the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

on the characteristic polynomial of a random unitary matrix we present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial z of a random n n unitary matrix as n first we show that evaluated at a finite set of distinct points is asymptotically a collection of i i d complex normal random variables this leads to a refinement of a recent central limit theorem due to keating and snaith and also explains the covariance structure of the eigenvalue counting function next we obtain a central limit theorem for ln z in a sobolev space of generalised functions on the unit circle in this limiting regime lower order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting gaussian measure large deviations results for ln z a evaluated at a finite set of distinct points can be obtained for for higher order scalings we obtain large deviations results for ln z a evaluated at a single point there is a phase transition at a ln n which only applies to negative deviations of the real part reflecting a switch from global to local conspiracy

1 f α noise in the fluctuations of the spectra of tridiagonal random matrices from the β hermite ensemble the 1 f ensuremath alpha noise displayed by the fluctuation of the n th unfolded eigenvalue where n plays the role of a discrete time was recently characterized for the classical gaussian ensembles of n ifmmode times else texttimes fi n random matrices it is investigated here for the ensuremath beta hermite ensemble by wavelet analysis of monte carlo simulated series both as a function of ensuremath beta and of n when ensuremath beta decreases from 1 to 0 for a given and large enough n the evolution from a 1 f noise at ensuremath beta 1 gaussian orthogonal ensemble goe to a 1 f 2 noise at ensuremath beta 0 gaussian diagonal ensemble gde is heterogeneous with a ensuremath sim 1 f 2 noise at the finest scales and a ensuremath sim 1 f noise at the coarsest ones

regularity conditions in the clt for linear eigenvalue statistics of wigner matrices abstract we show that the variance of centred linear statistics of eigenvalues of gue matrices remains bounded for large n for some classes of test functions less regular than lipschitz functions this observation is suggested by the limiting form of the variance which has previously been computed explicitly but it does not seem to appear in the literature we combine this fact with comparison techniques following tao vu and erdos yau et al and a littlewood paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the holder class c 1 2 in the case of matrices of gaussian convolution type we also give a variance bound which implies the clt for test functions in the sobolev space h 1 and c 1 for general wigner matrices satisfying moment conditions if the additional assumption of the test function being supported away from the edge of the spectrum is made we prove the clt for test functions of regularity h 1 2 l and h 1 2 for gue and johansson matrices respectively previous results on the clt impose the existence and continuity of at least one classical derivative

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

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

a generalization of the construction of the class operator it is shown that the construction of the class operator for su 2 is a partial case of a much more general problem that of decomposing an operator into components transforming under conjugation according to a given irreducible representation the problem is solved generally for arbitrary compact groups and some possibilities for extensions of this procedure to the case of non compact groups are indicated

correlations of the characteristic polynomials in the gaussian unitary ensemble or a singular hankel determinant we obtain large n asymptotics for products of powers of the absolute values of the characteristic polynomials in the gaussian unitary ensemble gue of n n matrices our results can also be interpreted as asymptotics of the determinant of a hankel matrix whose symbol is supported on the real line and possesses power like fisher hartwig singularities

weighted class operators a new approach to tensor operators for compact groups weighted class operators defined by averaging conjugation operators within fixed representation of a compact group against a weight function living on a conjugacy class are discussed and their connection to the tensor operators is studied making extensive use of the theory of induced representations we prove several results for weighted class operators and in particular identify them for weight functions being suitable matrix elements of the given group with irreducible tensor operators the results complement and extend those obtained earlier by the authors a orlowski and a strasburger j phys a 27 1971 1976 1994 and others

stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension the d dimensional fractional brownian motion fbm for short b t b 1 t b d t t r with hurst exponent 0 1 is a d dimensional centered self similar gaussian process with covariance e b i s b i t 1 2 i j lsl 2 ltl 2 t s 2 the long standing problem of defining a stochastic integration with respect to fbm and the related problem of solving stochastic differential equations driven by fbm has been addressed successfully by several different methods although in each case with a restriction on the range of either d or the case corresponds to the usual stochastic integration with respect to brownian motion while most computations become singular when gets under various threshhold values due to the growing irregularity of the trajectories as 0 we provide here a new method valid for any d and for 1 4 by constructing an approximation e t e 0 of fbm which allows to define iterated integrals and then applying the geometric rough path theory the approximation relies on the definition of an analytic process z on the cut plane z c r of which fbm appears to be a boundary value and allows to understand very precisely the well known see 5 but as yet a little mysterious divergence of levy s area for 1 4

freezing transition characteristic polynomials of random matrices and the riemann zeta function we argue that the freezing transition scenario previously explored in the statistical mechanics of 1 f noise random energy models also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large n ifmmode times else texttimes fi n random unitary matrices we postulate that our results extend to the extreme values taken by the riemann zeta function ensuremath zeta s over sections of the critical line s 1 2 it of constant length and present the results of numerical computations in support our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes random matrix theory and the theory of the riemann zeta function

random conformal weldings we construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle the homeomorphism is constructed using the exponential of x where x is the restriction of the 2 dimensional free field on the circle and the parameter is in the high temperature regime beta sqrt 2 the welding problem is solved by studying a non uniformly elliptic beltrami equation with a random complex dilatation for the existence a method of lehto is used this requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilatation restricted to dyadic cells of various scales for the uniqueness we invoke a result by jones and smirnov on conformal removability of holder curves our curves are closely related to sle for 4

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

new applications of the coherent state in calculating the class operator of the rotation group the authors show by exploiting the properties of the coherent state and the integration within the ordered product technique that it is possible to derive an explicit expression for the class operator of the su 2 group which has not been obtained before

universality for eigenvalue correlations at the origin of the spectrum we establish universality of local eigenvalue correlations in unitary random matrix ensembles frac 1 z n det m 2 alpha e n rm tr v m dm near the origin of the spectrum if v is even and if the recurrence coefficients of the orthogonal polynomials associated with x 2 e nv x have a regular limiting behavior then it is known from work of akemann et al and kanzieper and freilikher that the local eigenvalue correlations have universal behavior described in terms of bessel functions we extend this to a much wider class of confining potentials v our approach is based on the steepest descent method of deift and zhou for the asymptotic analysis of riemann hilbert problems this method was used by deift et al to establish universality in the bulk of the spectrum a main part of the present work is devoted to the analysis of a local riemann hilbert problem near the origin

on the evaluation of the class operator for the rotation group a simple derivation is given of the explicit value of the class operator for the rotation group so 3

quantum deformed poincare algebra on a two dimensional lattice we propose a definition of a poincare algebra for a two dimensional spacetime with one discretized dimension this algebra has the structure of a hopf algebra we use the link between onsager s uniformization of the ising model and the dispersion relation of a free particle in this spacetime together with the rapidity representation of the quantum deformation of the poincare enveloping algebra

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles