on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

weak solutions of the landau lifshitz bloch equation abstract the landau lifshitz bloch llb equation is a formulation of dynamic micromagnetics valid at all temperatures treating both the transverse and longitudinal relaxation components important for high temperature applications we study llb equation in case the temperature raised higher than the curie temperature the existence of weak solution is showed and its regularity properties are also discussed in this way we lay foundations for the rigorous theory of llb equation that is currently not available

computational studies for the stochastic landau lifshitz gilbert equation the stochastic landau lifshitz gilbert equation describes the thermally induced dynamics of magnetic moments in ferromagnetic materials solutions of this highly nonlinear stochastic pde are unit vector fields and satisfy an energy estimate these are crucial properties to construct a convergent discretization in space and time we propose a convergent finite element approximation of the problem based on the midpoint rule the numerical scheme preserves the underlying properties of the continuous problem further we construct a robust and efficient newton multigrid solver for the solution of the nonlinear systems associated with the discretized problems at each time level computational studies show the optimal convergence behavior of the scheme in the case of smooth solutions long time dynamics for finite ensembles of spins evidence the ergodicity of an invariant measure of the continuum model numerical experiments in two dimensions demonstrate pathwise finite time blow up behavior of the solution w

asymptotics for the fredholm determinant of the sine kernel on a union of intervals in the bulk scaling limit of the gaussian unitary ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the fredholm determinant of the sine kernel frac sin x y pi x y over this interval a formal asymptotic expansion for the determinant ass tends to infinity was obtained by dyson in this paper we replace a single interval of lengths bysj wherej is a union ofm intervals and present a proof of the asymptotics up to second order the logarithmic derivative with respect tos of the determinant equals a constant expressible in terms of hyperelliptic integrals timess plus a bounded oscillatory function ofs zero ifm 1 periodic ifm 2 and in general expressible in terms of the solution of a jacobi inversion problem pluso 1 also determined are the asymptotics of the trace of the resolvent operator which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none the proofs use ideas from orthogonal polynomial theory

martingale weak solutions of the stochastic landau lifshitz bloch equation abstract the present paper studies the stochastic landau lifshitz bloch equation which is recommended as the only valid model at temperature around the curie temperature and is especially important for the simulation of heat assisted magnetic recording we study the stochastic landau lifshitz bloch equation in the case that the temperature is raised higher than the curie temperature the global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed

the x ray problem revisited in this letter we revisit the x ray problem assuming point interaction between the conduction electrons and the instantaneously created core hole the latter s green s function can be represented as a fredholm determinant of certain wiener hopf operators acting on l2 0 t with discontinuous symbols here the symbols are the local conduction electron green s function in the frequency domain and t is the time the core hole spends in the system before removal in this situation the classical theory of singular integral equations usually employed in the literature to compute the large t asymptotics of the fredholm determinant ceased to be applicable a rigorous theory first put forward in the context of operator theory comes into play and universal constants are found in the asymptotics

existence of a unique solution and invariant measures for the stochastic landau lifshitz bloch equation abstract the landau lifshitz bloch equation perturbed by a space dependent noise was proposed in 10 as a model for evolution of spins in ferromagnetic materials at the full range of temperatures including the temperatures higher than the curie temperature in the case of a ferromagnet filling a bounded domain d r d d 1 2 3 we show the existence of strong in the sense of pdes martingale solutions furthermore in cases d 1 2 we prove uniqueness of pathwise solutions and the existence of invariant measures 1

markov chain models for threshold exceedances in recent research on extreme value statistics there has been an extensive development of threshold methods first in the univariate case and subsequently in the multivariate case as well in this paper an alternative methodology for extreme values of univariate time series is developed by assuming that the time series is markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions a new likelihood representation for threshold methods is presented which we apply to a markovian time series an important motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme events we address this issue by showing how a theory of compound poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest the methods are illustrated by application to temperature data

global weak solutions to a spatio temporal fractional landau lifshitz bloch equation abstract the present paper deals with global existence of weak solutions of a time space fractional landau lifshitz bloch equation involving the weak caputo derivative and a fractional laplacian we use faedo galerkin method with some commutator estimates in order to prove global existence of weak solutions for the model the uniqueness is also discussed in a special one dimensional case

powers of large random unitary matrices and toeplitz determinants we study the limiting behavior of tr u k n where u is an n x n random unitary matrix and k n is a natural number that may vary with n in an arbitrary way our analysis is based on the connection with toeplitz determinants the central observation of this paper is a strong szego limit theorem for toeplitz determinants associated to symbols depending on n in a particular way as a consequence of this result we find that for each fixed m n the random variables tru k j n min k j n n j 1 m converge to independent standard complex normals

critical scaling for monodromy fields the large scale asymptotics of the correlations for a family of two dimensional lattice field theories is calculated at the critical temperature

a finite element approximation for the stochastic landau lifshitz gilbert equation abstract the stochastic landau lifshitz gilbert llg equation describes the behaviour of the magnetisation under the influence of the effective field containing random fluctuations we first transform the stochastic llg equation into a partial differential equation with random coefficients without the ito term the resulting equation has time differentiable solutions we then propose a convergent linear scheme for the numerical solution of the reformulated equation as a consequence we show the existence of weak martingale solutions to the stochastic llg equation a salient feature of this scheme is that it does not involve solving a system of nonlinear algebraic equations and that no condition on time and space steps is required when 1 2 1 numerical results are presented to show the applicability of the method

on a class of integral operators on a half space with discontinuous symbol this paper is concerned with the spectral asymptotics of the integral operator a on l2 with kernel 2 n ei x y x d where and are regions in rn and is a large parameter more precisely we consider the asymptotics as of tr a for suitable functions f the first order asymptotics are classical and there is a conjectured second order result the conjecture is proved here if one of the regions or is a half space

