yang mills measure and the master field on the sphere we study the yang mills measure on the sphere with unitary structure group in the limit where the structure group has high dimension we show that the traces of loop holonomies converge in probability to a deterministic limit which is known as the master field on the sphere the values of the master field on simple loops are expressed in terms of the solution of a variational problem we show that given its values on simple loops the master field is characterized on all loops of finite length by a system of differential equations known as the makeenko migdal equations we obtain a number of further properties of the master field on specializing to families of simple loops our results identify the high dimensional limit in non commutative distribution of the brownian bridge in the group of unitary matrices starting and ending at the identity

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

product of random projections jacobi ensembles and universality problems arising from free probability we consider the product of two independent randomly rotated projectors the square of its radial part turns out to be distributed as a jacobi ensemble we study its global and local properties in the large dimension scaling relevant to free probability theory we establish asymptotics for one point and two point correlation functions as well as properties of largest and smallest eigenvalues

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

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

simple expressions for a function occurring in diffraction theory the maliuzhinets integral function arises in connection with diffraction by a half plane two simple expressions are derived which when used in conjunction with known identities serve to approximate the function to a high degree of accuracy throughout the entire complex plane

general solution for a class of diffraction problems using a special mathematical technique previously developed by malyuzhinets the general solution of two dimensional plane wave diffraction problems in an angular region of any angle with boundary conditions containing spatial derivatives of arbitrary orders is constructed

central limit theorem for the heat kernel measure on the unitary group we prove that for a finite collection of real valued functions f1 fn on the group of complex numbers of modulus 1 which are derivable with lipschitz continuous derivative the distribution of trf1 trfn under the properly scaled heat kernel measure at a given time on the unitary group u n has gaussian fluctuations as n tends to infinity with a covariance for which we give a formula and which is of order n 1 in the limit where the time tends to infinity we prove that this covariance converges to that obtained by p diaconis and s n evans in a previous work on uniformly distributed unitary matrices finally we discuss some combinatorial aspects of our results

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

local eigenvalue density for general manova matrices we consider random n n matrices of the form begin aligned left xx yy right frac 1 2 yy left xx yy right frac 1 2 end aligned where x and y have independent entries with zero mean and variance one these matrices are the natural generalization of the gaussian case which are known as manova matrices and which have joint eigenvalue density given by the third classical ensemble the jacobi ensemble we show that away from the spectral edge the eigenvalue density converges to the limiting density of the jacobi ensemble even on the shortest possible scales of order 1 n up to logn factors this result is the analogue of the local wigner semicircle law and the local marchenko pastur law for general manova matrices

liberation of projections abstract we study the liberation process for projections p q p t q u t p u t q where u t is a free unitary brownian motion freely independent from p q its action on the operator valued angle q p t q between the projections induces a flow on the corresponding spectral measures t we prove that the cauchy transform of the measure satisfies a holomorphic pde we develop a theory of subordination for the boundary values of this pde and use it to show that the spectral measure t possesses a piecewise analytic density for any t 0 and any initial projections of trace 1 2 we us this to prove the unification conjecture for free entropy and information in this trace 1 2 setting

local structure of the self focusing singularity of the nonlinear schro dinger equation abstract for the cubic schrodinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d 2 tod 2

diffraction by a thin dielectric half plane a uniform diffraction coefficient is derived for a thin dielectric half plane illuminated by an h polarized incident plane wave this is accomplished by modeling the half plane using a pair of resistive and modified conductive sheets and then solving the resulting integral equations via the wiener hopf technique in contrast to a recently published solution the one presented here satisfies reciprocity and the numerical results which are included confirm its accuracy

derivation and application of a class of generalized boundary conditions electromagnetic scattering boundary conditions involving higher order derivatives are presented for simulating surfaces whose reflection coefficients are known analytically numerically or experimentally procedures for determining the coefficients of the derivatives are discussed along with the effect of displacing the surface where the boundary conditions are applied provided the coefficients satisfy a duality relation equivalent forms of the boundary conditions involving tangential field components are deduced and these provide the natural extension to nonplanar surfaces as an illustration a metal backed uniform dielectric layer is simulated it is shown that fourth order conditions are capable of providing an accurate simulation for layers at least a quarter of a wavelength in thickness

