on the joint distribution of the maximum and its position of the airy2 process minus a parabola the maximal point of the airy2 process minus a parabola is believed to describe the scaling limit of the end point of the directed polymer in a random medium this was proved to be true for a few specific cases recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained one by moreno flores quastel and remenik and the other by schehr the first formula is given in terms of the airy function and an associated operator and the second formula is expressed in terms of the lax pair equations of the painleve ii equation we give a direct proof that these two formulas are the same

vicious walks with a wall noncolliding meanders and chiral and bogoliubov de gennes random matrices spatially and temporally inhomogeneous evolution of one dimensional vicious walkers with wall restriction is studied we show that its continuum version is equivalent with a noncolliding system of stochastic processes called brownian meanders here the brownian meander is a temporally inhomogeneous process introduced by yor as a transform of the bessel process that is the motion of radial coordinate of the three dimensional brownian motion represented in spherical coordinates it is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of gaussian ensembles of bogoliubov de gennes hamiltonians of the mean field theory of superconductivity which have a particle hole symmetry we report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type c to type ci in the nonstandard classes of random matrix theory of altland and zirnbauer the relation between the noncolliding systems of the generalized meanders of yor which are associated with the even dimensional bessel processes and the chiral random matrix theory is also clarified

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

global l1 theory and regularity for the 3d nonlinear wigner poisson fokker planck system abstract a global existence uniqueness and regularity theorem is proved for the simplest markovian wigner poisson fokker planck model incorporating friction and dissipation mechanisms the proof relies on green function and energy estimates under mild formulation making essential use of the husimi function and the elliptic regularization of the fokker planck operator

noncolliding brownian motion with drift and time dependent stieltjes wigert determinantal point process using the determinantal formula of biane bougerol and o connell we give multitime joint probability densities to the noncolliding brownian motion with drift where the number of particles is finite we study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well ordered with a scale we show that at each time t 0 the single time probability density of particle system is exactly transformed to the biorthogonal stieltjes wigert matrix model in the chern simons theory introduced by dolivet and tierz here one parameter extensions extensions of the stieltjes wigert polynomials which are themselves q extensions of the hermite polynomials play an essential role the two parameters a and of the process combined with time t are mapped to the parameters q and of the biorthogonal polynomials by the transformation of normalization factor of our probability density the partition function of the chern simons matrix model is readily obtained we study the determinantal structure of the matrix model and prove that at each time t 0 the present noncolliding brownian motion with drift is a determinantal point process in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel using the obtained correlation kernel we study time evolution of the noncolliding brownian motion with drift

anomalous continuous time random walk spectral diffusion in coherent third order optical response recursive relations are derived for nonlinear optical response functions of a two level chromophore with stochastic frequency fluctuations described by a continuous time random walk stationary ensembles are constructed and signatures of anomalous relaxation in the photon echo signal are discussed for a two state jump modulation with a power law waiting time density function psi t approximately t alpha 1 stretched exponential decay of the photon echo signal is predicted for 0 alpha 1 and power law asymptotics for 1 alpha 2

boundedness of the gradient of a solution and wiener test of order one for the biharmonic equation the behavior of solutions to the biharmonic equation is well understood in smooth domains in the past two decades substantial progress has also been made for the polyhedral domains and domains with lipschitz boundaries however very little is known about higher order elliptic equations in the general setting

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

quantum smoluchowski equation a systematic study the strong friction regime at low temperatures is analyzed systematically starting from the formally exact path integral expression for the reduced dynamics this quantum smoluchowski regime allows for a type of semiclassical treatment in the inverse friction strength so that higher order quantum corrections to the original quantum smoluchowski equation j ankerhold p pechukas and h grabert phys rev lett 87 086802 2001 j ankerhold and h grabert phys rev lett 101 119903 2008 can be derived drift and diffusion coefficients are determined by the equilibrium distribution in position and are directly related to the corresponding action of extremal paths and fluctuations around them it is shown that the inclusion of higher order corrections reproduces the quantum enhancement above crossover for the decay rate out of a metastable well exactly

the height of watermelons with wall we derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall this generalizes a famous result of de bruijn et al 1972 graph theory and computing new york academic pp 15 22 on the average height of planted plane trees and results by fulmek 2007 electron j combin 14 r64 and katori et al 2008 j stat phys 131 1067 83 on the expected value and higher moments respectively of the height distribution of watermelons with two branches the asymptotics for the moments depend on the analytic behaviour of certain multidimensional dirichlet series in order to obtain this information we prove a reciprocity relation satisfied by the derivatives of one of jacobi s theta functions which generalizes the well known reciprocity law for jacobi s theta functions

