oscillation criteria for second order linear difference equations in this paper some new oscillation criteria are obtained for the second order linear difference email protected 2x n 1 p nx n 0 n 0 1 2 where dx n x n 1 x n is the forward difference operator p n is a sequence of nonnegative real numbers our results improve the know results in the literature and an example is given to demonstrate the advantage of our results

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition the one dimensional totally asymmetric simple exclusion process tasep is considered we study the time evolution property of a tagged particle in the tasep with the step initial condition calculated is the multi time joint distribution function of its position using the relation of the dynamics of the tasep to the schur process we show that the function is represented as the fredholm determinant we also study the scaling limit the universality of the largest eigenvalue in the random matrix theory is realized in the limit when the hopping rates of all particles are the same it is found that the joint distribution function converges to that of the airy process after the time at which the particle begins to move on the other hand when there are several particles with small hopping rate in front of a tagged particle the limiting process changes at a certain time from the airy process to the process of the largest eigenvalue in the hermitian multi matrix model with external sources

step fluctuations for a faceted crystal a statistical mechanics model for a faceted crystal is the 3d ising model at zero temperature it is assumed that in one octant all sites are occupied by atoms the remaining ones being empty allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds if v denotes the number of atoms removed then the grand canonical boltzmann weight is q v 0 q 1 as shown by cerf and kenyon in the limit q 1 a deterministic shape is attained which has the three facets 100 010 001 and a rounded piece interpolating between them we analyse the step statistics as q 1 in the rounded piece it is given by a determinantal process based on the discrete sine kernel exactly at the facet edge the steps have more space to meander their statistics is again determinantal but this time based on the airy kernel in particular the border step is well approximated by the airy process which has been obtained previously in the context of growth models our results are based on the asymptotic analysis for space time inhomogeneous transfer matrices

universality properties of gelfand tsetlin patterns a standard gelfand tsetlin pattern of depth n is a configuration of particles in 1 ldots n times mathbb r for each r in 1 ldots n r times mathbb r is referred to as the rth level of the pattern a standard gelfand tsetlin pattern has exactly r particles on each level r and particles on adjacent levels satisfy an interlacing constraint probability distributions on the set of gelfand tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random hermitian matrices of size n we consider such probability spaces when the distribution of the matrix is unitarily invariant prove a determinantal structure for a broad subclass and calculate the correlation kernel in particular we consider the case where the eigenvalues of the random matrix are fixed this corresponds to choosing uniformly from the set of gelfand tsetlin patterns whose nth level is fixed at the eigenvalues of the matrix fixing q n in 1 ldots n and letting n under the assumption that frac q n n to alpha in 0 1 and the empirical distribution of the particles on the nth level converges weakly the asymptotic behaviour of particles on level q n is relevant to free probability theory saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the sine kernel

gibbs ensembles of nonintersecting paths we consider a family of determinantal random point processes on the two dimensional lattice and prove that members of our family can be interpreted as a kind of gibbs ensembles of nonintersecting paths examples include probability measures on lozenge and domino tilings of the plane some of which are non translation invariant the correlation kernels of our processes can be viewed as extensions of the discrete sine kernel and we show that the gibbs property is a consequence of simple linear relations satisfied by these kernels the processes depend on infinitely many parameters which are closely related to parametrization of totally positive toeplitz matrices

the arctic circle boundary and the airy process we prove that the appropriately rescaled boundary of the north polar region in the aztec diamond converges to the airy process the proof uses certain determinantal point processes given by the extended krawtchouk kernel we also prove a version of propp s conjecture concerning the structure of the tiling at the center of the aztec diamond 1 introduction and results domino tilings of the aztec diamond were introduced in 8 9 asymptotic properties of random domino tilings of the aztec diamond have been studied in 5 12 15 in particular in 12 the existence of the so called arctic circle was proved the arctic circle is the asymptotic boundary of the disordered so called temperate region of the tiling outside this boundary the tiling forms a completely regular brick wall pattern the methods in 12 combined with the results in 13 show that the fluctuations of the point of intersection of the boundary of the temperate region with a line converge to the tracy widom distribution of random matrix theory in this paper we extend this result to show that the fluctuations of the boundary around the arctic circle converges to the airy process introduced in 23 the paper is a continuation of the approach used in 14 and 15 where certain point processes with determinantal correlation functions 24 and the krawtchouk ensemble were used we will use the general techniques developed in 16 and investigate an extended point process which also has determinantal correlation functions given by a kernel which we call the extended krawtchouk kernel the aztec diamond an of order n is the union of all lattice squares m m 1 l l 1 m l 2 z that lie inside the region x1 y1 x1 y1