the norm of polynomials in large random and deterministic matrices let bf x n x 1 n ldots x p n be a family of n n independent normalized random matrices from the gaussian unitary ensemble we state sufficient conditions on matrices bf y n y 1 n ldots y q n possibly random but independent of xn for which the operator norm of p bf x n bf y n bf y n converges almost surely for all polynomials p limits are described by operator norms of objects from free probability theory taking advantage of the choice of the matrices yn and of the polynomials p we get for a large class of matrices the no eigenvalues outside a neighborhood of the limiting spectrum phenomena we give examples of diagonal matrices yn for which the convergence holds convergence of the operator norm is shown to hold for block matrices even with rectangular gaussian blocks a situation including non white wishart matrices and some matrices encountered in mimo systems

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

multivariate analysis and jacobi ensembles largest eigenvalue tracy widom limits and rates of convergence abstract let a and b be independent central wishart matrices in p variables with common covariance and having m and n degrees of freedom respectively the distribution of the largest eigenvalue of a b 1 b has numerous applications in multivariate statistics but is difficult to calculate exactly suppose that m and n grow in proportion to p we show that after centering and scaling the distribution is approximated to second order o p 2 3 by the tracy widom law the results are obtained for both complex and then real valued data by using methods of random matrix theory to study the largest eigenvalue of the jacobi unitary and orthogonal ensembles asymptotic approximations of jacobi polynomials near the largest zero play a central role

the effects of normal dispersion on collapse events abstract it is shown analytically that positive or normal dispersion arrests at least the initial stage of the critical self similar collapse events associated with the nonlinear schrodinger equation this is done by pertubing a wave packet that undergoes critical self similar collapse in d dimensions by adding weak positive dispersion in one additional direction singular perturbation analysis then yields a closed set of simple equations for the parameters of the collapsing mode from which it is clear that in the absence of normal dispersion the self similar collapse will continue while in its presence the collapse attractor disappears direct numerical integrations are in excellent agreement with the predictions of the simple reduced equations

the spectral edge of unitary brownian motion the brownian motion u n t t ge 0 on the unitary group converges as a process to the free unitary brownian motion u t t ge 0 as n rightarrow infty in this paper we prove that it converges strongly as a process not only in distribution but also in operator norm in particular for a fixed time t 0 we prove that the unitary brownian motion has a spectral edge there are no outlier eigenvalues in the limit we also prove an extension theorem any strongly convergent collection of random matrix ensembles independent from a unitary brownian motion also converge strongly jointly with the brownian motion we give an application of this strong convergence to the jacobi process

schur weyl duality and the heat kernel measure on the unitary group abstract we investigate a relation between the brownian motion on the unitary group and the most natural random walk on the symmetric group based on schur weyl duality we use this relation to establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure this expectation turns out to be the generating series of certain paths in the cayley graph of the symmetric group using our expansion we recover asymptotic results of xu biane and voiculescu we give an interpretation of our main expansion in terms of random ramified coverings of a disk

nonlinearity saturation as a singular perturbation of the nonlinear schroźdinger equation saturation of a kerr type nonlinearity in the nonlinear schro dinger equation nlse can be regarded as a singular perturbation which regularizes the well known blow up phenomenon in the cubic nlse an asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions in one dimension this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum in two dimensions we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast scale operator this leads to an outer solution with a free logarithmic singularity whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves however for a certain value of wave power we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field we test our results by numerical simulation of both the full equation and free boundary reductions

te scattering by a two dimensional groove in a ground plane using higher order boundary conditions an application of a third order generalized boundary condition to scattering by a filled rectangular groove is presented deficiencies of such higher order boundary conditions are addressed and a correction for the present case is proposed as part of the process of examining and correcting the accuracy of the proposed generalized boundary conditions an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition

the large n limit of the segal bargmann transform on un we study the two parameter segal bargmann transform b n on the unitary group un for large n acting on matrix valued functions that are equivariant under the adjoint action of the group the transform has a meaningful limit gs t as n which can be identified as an operator on the space of complex laurent polynomials we introduce the space of trace polynomials and use it to give effective computational methods to determine the action of the heat operator and thus the segal bargmann transform we prove several concentration of measure and limit theorems giving a direct connection from the finite dimensional transform b n to its limit gs t we characterize the operator gs t through its inverse action on the standard polynomial basis finally we show that in the case s t the limit transform gt t is the free hall transform g t introduced by biane