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients dans r n 1 on considere l equation parabolique d ordre 2 lu div acx t x u d t u 0 ou a x t a ij x t est une fonction matricielle symetrique reelle sur r n 1 a entrees c

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

brownian gibbs property for airy line ensembles consider a collection of n brownian bridges b i n n to mathbb r b i n b i n 0 1 i n conditioned not to intersect the edge scaling limit of this system is obtained by taking a weak limit as n of the collection of curves scaled so that the point 0 21 2 n is fixed and space is squeezed horizontally by a factor of n 2 3 and vertically by n 1 3 if a parabola is added to each of the curves of this scaling limit an x translation invariant process sometimes called the multi line airy process is obtained we prove the existence of a version of this process which we call the airy line ensemble in which the curves are almost surely everywhere continuous and non intersecting this process naturally arises in the study of growth processes and random matrix ensembles as do related processes with wanderers and outliers we formulate our results to treat these relatives as well note that the law of the finite collection of brownian bridges above has the property called the brownian gibbs property of being invariant under the following action select an index 1 k n and erase b k on a fixed time interval a b n n then replace this erased curve with a new curve on a b according to the law of a brownian bridge between the two existing endpoints a b k a and b b k b conditioned to intersect neither the curve above nor the one below we show that this property is preserved under the edge scaling limit and thus establish that the airy line ensemble has the brownian gibbs property an immediate consequence of the brownian gibbs property is a confirmation of the prediction of m prahofer and h spohn that each line of the airy line ensemble is locally absolutely continuous with respect to brownian motion we also obtain a proof of the long standing conjecture of k johansson that the top line of the airy line ensemble minus a parabola attains its maximum at a unique point this establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi line airy process and readily yields several other interesting properties of this process

maximum distributions of bridges of noncolliding brownian paths one dimensional brownian motion starting from the origin at time t 0 conditioned to return to the origin at time t 1 and to stay positive during time interval 0 t 1 is called the bessel bridge with duration 1 we consider an n particle system of such bessel bridges conditioned never to collide with each other in 0 t 1 which is the continuum limit of the vicious walk model in watermelon configuration with a wall distributions of maximum values of paths attained in the time interval t epsilon 0 1 are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane for the outermost path the distribution function of maximum value is exactly determined for general n we show that the present n path system of noncolliding bessel bridges is realized as the positive eigenvalue process of the 2nx2n matrix valued brownian bridge in the symmetry class c using this fact computer simulations are performed and numerical results on the n dependence of the maximum value distributions of the inner paths are reported the present work demonstrates that the extreme value problems of noncolliding paths are related to random matrix theory the representation theory of symmetry and number theory

star unitary transformations from dynamics to irreversibility and stochastic behavior we consider a simple model of a classical harmonic oscillator coupled to a field in standard approaches langevin type equations for bare particles are derived from hamiltonian dynamics these equations contain memory terms and are time reversal invariant in contrast the phenomenological langevin equations have no memory terms they are markovian equations and give a time evolution split in two branches semigroups each of which breaks time symmetry a standard approach to bridge dynamics with phenomenology is to consider the markovian approximation of the former in this paper we present a formulation in terms of dressed particles which gives exact markovian equations we formulate dressed particles for poincare nonintegrable systems through an invertible transformation operator lambda introduced by prigogine and co workers lambda is obtained by an extension of the canonical unitary transformation operator u that eliminates interactions for integrable systems our extension is based on the removal of divergences due to poincare resonances which breaks time symmetry the unitarity of u is extended to star unitarity for lambda we show that lambda transformed variables have the same time evolution as stochastic variables obeying langevin equations and that lambda transformed distribution functions satisfy exact fokker planck equations the effects of gaussian white noise are obtained by the nondistributive property of lambda with respect to products of dynamical variables