comparison theorems and oscillation criteria for difference equations abstract first we establish the equivalence of the oscillation of the delay difference equation x n p n x n k 0 and the second order difference equation 2 y n 1 2 k 1 k k k 1 p n k k k 1 k 1 y n 0 where pn is a sequence of nonnegative real numbers and k is a positive integer next we obtain some sharp conditions for oscillations and nonoscillations of the first equation

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

fermionic construction of tau functions and random processes abstract tau functions expressed as fermionic expectation values e date m jimbo m kashiwara t miwa transformation groups for soliton equations in m jimbo t miwa eds nonlinear integrable systems classical theory and quantum theory world scientific 1983 pp 39 120 are shown to provide a natural and straightforward description of a number of random processes and statistical models involving hard core configurations of identical particles on the integer lattice like a discrete version simple exclusion processes asep nonintersecting random walkers lattice coulomb gas models and others as well as providing a powerful tool for combinatorial calculations involving paths between pairs of partitions we study the decay of the initial step function within the discrete asep d asep model as an example for constant hopping rates we obtain vershik kerov type of asymptotic configuration of particles

airy processes with wanderers and new universality classes consider n m nonintersecting brownian bridges with n of them leaving from 0 at time t 1 and returning to 0 at time t 1 while the m remaining ones wanderers go from m points a i to m points b i first we keep m fixed and we scale a i b i appropriately with n in the large n limit we obtain a new airy process with wanderers in the neighborhood of root 2n the approximate location of the rightmost particle in the absence of wanderers this new process is governed by an airy type kernel with a rational perturbation letting the number m of wanderers tend to infinity as well leads to two pearcey processes about two cusps a closing and an opening cusp the location of the tips being related by an elliptic curve upon tuning the starting and target points one can let the two tips of the cusps grow very close this leads to a new process which might be governed by a kernel represented as a double integral involving the exponential of a quintic polynomial in the integration variables

the arctic curve of the domain wall six vertex model the problem of the form of the arctic curve of the six vertex model with domain wall boundary conditions in its disordered regime is addressed it is well known that in the scaling limit the model exhibits phase separation with regions of order and disorder sharply separated by a smooth curve called the arctic curve to find this curve we study a multiple integral representation for the emptiness formation probability a correlation function devised to detect spatial transition from order to disorder we conjecture that the arctic curve for arbitrary choice of the vertex weights can be characterized by the condition of condensation of almost all roots of the corresponding saddle point equations at the same known value in explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle point equations the arctic curve is obtained in parametric form and appears to be a non algebraic curve in general it turns into an algebraic one in the so called root of unity cases the arctic curve is also discussed in application to the limit shape of q enumerated with 0 q 4 large alternating sign matrices in particular as q 0 the limit shape tends to a nontrivial limiting curve given by a relatively simple equation

universal singularity at the closure of a gap in a random matrix theory we consider a hamiltonian h h 0 v in which h 0 is a given non random hermitian matrix and v is an n times n hermitian random matrix with a gaussian probability distribution we had shown before that dyson s universality of the short range correlations between energy levels holds at generic points of the spectrum independently of h 0 we consider here the case in which the spectrum of h 0 is such that there is a gap in the average density of eigenvalues of h which is thus split into two pieces when the spectrum of h 0 is tuned so that the gap closes a new class of universality appears for the energy correlations in the vicinity of this singular point

non colliding paths in the honeycomb dimer model and the dyson process in this paper we describe a natural family of random non intersecting discrete paths in the dimer model on the honeycomb lattice we show that when the dimer model is going to freeze this family of paths after a proper rescaling converges to the extended sine process obtained traditionally as the limit of the dyson model when the number of particles goes to infinity

