matricesz g c dadbexp tr a2 b2 2cab g n a4 b4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

large deviations asymptotics for spherical integrals consider the spherical integral i n dn en exp n tr udnu en dm n u where m n denote the haar measure on the orthogonal group on when 1 and on the unitary group un when 2 and dn en are diagonal real matrices whose spectral measures converge to d e in this paper we prove the existence and represent as solution to a variational problem the limit i d e lim n 2 log i n dn en this limit appears in so called matrix models but also in the evaluation of large deviations of the spectral measure of generalized wishart matrices our technique is based on stochastic calculus large deviations and elements from free probability

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

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

macdonald s identities and the large n limit of ym 2 on the cylinder the purpose of this paper is to determine the large n asymptotics of the free energy fn a u a of ym2 two dimensional yang mills theory with gauge group gn su n on a cylinder where a is a so called principal element of type mathematically where hgn is the central heat kernel of gn we find that where is an explicit quadratic functional in the limit distribution d of eigenvalues of un depending only on the integral geometry of su 2 the factor of n contradicts some predictions in the physics literature on the large n limit of ym2 on the cylinder due to gross matytsin kazakov wynter and others

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

addendum to large deviations asymptotics for spherical integrals abstract we correct an omission in guionnet and zeitouni j funct anal 188 2002 461 and improve the main result there to a full large deviations trajectorial result

discussions around voiculescu s free entropies abstract in this article we set two analogous definitions of the free entropies and introduced by voiculescu invent math 118 1994 411 132 1998 189 we discuss their relations improving the preceding results obtained in cabanal duvillard and guionnet ann probab 2001 to appear where a bound on the microstates entropy was established

the spectrum of heavy tailed random matrices let x n be an n n random symmetric matrix with independent equidistributed entries if the law p of the entries has a finite second moment it was shown by wigner 14 that the empirical distribution of the eigenvalues of x n once renormalized by sqrt n converges almost surely and in expectation to the so called semicircular distribution as n goes to infinity in this paper we study the same question when p is in the domain of attraction of an stable law we prove that if we renormalize the eigenvalues by a constant a n of order n frac 1 alpha the corresponding spectral distribution converges in expectation towards a law mu alpha which only depends on we characterize mu alpha and study some of its properties it is a heavy tailed probability measure which is absolutely continuous with respect to lebesgue measure except possibly on a compact set of capacity zero

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

diversity multiplexing tradeoff for the mimo static half duplex relay in this paper we investigate the diversity multiplexing tradeoff dmt of the multiple antenna mimo static half duplex relay channel a general expression is derived for the dmt upper bound which can be achieved by a compress and forward protocol at the relay under certain assumptions the dmt expression is given as the solution of a minimization problem in general and an explicit expression is found when the relay channel is symmetric in terms of number of antennas i e the source and the destination have n antennas each and the relay has m antennas it is observed that the static half duplex dmt matches the full duplex dmt when the relay has a single antenna and is strictly below the full duplex dmt when the relay has multiple antennas besides the derivation of the upper bound involves a new asymptotic study of spherical integrals that is integrals with respect to the haar measure on the unitary group u n which is a topic of mathematical interest in itself

universality of correlation functions of hermitian random matrices in an external field the behavior of correlation functions is studied in a class of matrix models characterized by a measure exp s containing a potential term and an external source term s n tr v m ma in the large n limit the short distance behavior is found to be identical to the one obtained in previously studied matrix models thus extending the universality of the level spacing distribution the calculation of correlation functions involves finite n determinant formulae reducing the problem to the large n asymptotic analysis of a single kernel k this is performed by an appropriate matrix integral formulation of k multi matrix generalizations of these results are discussed