universality of the double scaling limit in random matrix models we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit the limiting kernels are constructed out of functions associated with the second painleve equation this extends a result of bleher and its for the special case of a critical quartic potential the two main tools we use are equilibrium measures and riemann hilbert problems in our treatment of equilibrium measures we allow a negative density near the critical point which enables us to treat all cases simultaneously the asymptotic analysis of the riemann hilbert problem is done with the deift zhou steepest descent analysis for the construction of a local parametrix at the critical point we introduce a modification of the approach of baik deift and johansson so that we are able to satisfy the required jump properties exactly c 2005 wiley periodicals inc

a square root method for the density matrix and its applications to lindblad operators abstract the evolution of open systems subject to both hamiltonian and dissipative forces is studied by writing the nm element of the time t dependent density matrix in the form nm t 1 a 1 a n t m t the so called square root factors the t s are non square matrices and are averaged over a systems of the ensemble this square root description is exact evolution equations are then postulated for the t factors such as to reduce to the lindblad type evolution equations for the diagonal terms in the density matrix for the off diagonal terms they differ from the lindblad equations the square root factors t are not unique and the equations for the t s depend on the specific representation chosen two criteria can be suggested for fixing the choice of t s one is simplicity of the resulting equations and the other has to do with the reduction of the difference between the t formalism and the lindblad equations when the method is tested on cases which have been previously treated by other methods our results agree with them examples chosen are i molecular systems such that are either periodically driven near level degeneracies for which we calculate the decoherence occurring in multiple landau zener transitions or else when undergoing descent around conical intersections in the potential surfaces ii formal dissipative systems with lindblad type operators representing either a non markovian process or a two state system coupled to bosons attractive features of the present factorization method are complete positivity the no higher than linear increase of the implementation effort with the number of states involved and the introduction of randomness only at the start of the process

transformed number states from a generalized bogoliubov transformation and their relationships to the eigenstates of the kossakowski lindblad equation we use a generalized bogoliubov transformation to construct a set of transformed occupation number states of the harmonic oscillator in the liouville space general expressions for the expansion coefficients of these states in the bare number basis are obtained and the properties of these states investigated the transformation is non dynamical in nature and divides the transformed basis into disconnected correlation subspaces under the transformation the basis is complete and orthonormal elements of the basis are in general mixed states and the state with the lowest number indices is the thermal vacuum since the kossakowski lindblad kl equation remains form invariant under the same transformation the transformation parameter can be identified to the thermal parameter of the master equation the transformed states then acquire thermal property through this connection we also work out the explicit expressions of the transformation matrices between the transformed states and the eigenstates of the kl equation for different temperatures

asymptotics of the airy kernel determinant the authors use riemann hilbert methods to compute the constant that arises in the asymptotic behavior of the airy kernel determinant of random matrix theory

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

symmetry of bilinear master equations for a quantum oscillator abstract we study the most general continuous transformation on the generators of bilinear master equations of a quantum oscillator we find that transformation operators that preserve the hermiticity of density operators and conserve the probability of reduced dynamics should be adjoint symmetric and they are not limited to the pure product of unitary operators in the bra and ket space but could be a mixture of them we need to include the more general transformation operators to explore the full symmetry of generic reduced dynamics we discuss how the operators are related to those considered in previous works and illustrate how they leave the reduced dynamics form invariant or map one into the other the positive semidefinite requirement on the density operator can be imposed to give a valid range of transformation parameters

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

distribution of the time at which n vicious walkers reach their maximal height we study the extreme statistics of n nonintersecting brownian motions vicious walkers over a unit time interval in one dimension using path integral techniques we compute exactly the joint distribution of the maximum m and of the time m at which this maximum is reached we focus in particular on nonintersecting brownian bridges watermelons without wall and nonintersecting brownian excursions watermelons with a wall we discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium dprm with one free end point on the other hand we also check our results using numerical simulations of dyson s brownian motion and confront them with numerical simulations of the polynuclear growth model png and of a model of dprm on a discrete lattice some of the results presented here were announced in a recent letter j rambeau and g schehr europhys lett 91 60006 2010

double scaling limit in the random matrix model the riemann hilbert approach we prove the existence of the double scaling limit in the unitary matrix model with quartic interaction and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the function for the hastings mcleod solution to the painleve ii equation the proof is based on the riemann hilbert approach and the central point of the proof is an analysis of analytic semiclassical asymptotics for the function at the critical point in the presence of four coalescing turning points 2003 wiley periodicals inc

