universality of the stochastic airy operator we introduce a new method for studying universality of random matrices let t n be the jacobi matrix associated to the dyson beta ensemble with uniformly convex polynomial potential we show that after scaling tn converges to the stochastic airy operator in particular the top edge of the dyson beta ensemble and the corresponding eigenvectors are universal as a byproduct these ideas lead to conjectured operator limits for the entire family of soft edge distributions c 2015 wiley periodicals inc

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

bulk universality of general β ensembles with non convex potential we prove the bulk universality of the ensembles with non convex regular analytic potentials for any 0 this removes the convexity assumption appeared in the earlier work p bourgade l erdos and h t yau universality of general ensembles preprint arxiv 0907 5605 2011 the convexity condition enabled us to use the logarithmic sobolev inequality to estimate events with small probability the new idea is to introduce a convexified measure so that the local statistics are preserved under this convexification

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

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

interacting particle systems at the edge of multilevel dyson brownian motions abstract we study the joint asymptotic behavior of spacings between particles at the edge of multilevel dyson brownian motions when the number of levels tends to infinity despite the global interactions between particles in multilevel dyson brownian motions we observe a decoupling phenomenon in the limit the global interactions become negligible and only the local interactions remain the resulting limiting objects are interacting particle systems which can be described as brownian versions of certain totally asymmetric exclusion processes this is the first appearance of a particle system with local interactions in the context of general random matrix models

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

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

edge statistics for a class of repulsive particle systems we study a class of interacting particle systems on mathbb r which was recently investigated by gotze and venker ann probab 42 6 2207 2242 2014 doi 10 1214 13 aop844 these ensembles generalize eigenvalue ensembles of hermitian random matrices by allowing different interactions between particles although these ensembles are not known to be determinantal one can use the stochastic linearization method of gotze and venker ann probab 42 6 2207 2242 2014 doi 10 1214 13 aop844 to represent them as averages of determinantal ones our results describe the transition between universal behavior in the regime of the tracy widom law and non universal behavior for large deviations of the rightmost particle moreover a detailed analysis of the transition that occurs in the regime of moderate deviations is provided we also compare our results with the corresponding ones obtained recently for determinantal ensembles eichelsbacher et al in symmetry integr geom methods appl 12 18 2016 doi 10 3842 sigma 2016 093 schuler in moderate large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles ph d thesis university of bayreuth 2015 in particular we discuss how the averaging effects the leading order behavior in the regime of large deviations in the analysis of the averaging procedure we use detailed asymptotic information on the behavior of christoffel darboux kernels that is uniform for perturbative families of weights such results have been provided by k schubert k schuler and the authors in kriecherbauer et al markov process relat fields 21 3 part 2 639 694 2015

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

eigenvalue distributions of large hermitian matrices wigner s semi circle law and a theorem of kac murdock and szego abstract wigner s semi circle law describes the eigenvalue distribution of certain large random hermitian matrices a new proof is given for the case of gaussian matrices that involves reducing a random matrix to tridiagonal form by a method that is well known as a technique for numerical computation of eigenvalues the result is a generalized toeplitz matrix whose eigenvalue distribution can be found using a theorem of kac murdock and szego a new and more elementary proof of the latter is also given the arguments use only direct l 2 estimates rather than the transform methods or moment calculations employed previously

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

universality of the stochastic bessel operator we establish universality at the hard edge for general beta ensembles assuming that the background potential v is a polynomial such that x mapsto v x 2 is strongly convex beta ge 1 and the dimension difference parameter a ge 0 the method rests on the corresponding tridiagonal matrix models showing that their appropriate continuum scaling limit is given by the stochastic bessel operator as conjectured in edelman and sutton j stat phys 127 1121 1165 2007 and rigorously established in ramirez and rider commun math phys 288 887 906 2009 the latter characterizes the hard edge in the case of linear potential and all beta the classical beta laguerre ensembles

some remarks about semiclassical trace invariants and quantum normal forms in this paper we explore the connection between semi classical and quantum birkhoff canonical forms bcf for schrodinger operators in particular we give a non symbolic operator theoretic derivation of the quantum birkhoff canonical form and provide an explicit recipe for expressing the quantum bcf in terms of the semi classical bcf

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

from random matrices to stochastic operators we propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators three particular stochastic operators commonly arise each associated with a familiar class of local eigenvalue behavior the stochastic airy operator displays soft edge behavior associated with the airy kernel the stochastic bessel operator displays hard edge behavior associated with the bessel kernel the article concludes with suggestions for a stochastic sine operator which would display bulk behavior associated with the sine kernel

the sine _ beta operator we show that text sine beta the bulk limit of the gaussian beta ensembles is the spectrum of a self adjoint random differential operator begin aligned f mapsto 2 r t 1 left begin array cc 0 tfrac d dt tfrac d dt 0 end array right f qquad f 0 1 rightarrow mathbb r 2 end aligned where r t is the positive definite matrix representation of hyperbolic brownian motion with variance 4 beta in logarithmic time the result connects the montgomery dyson conjecture about the text sine 2 process and the non trivial zeros of the riemann zeta function the hilbert polya conjecture and de brange s attempt to prove the riemann hypothesis we identify the brownian carousel as the sturm liouville phase function of this operator

a novel wideband fmm for fast integral equation solution of multiscale problems in electromagnetics in this paper we propose a novel scheme to accelerate integral equation solvers when applied to multiscale problems these class of problems exhibit multiple length frequency scales and arise when analyzing scattering radiation from realistic structures where dense discretization is necessary to accurately capture geometric features solutions to the discretized integral equations due to these structures is challenging due to their high computational cost and ill conditioning of the resulting matrix system the focus of this paper is on ameliorating the computational cost our approach will rely on exploiting the recently developed accelerated cartesian expansion ace algorithm to arrive at a method that is stable and efficient at low frequencies these will then be integrated with the well known fast multipole method thus forming a scheme that is wideband rigorous convergence estimates of this method are derived and convergence and efficiency of the overall fast method is demonstrated these are then integrated into an existing integral equation solver whose efficiency is demonstrated for some practical problems

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and