boxed plane partitions as an exactly solvable boson model plane partitions naturally appear in many problems of statistical physics and quantum field theory for instance in the theory of faceted crystals and of topological strings on calabi yau threefolds in this paper a connection is made between the exactly solvable model with the boson dynamical variables and a problem of enumeration of boxed plane partitions three dimensional young diagrams placed into a box of a finite size the scalar product of the introduced model gives the macmahon enumeration formula while the correlation functions of the model may be considered as the generating functionals of the young diagrams with the fixed heights of its certain columns the evaluation of the scalar product and the correlation functions is based on the yang baxter algebra the analytical answers are obtained in terms of determinants and they can also be expressed through the schur functions

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

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

infinite hierarchies of nonlinearly dependent periodic orbits quadratic maps are used to show explicitly that the skeleton of unstable periodic orbits underlying classical and quantum dynamics is stratified into a doubly infinite hierarchy of orbits inherited from a set of basic seeds through certain nonlinear transformations t alpha x the hierarchy contains nonunique substructurings which arise from the different possibilities of sequencing the transformations t alpha x the structuring of the orbital skeleton is shown to be generic for abelian equations i e for all dynamical systems generated by iterating rational functions

a note correcting the proof of a lemma in a recent paper abstract a nonoscillation criterion for a second order linear difference equation is established correcting a result in 1

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

accumulation points in nonlinear parameter lattices abstract in 1963 myrberg determined a period doubling cascade of the quadratic map to accumulate at 1 401155189 as found later the geometric way with which model parameters approach this value has universal behavior and several characteristic exponents associated with it in the present paper we discuss the existence of an infinite number of points characterized by the simultaneous accumulation of two or more bifurcation cascades we present an accurate numerical determination of the vertices of a doubly infinite nonlinear lattice which lead to a point of double accumulation in addition we discuss the number theoretic nature of irrationalities characterizing vertices novel classes of universality with characteristic exponents are conjectured to exist near points of multiple accumulations

an oscillation criterion for a delay difference equation by using the method of yu zhang and qian in j s yu b g zhang x z qian oscillation of delay difference equations with oscillating coefficients j math anal appl 177 1993 432 444 we prove another oscillation criterion for all solutions of the following linear delay difference equation x n 1 x n p nx n k 0 n 0 1 2 where p n is a sequence of real numbers and k is a positive integer

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

oscillation and comparison results for delay difference equations abstract we establish a necessary and sufficient condition for the existence of positive solutions and obtain sturm type comparison theorem for the difference equation x n 1 x n i 1 m pi n x n ki o n 0 we also obtain the following comparison result equation oscillates if and only if x n 1 x n i 1 m qi n x n ki o n 0 oscillates provided n lim pi n qi i 1 2 m

tacnode gue minor processes and double aztec diamonds we study determinantal point processes arising in random domino tilings of a double aztec diamond a region consisting of two overlapping aztec diamonds at a turning point in a single aztec diamond where the disordered region touches the boundary the natural limiting process is the gue minor process increasing the size of a double aztec diamond while keeping the overlap between the two aztec diamonds finite we obtain a new determinantal point process which we call the tacnode gue minor process this process can be thought of as two colliding gue minor processes as part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode gue minor process we find the inverse kasteleyn matrix for the dimer model version of the double aztec diamond

double aztec diamonds and the tacnode process discrete and continuous non intersecting random processes have given rise to critical infinite dimensional diffusions like the airy process the pearcey process and variations thereof it has be

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

random skew plane partitions and the pearcey process we study random skew 3d partitions weighted by qvol and specifically the q 1 asymptotics of local correlations near various points of the limit shape we obtain sine kernel asymptotics for correlations in the bulk of the disordered region airy kernel asymptotics near a general point of the frozen boundary and a pearcey kernel asymptotics near a cusp of the frozen boundary

crises and hysteresis in coupled logistic maps a new type of crisis is found in coupled logistic maps in this crisis several chaotic attractors undergo cyclic collisions with their basin boundaries and merge into a bigger one whose chaotic dynamical behavior exhibits cyclic transitions among its components with a well defined overall period