entropic uncertainty bound for open pointer based simultaneous measurements of conjugate observables we discuss the information entropy for a general open pointer based simultaneous measurement and show how it is bound from below this entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics furthermore the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results

exact markovian kinetic equation for a quantum brownian oscillator we derive an exact markovian kinetic equation for an oscillator linearly coupled to a heat bath describing quantum brownian motion our work is based on the subdynamics formulation developed by prigogine and collaborators the space of distribution functions is decomposed into independent subspaces that remain invariant under liouville dynamics for integrable systems in poincar e s sense the invariant subspaces follow the dynamics of uncoupled renormalized particles in contrast for nonintegrable systems the invariant subspaces follow a dynamics with broken time symmetry involving generalized functions this result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces we comment on the relation between our markovian kinetic equation and the hu paz zhang equation

a wiener test a la landis for evolutive hormander operators abstract in this paper we prove a wiener type characterization of boundary regularity in the spirit of a classical result by landis for a class of evolutive hormander operators we actually show the validity of our criterion for a larger class of degenerate parabolic operators with a fundamental solution satisfying suitable two sided gaussian bounds our condition is expressed in terms of a series of balayages or as it turns out to be equivalently riesz potentials

su 1 1 lie algebraic approach to linear dissipative processes in quantum optics an su 1 1 lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems the liouville space formulation thermo field dynamics and the disentanglement theorem of su 1 1 lie algebra play essential roles in this formulation in the liouville space the time evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of su 1 1 lie algebra and the thermal state condition of thermo field dynamics the presented formulation is used for investigating a dissipative nonlinear oscillator the quantum mechanical model of phase modulation and the photon echo in the localized electron phonon system this algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems

polyharmonic capacity and wiener test of higher order in the present paper we establish the wiener test for boundary regularity of the solutions to the polyharmonic operator we introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point in the case of the laplacian the test for regularity of a boundary point is the celebrated wiener criterion of 1924 it was extended to the biharmonic case in dimension three by mayboroda and maz ya invent math 175 2 287 334 2009 as a preliminary stage of this work in mayboroda and maz ya invent math 196 1 168 2014 we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains accompanied by sharp estimates on the green function the present work pioneers a new version of capacity and establishes the wiener test in the full generality of the polyharmonic equation of arbitrary order

eigenvalues of the liouvillians of quantum master equation for a harmonic oscillator abstract the eigenvalues of the liouvillians of markovian master equation for a harmonic oscillator have a generic form the liouvillians considered are quadratic in the position coordinates or creation and annihilation operators as well as having positive renormalized frequencies we prove this by showing that a generic liouvillian of this form can be similarly related to the liouvillian of the kossakowski lindblad equation whose eigenvalues are already known the left and right eigenfunctions of the generic liouvillian also form a complete and biorthogonal set examples of similarly related right eigenfunctions are given

the petrovskiĭ criterion and barriers for degenerate and singular p parabolic equations in this paper we obtain sharp petrovski criteria for the p parabolic equation both in the degenerate case p 2 and the singular case 1 p 2 we also give an example of an irregular boundary point at which there is a barrier thus showing that regularity cannot be characterized by the existence of just one barrier

nonintersecting brownian motions on the half line and discrete gaussian orthogonal polynomials we study the distribution of the maximal height of the outermost path in the model of n nonintersecting brownian motions on the half line as n showing that it converges in the proper scaling to the tracy widom distribution for the largest eigenvalue of the gaussian orthogonal ensemble this is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the airy2 process minus a parabola our proof is based on riemann hilbert analysis of a system of discrete orthogonal polynomials with a gaussian weight in the double scaling limit as this system approaches saturation we consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point

wiener s criterion for the heat equation this paper establishes a necessary and sufficient condition for a boundary point of an arbitrary bounded open subset of r 1 to be regular for the heat equation our criterion is in the sense described below the exact analogue of winner s test w for boundary regularity of harmonic functions a careful statement of our result requires that we first of all introduce a great deal of terminology let 2 q r 1 n 1 denote any bounded open subset and 8 2 its topological boundary we will often write a typical point z e r 1 as z x t xer te r if ue c2 q r we may define the heat operator

toda flows with infinitely many variables abstract the toda flow and related flows extend naturally to operators in hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties