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

entanglement entropy and berezin toeplitz operators we consider berezin toeplitz operators on compact kahler manifolds whose symbols are characteristic functions when the support of the characteristic function has a smooth boundary we prove a two term weyl law the second term being proportional to the riemannian volume of the boundary as a consequence we deduce the area law for the entanglement entropy of integer quantum hall states another application is for the determinantal processes with correlation kernel the bergman kernels of a positive line bundle we prove that the number of points in a smooth domain is asymptotically normal

eigenvalue distributions of random unitary matrices let u be an n n random matrix chosen from haar measure on the unitary group for a fixed arc of the unit circle let x be the number of eigenvalues of m which lie in the specified arc we study this random variable as the dimension n grows using the connection between toeplitz matrices and random unitary matrices and show that x e x var x 1 2 is asymptotically normally distributed in addition we show that for several fixed arcs i1 im the corresponding random variables are jointly normal in the large n limit

counting free fermions on a line a fisher hartwig asymptotic expansion for the toeplitz determinant in the double scaling limit we derive an asymptotic expansion for a wiener hopf determinant arising in the problem of counting one dimensional free fermions on a line segment at zero temperature this expansion is an extension of the result in the theory of toeplitz and wiener hopf determinants known as the generalized fisher hartwig conjecture the coefficients of this expansion are conjectured to obey certain periodicity relations which renders the expansion explicitly periodic in the counting parameter we present two methods to calculate these coefficients and verify the periodicity relations order by order the matrix riemann hilbert problem and the painleve v equation we show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients

a strong central limit theorem for a class of random surfaces this paper is concerned with d 2 dimensional lattice field models with action v nabla phi cdot where v mathbf r d rightarrow mathbf r is a uniformly convex function the fluctuations of the variable phi 0 phi x are studied for large x via the generating function given by g x mu ln langle e mu phi 0 phi x rangle a in two dimensions g x mu partial 2g x mu partial mu 2 is proportional to ln vert x vert the main result of this paper is a bound on g x mu partial 3 g x mu partial mu 3 which is uniform in vert x vert for a class of convex v the proof uses integration by parts following helffer sjostrand and witten and relies on estimates of singular integral operators on weighted hilbert spaces

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

trace formulas for toeplitz matrices with piecewise continuous symbols asymptotically for certain functions f and for 4 a piecewise continuous function the more general problem has been studied for smooth 4 in 9 and also for many other classes of operators 7 8 lo it of course yields information about determinants by choosing f to be the logarithm and gives information about the behavior of the eigenvalues of t closely related to the matrix t q5 is the toeplitz matrix t hz hz where hp felp s if o k o

from gap probabilities in random matrix theory to eigenvalue expansions we present a method to derive asymptotics of eigenvalues for trace class integral operators acting on a single interval which belongs to the ring of integrable operators its et al 1990 int j mod phys b 4 1003 37 our emphasis lies on the behavior of the spectrum of k as and i is fixed we show that this behavior is intimately linked to the analysis of the fredholm determinant as and in a stokes type scaling regime concrete asymptotic formulae are obtained for the eigenvalues of airy and bessel kernels in random matrix theory

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

toeplitz and wiener hopf determinants with piecewise continuous symbols abstract the asymptotics for determinants of toeplitz and wiener hopf operators with piecewise continuous symbols are obtained in this paper if w is the wiener hopf operator defined on l2 0 with piecewise continuous symbol having a finite number of discontinuities at r then under appropriate conditions it is shown that det w g r2k where g exp log 0 r 1 2 log r r and k is a completely determined constant an analogous result is obtained for toeplitz operators the main point of the paper is to obtain a result in the wiener hopf case since the toeplitz case had been treated earlier in the toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known fortunately in the toeplitz case such asymptotics were known the difficulty in the wiener hopf case is that there was not a single singular case where the determinant was explicitly known this problem was overcome by using the fact that wiener hopf determinants when discretized become toeplitz determinants whose entries depend on the size of the matrix no theorem on toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results

large deformations of the tracy widom distribution i non oscillatory asymptotics we analyze the left tail asymptotics of deformed tracy widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 gamma in 0 1 as gamma varies a transition from tracy widom statistics gamma 1 to classical weibull statistics gamma 0 was observed in the physics literature by bohigas et al phys rev e 79 031117 2009 we provide a description of this transition by rigorously computing the leading order left tail asymptotics of the thinned goe gue and gse tracy widom distributions in this paper we obtain the asymptotic behavior in the non oscillatory region with gamma in 0 1 fixed for the goe gue and gse distributions and gamma uparrow 1 at a controlled rate for the gue distribution this is the first step in an ongoing program to completely describe the transition between tracy widom and weibull statistics as a corollary to our results we obtain a new total integral formula involving the ablowitz segur solution to the second painleve equation