diffraction by a symmetric material junction simulated with generalized sheet transition conditions when diffraction by a material discontinuity in a thick dielectric ferrite slab is treated by the dual integral equation approach a variation of the weiner hopf method the resulting solution displays unknown constants characteristic of the incompleteness associated with the generalized transition conditions employed in modeling a thick slab the constants can be shown to depend on the geometry and properties of the discontinuity their explicit determination by enforcing field continuity across the slab junction is treated the field internal to the slab is determined by first expressing the field as an expansion of discrete and continuous eigenmodes valid in the exterior and interior slab region the expansion or mode coefficients are then identified by recasting the weiner hopf solution in a suitable form enabling the solution constants to be numerically computer by matching the field expansions to the left and right of the slab discontinuity at a discrete set of points

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

wigner chaos and the fourth moment we prove that a normalized sequence of multiple wigner integrals in a fixed order of free wigner chaos converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2 the fourth moment of the semicircular law this extends to the free probabilistic setting some recent results by nualart and peccati on characterizations of central limit theorems in a fixed order of gaussian wiener chaos our proof is combinatorial analyzing the relevant noncrossing partitions that control the moments of the integrals we can also use these techniques to distinguish the first order of chaos from all others in terms of distributions we then use tools from the free malliavin calculus to give quantitative bounds on a distance between different orders of chaos when applied to highly symmetric kernels our results yield a new transfer principle connecting central limit theorems in free wigner chaos to those in gaussian wiener chaos we use this to prove a new free version of an important classical theorem the breuer major theorem

a uniform geometrical theory of diffraction for an imperfectly conducting half plane diffraction tensors are presented in the context of the uniform geometrical theory of diffraction utd for the high frequency scattering by an impedance half plane at normal and oblique skew incidence these are based on the exact wiener hopf solution and were derived according to the utd ansatz in addition unlike previous uniform diffraction coefficients the ones given here reduce to the known utd diffraction coefficients for the perfectly conducting case the coefficients are explicit and therefore appropriate for practical applications several scattering patterns are also presented and compared to a previous heuristic solution

free convolution operators and free hall transform abstract we define an extension of the polynomial calculus on a w probability space by introducing an algebra c x i i i which contains polynomials this extension allows us to define transition operators for additive and multiplicative free convolution it also permits us to characterize the free segal bargmann transform and the free hall transform introduced by biane in a manner which is closer to classical definitions finally we use this extension of polynomial calculus to prove two asymptotic results on random matrices the convergence for each fixed time as n tends to of the distribution of the brownian motion on the linear group gl n c to the distribution of a free multiplicative circular brownian motion and the convergence of the classical hall transform on u n to the free hall transform

similarity structure of wave collapse abstract similarity transformations of the cubic schrodinger equation cse are investigated the transformations are used to remove the explicit time variation in the cse and reduce it to differential equations in the spatial variables only two different methods for similarity reduction are employed and the significance of similarity in the evolution of a collapsing wave packet is investigated numerical solutions in radial symmetry demonstrate that the similarity behaviour is local in space and time and that some similarity solutions must be classified as improper solutions the nature of the collapsing singularity is reexamined

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star

em plane wave diffraction by a a planar junction of two thin material half planes oblique incidence a uniform geometrical theory of diffraction utd solution is developed for analyzing the high frequency diffraction of an obliquely incident plane wave by a two part thin planar transparent material slab the solution is obtained by appropriately combining two solutions for the two related configurations involving perfectly conducting electric and magnetic ground plane bisections of the original slab the analysis is based on the wiener hopf technique and each of the grounded material half planes is assumed to be electrically thin so that it can be modeled by a generalized impedance boundary condition of 0 t where t is the corresponding slab thickness to solve the boundary value problem completely an additional condition related to the field behavior at the junction of the two material half planes is imposed this junction condition is determined by matching an approximate quasi static solution which is developed in the proximity of the discontinuity with the corresponding external wiener hopf solution in the common region of overlap the solution thus obtained automatically satisfies reciprocity