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

units remarkable points in dynamical systems abstract in number theory units are very special numbers characterized by having their norm equal to unity so in the real quadratic field z 3 the number of 2 3 0 2679491924 is a unit because 2 3 2 3 1 in this paper we determine precisely the numerical values of the coordinates of some points defined by multiple intersections of domains of stability in the parameter space of the henon map and in all cases considered for which analytical calculations were feasible find that such intersection points are invariably defined by units and by simple functions of units the very special points defined by units are analogous to the familiar multicritical points in phase diagrams some simple consequences of the precise dynamics on the ground fields enforced by the equations of motion are discussed

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

infinite systems of noncolliding generalized meanders and riemann liouville differintegrals yor s generalized meander is a temporally inhomogeneous modification of the 2 1 dimensional bessel process with 1 in which the inhomogeneity is indexed by kappa in 0 2 nu 1 we introduce the noncolliding particle systems of the generalized meanders and prove that they are pfaffian processes in the sense that any multitime correlation function is given by a pfaffian in the infinite particle limit we show that the elements of matrix kernels of the obtained infinite pfaffian processes are generally expressed by the riemann liouville differintegrals of functions comprising the bessel functions j used in the fractional calculus where orders of differintegration are determined by as special cases of the two parameters the present infinite systems include the quaternion determinantal processes studied by forrester nagao and honner and by nagao which exhibit the temporal transitions between the universality classes of random matrix theory

random walks and random permutations a connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the gue random hermitian matrices the scaling of the distribution gives the maximum mean displacement after t time steps as 2t 1 2 with standard deviation proportional to 1 3 the exponent 1 3 is typical of a large class of two dimensional growth problems

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

vicious walkers and young tableaux i without walls we rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of young tableaux and integer partitions in this way we provide a proof of a result previously only conjectured for the total number of stars

eynard mehta theorem schur process and their pfaffian analogs we give simple linear algebraic proofs of the eynard mehta theorem the okounkov reshetikhin formula for the correlation kernel of the schur process and pfaffian analogs of these results we also discuss certain general properties of the spaces of all determinantal and pfaffian processes on a given finite set

asymptotics of random lozenge tilings via gelfand tsetlin schemes a gelfand tsetlin scheme of depth n is a triangular array with m integers at level m m 1 ldots n subject to certain interlacing constraints we study the ensemble of uniformly random gelfand tsetlin schemes with arbitrary fixed n th row we obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble and also of its q deformation this provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice or equivalently of random stepped surfaces we work with a class of polygons which allows arbitrarily large number of sides we show that the local limit behavior of random tilings as all dimensions of the polygon grow is directed by ergodic translation invariant gibbs measures the slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by kenyon and okounkov acta math 199 2 263 302 2007 we also prove that at the edge of the limit shape the asymptotic behavior of random tilings is given by the airy process in particular our results cover the most investigated case of random boxed plane partitions when the polygon is a hexagon

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

exact solutions for kpz type growth processes random matrices and equilibrium shapes of crystals abstract three models from statistical physics can be analyzed by employing space time determinantal processes 1 crystal facets in particular the statistical properties of the facet edge and equivalently tilings of the plane 2 one dimensional growth processes in the kardar parisi zhang universality class and directed last passage percolation 3 random matrices multi matrix models and dyson s brownian motion we explain the method and survey results of physical interest

remarks on oscillation of second order linear difference equation in this remark we shall show that the lemma 1 and parts of the proof of theorem 1 in the paper oscillation criteria for second order linear difference equations applied mathematics and computation 145 2003 591 601 are incorrect

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

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

asymptotics for the covariance of the airy2 process in this paper we compute some of the higher order terms in the asymptotic be havior of the two point function p a2 0 s1 a2 t s2 extending the previous work of adler and van moerbeke arxiv math pr 0302329 ann probab 33 1326 1361 2005 a nd widom j stat phys 115 1129 1134 2004 we prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the painleve ii function q and its derivative q further for up to tenth order we give this asymptotic approximation as a linear combination of the tracy widom gue density function f2 and its derivatives as a corollary to this the asymptotic covariance is expressed up to tenth order in terms of the moments of the tracy widom gue distribution

fractal and riddled basins arithmetic signatures in the parameter space of two coupled quadratic maps we investigate the parameter space of two coupledquad ratic logistic maps of special interest is the analytical characterization of the precursors leading to riddled basins we delimit stability domains for orbits with the two lowest periods in addition we study the singularities of the phase space surfaces obtainedby eliminating all parameters from the equations of motion c 2001 elsevier science b v all rights reserved

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

dynamical scaling implications of ferrari prahofer and spohn s remarkable spatial scaling results for facet edge fluctuations spurred by theoretical predictions from ferrari et al phys rev e 69 035102 r 2004 we rederived and extended their result heuristically with experimental colleagues we used stm line scans to corroborate their prediction that the fluctuations of the step bounding a facet exhibit scaling properties distinct from those of isolated steps or steps on vicinal surfaces the correlation functions was shown to go as t 0 15 3 decidedly different from the t 0 26 2 behavior for fluctuations of isolated steps

semicycles of solutions of nonlinear difference equations with several delays abstract in this paper we study the semicycles of oscillatory solutions of nonlinear difference equation with several delays where pi n is a real sequence with pi n 0 for all large n l mi kij are positive integers ij i 1 2 l j 1 2 mi are nonnegative real numbers with mij 1 ij 1 i 1 2 l upper bound of numbers of terms of semicycles are determined

non colliding brownian motions and the extended tacnode process we consider non colliding brownian motions with two starting points and two endpoints the points are chosen so that the two groups of brownian motions just touch each other a situation that is referred to as a tacnode the extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by delvaux kuijlaars and zhang the form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by adler ferrari and van moerbeke we also obtain the correlation kernel for a finite number of non colliding brownian motions starting at two points and ending at arbitrary points

oscillations of delay difference equations in a critical state consider the first order delay difference equation xn i 1m pi n xn ki 0 n 0 1 2 in the case where lim infn pi n pi 0 ki 0 i 1 2 a necessary and sufficient condition for the oscillation of all solutions of the above equation is established in the critical state that the corresponding limiting equation xn i 1m pixn ki 0 n 0 1 2 admits a nonoscillatory solution it is to be pointed out that there is no result in this critical state for a delay difference equation with more than one delay argument

correlation function of the schur process with a fixed final partition we consider a generalization of the schur process in which a partition evolves from the empty partition into an arbitrary fixed final partition we obtain a double integral representation of the correlation kernel for a special final partition with only one row the edge scaling limit is also discussed by the use of the saddle point analysis if we appropriately scale the length of the row the limiting correlation kernel changes from the extended airy kernel

arctic curves of the six vertex model on generic domains the tangent method we revisit the problem of determining the arctic curve in the six vertex model with domain wall boundary conditions we describe an alternative method by which we recover the previously conjectured analytic expression in the square domain we adapt the method to work for a large class of domains and for other models exhibiting limit shape phenomena we study in detail some examples and derive in particular the arctic curve of the six vertex model in a triangoloid domain at the ice point

fluctuations of an atomic ledge bordering a crystalline facet when a high symmetry facet joins the rounded part of a crystal the step line density vanishes as square root of r with r denoting the distance from the facet edge this means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation we investigate the statistical properties of the border ledge fluctuations in the scaling regime they turn out to be non gaussian and related to the edge statistics of gaussian unitary ensemble multimatrix models

level spacing of random matrices in an external source in an earlier work we had considered a gaussian ensemble of random matrices in the presence of a given external matrix source the measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials or on the coulomb gas representation are not available nevertheless the n point correlation functions are still given in terms of the determinant of a kernel known through an explicit integral representation this kernel is no longer symmetric though and is not readily accessible to standard methods in particular finding the level spacing probability is always a delicate problem in fredholm theory and we have to reconsider the problem within our model we find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state the problem is solved through coupled non linear differential equations which turn out to form a hamiltonian system as a result we find that the level spacing probability p s behaves like exp c s 8 over 3 for large spacing s this is consistent with the asymptotic behavior exp c s 2 beta 2 whenever the density of state behaves near the edge as rho lambda sim lambda beta

nonexistence and existence of positive solutions for difference equations with unbounded delay consider the delay difference equation xn 1 xn pnx n 0 0 1 2 where n z n n for n n and limn n pn r some existence results of positive solutions and oscillation criteria for this equation are obtained

spectral form factor in a random matrix theory in the theory of disordered systems the spectral form factor s ensuremath tau the fourier transform of the two level correlation function with respect to the difference of energies is linear for ensuremath tau mathrm ensuremath tau mathrm c and constant for ensuremath tau g mathrm ensuremath tau mathrm c near zero and near mathrm ensuremath tau mathrm c it exhibits oscillations which have been discussed in several recent papers in problems of mesoscopic fluctuations and quantum chaos a comparison is often made with a random matrix theory it turns out that even in the simplest gaussian unitary ensemble these oscillations have not yet been studied there for random matrices the two level correlation function ensuremath rho ensuremath lambda 1 ensuremath lambda 2 exhibits several well known universal properties in the large n limit its fourier transform is linear as a consequence of the short distance universality of ensuremath rho ensuremath lambda 1 ensuremath lambda 2 however the crossover near zero and mathrm ensuremath tau mathrm c requires one to study these correlations for finite n for this purpose we use an exact contour integral representation of the two level correlation function which allows us to characterize these crossover oscillatory properties this representation is then extended to the case in which the hamiltonian is the sum of a deterministic part mathrm h 0 and of a gaussian random potential v finally we consider the extension to the time dependent case

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

dominos and the gaussian free field we define a scaling limit of the height function on the domino tiling model dimer model on simply connected regions in z 2 and show that it is the massless free field a gaussian process with independent coefficients when expanded in the eigenbasis of the laplacian

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

oscillation criteria for difference equations with unbounded delay consider the delay difference equation n 1 n pn n 0 n 0 1 2 where n z is nondecreasing n n for n n and limn n pn is a nonnegative sequence some oscillation criteria for this equation are obtained

a cell population model described by impulsive pdes existence and numerical approximation this paper considers a cell population model described by impulsive pdes existence and numerical approximation of solutions are investigated by showing that the related operator a x ddx on c0 generates a strongly conitunous semigroups under general assumption on a x we prove the existence of the solution of the nonlinear problem with impulses and given approximation of the solution of the homogenuous equation

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

scaling limit of vicious walks and two matrix model we consider the diffusion scaling limit of the one dimensional vicious walker model of fisher and derive a system of nonintersecting brownian motions the spatial distribution of n particles is studied and it is described by use of the probability density function of eigenvalues of nxn gaussian random matrices the particle distribution depends on the ratio of the observation time t and the time interval t in which the nonintersecting condition is imposed as t t is going on from 0 to 1 there occurs a transition of distribution which is identified with the transition observed in the two matrix model of pandey and mehta despite of the absence of matrix structure in the original vicious walker model in the diffusion scaling limit accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution

oscillation and nonoscillation for second order linear difference equations new oscillation and nonoscillation theorems are obtained for the second order linear difference equation 2xn 1 pnxn 0 where if pn n 1 is a real sequence with pn 0

polynuclear growth model with external source and random matrix model with deterministic source we present a random matrix interpretation of the distribution functions which have appeared in the study of the one dimensional polynuclear growth png model with external sources it is shown that the distribution goe2 which is defined as the square of the gaussian orthogonal ensemble goe tracy widom distribution can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source which have been studied in a different context previously compared to the original interpretation of the goe2 as the square of goe ours has an advantage in that it can also describe the transition from the gaussian unitary ensemble gue tracy widom distribution to the goe2 we further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain noncolliding brownian motion model and the multilayer png model with an external source this provides us with a multimatrix model interpretation of the multipoint height distributions of the png model with an external source

universality of correlation functions of hermitian random matrices in an external field the behavior of correlation functions is studied in a class of matrix models characterized by a measure exp s containing a potential term and an external source term s n tr v m ma in the large n limit the short distance behavior is found to be identical to the one obtained in previously studied matrix models thus extending the universality of the level spacing distribution the calculation of correlation functions involves finite n determinant formulae reducing the problem to the large n asymptotic analysis of a single kernel k this is performed by an appropriate matrix integral formulation of k multi matrix generalizations of these results are discussed

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process