solutions of generic bilinear master equations for a quantum oscillator positive and factorized conditions on stationary states abstract we obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the gaussian form the well behavedness and positive semidefiniteness of the stationary states could be characterized by a three dimensional minkowski vector by requiring the stationary states to satisfy a factorized condition we obtain a generic class of master equations that includes the well known ones and their generalizations some of which are completely positive a further subset of the master equations with the gibbs states as stationary states is also obtained for master equations with not completely positive generators an analysis on the stationary states for a given initial state suggests conditions on the coefficients of the master equations that generate positive evolution

nonintersecting brownian motions on the unit circle we consider an ensemble of n nonintersecting brownian particles on the unit circle with diffusion parameter n 1 2 which are conditioned to begin at the same point and to return to that point after time t but otherwise not to intersect there is a critical value of t which separates the subcritical case in which it is vanishingly unlikely that the particles wrap around the circle and the supercritical case in which particles may wrap around the circle in this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution we also give a streamlined approach to identifying the pearcey and tacnode processes in scaling limits the formula of the tacnode correlation kernel is new and involves a solution to a lax system for the painleve ii equation of size 2 2 the proofs are based on the determinantal structure of the ensemble asymptotic results for the related system of discrete gaussian orthogonal polynomials and a formulation of the correlation kernel in terms of a double contour integral

quantum moment hydrodynamics and the entropy principle this paper presents how a non commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models our starting point is the moment method which consists in integrating the quantum liouville equation with respect to momentum p against a given vector of monomials of p like in the classical case the so obtained moment system is not closed inspired from levermore s procedure in the classical case 26 we propose to close the moment system by a quantum wigner distribution function which minimizes the entropy subject to the constraint that its moments are given in contrast to the classical case the quantum entropy is defined globally and not locally as the trace of an operator therefore the relation between the moments and the lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators instead of purely local ones in the classical case in the present paper we discuss some practical aspects and consequences of this nonlocal feature

nonintersecting brownian bridges on the unit circle with drift abstract nonintersecting brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers we investigate the effect of adding a drift term to walkers on the circle conditioned to start and end at the same position for each return time t 2 we show there exists a critical drift c t such that if 2 m t c t for some integer m then the expected winding number for each walker is asymptotically m in addition we compute the asymptotic distribution of total winding numbers in the double scaling regime in which the expected number of walkers with winding number not equal to m is finite the method of proof is riemann hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights this is the first asymptotic analysis of such a class of polynomials we determine asymptotic formulas for these polynomials as the degree of the polynomial grows large and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

exact distribution of the maximal height of p vicious walkers using path integral techniques we compute exactly the distribution of the maximal height hp of p nonintersecting brownian walkers over a unit time interval in one dimension both for excursions p watermelons with a wall and bridges p watermelons without a wall for all integer p or 1 for large p we show that approximately square root 2p excursions whereas approximately square root p bridges our exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones in addition our method establishes a physical connection between vicious walkers and random matrix theory

wiener s criterion for parabolic equations with variable coefficients and its consequences dans un ensemble borne de r n 1 on etudie la regularite des points limites pour le probleme de dirichlet pour un operateur parabolique a coefficients lisses on donne une caracterisation geometrique de ces points limites qui sont reguliers

quantum theory of non integrable systems in 1889 h poincare introduced a basic distinction between integrable and non integrable dynamical systems this distinction refers to the role of resonances which may lead to divergences a specially important class of non integrable systems are large poincare systems lps which have a continuous spectrum and present continuous sets of resonances as has been shown earlier lps play an essential role both in classical and quantum mechanics essentially all nontrivial problems of field theory as well as kinetic theory belong this class we consider here the well known friedrichs model in which an unstable discrete level is coupled to a continuum poincare s theorem prevents the existence of solutions of the eigenvalue problem associated to the hamiltonian which would be analytic in the coupling constant

reduced dynamics need not be completely positive the reduced dynamics of a quantum system in contact with a reservoir is generally thought to be completely positive this is certainly true if product initial conditions are used to define the dynamics we show that with correlated initial conditions it need not be so in this case the dynamics can properly be defined only on a subset of initial system states extension by linearity to all possible initial states is trivially possible but the extension may not be physically realizable and may not even be positive let alone completely positive

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight