conditional intensity and gibbsianness of determinantal point processes the papangelou intensities of determinantal or fermion point processes are investigated these exhibit a monotonicity property expressing the repulsive nature of the interaction and satisfy a bound implying stochastic domination by a poisson point process we also show that determinantal point processes satisfy the so called condition which is a general form of gibbsianness under a continuity assumption the gibbsian conditional probabilities can be identified explicitly

unambiguous 1 uniform morphisms a morphism s is unambiguous with respect to a word a if there is no other morphism t that maps a to the same image as s in the present paper we study the question of whether for any given word there exists an unambiguous 1 uniform morphism i e a morphism that maps every letter in the word to an image of length 1

a connection formula for the second painleve transcendent we consider a particular case of the second painleve transcendent y xy 2y 3 it is known that if y x kai x as x then if 0 k 1 y x sim d left x right raise0 5ex hbox scriptstyle 1 kern 0 1em kern 0 15em lower0 25ex hbox scriptstyle 4 sin frac 2 3 left x right 3 2 frac 3 4 d2 ell n left x right c asx to infty where d k and c k are the connection formulae for this nonlinear ordinary differential equation

intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators a theory of intrinsic fluctuations is developed of a phase ordering parameter for large populations of weakly and uniformly coupled limit cycle oscillators with distributed native frequencies in particular it is shown that the intensity as well as the correlation time of fluctuations exhibit power law divergence at the onset of mutual entrainment with critical exponents which depend on whether the coupling strength approaches the threshold from below or above this peculiar feature is demonstrated by numerical simulations mainly through finite size scaling analyses in the course of exploring its origin we encounter a new concept termed a correlation frequency which provides a natural interpretation of the finite size scaling laws a comment is given on a recent theory by kuramoto and nishikawa to clarify why it contradicts our results

analytical depoissonization and its applications in combinatorics and analysis of algorithms a poisson version of a problem henceforth called a poisson model or poissonization is often easier to solve than the original one which we name here the bernoulli model poissonization is a technique that replaces the original input e g think of balls thrown into urns by a poisson process e g think of balls arriving according to a poisson process into urns more precisely analytical poisson transform maps a sequence e g characterizing the bernoulli model into a generating function of a complex variable however after poissonization one must depoissonize in order to translate the results of the poisson model into the original i e bernoulli model we present in this paper several analytical depoissonization results that fall into the following general scheme if the poisson transform has an appropriate growth in the complex plane then an asymptotic expansion of the sequence can be expressed in terms of the poisson transform and its derivatives evaluated on the real line not surprisingly actual formulations of depoissonization results depend on the nature of the growth of the poisson transform and thus we have polynomial and exponential depoissonization theorems normalization e g as in the central limit theorem introduces another twist that led us to formulate the so called diagonal depoissonization theorems finally we illustrate our results on numerous examples from combinatorics and the analysis of algorithms and data structures e g combinatorial assemblies digital trees multiaccess protocols probabilistic counting selecting a leader data compression etc authors abstract

associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise we study associative memory based on temporal coding in which successful retrieval is realized as an entrainment in a network of simple phase oscillators with distributed natural frequencies under the influence of white noise the memory patterns are assumed to be given by uniformly distributed random numbers on 0 2 ensuremath pi so that the patterns encode the phase differences of the oscillators to derive the macroscopic order parameter equations for the network with an extensive number of stored patterns we introduce an effective transfer function by assuming a fixed point equation of the form of the thouless anderson palmer equation which describes the time averaged output as a function of the effective time averaged local field properties of the networks associated with synchronization phenomena for a discrete symmetric natural frequency distribution with three frequency components are studied based on the order parameter equations and are shown to be in good agreement with the results of numerical simulations two types of retrieval states are found to occur with respect to the degree of synchronization when the size of the width of the natural frequency distribution is changed

dynamic mean field models h theorem for stochastic processes and basins of attraction of stationary processes abstract stochastic processes of dynamic mean field models are studied in terms of nonlinear fokker planck equations we show that h theorems for single time point distributions can be used to derive h theorems for hierarchies of joint distributions in doing so we prove the convergence of transient stochastic processes to stationary ones for multistable systems we furthermore determine the basins of attraction of these stationary processes our results are illustrated by means of a mean field model exhibiting bistability

moment formulae for general point processes abstract the goal of this paper is to generalize most of the moment formulae obtained in 12 more precisely we consider a general point process and show that the quantities relevant to our problem are the so called papangelou intensities when the papangelou intensities of are well defined we show some general formulae to recover the moment of order n of the stochastic integral of the point process we will use these extended results to introduce a divergence operator and study a random transformation of the point process

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

operators associated with soft and hard spectral edges from unitary ensembles abstract using hankel operators and shift invariant subspaces on hilbert space this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices such tracy widom operators are realized as controllability operators for linear systems and are reproducing kernels for weighted hardy spaces known as sonine spaces periodic solutions of hill s equation give a new family of tracy widom type operators this paper identifies a pair of unitary groups that satisfy the von neumann weyl anti commutation relations and leave invariant the subspaces of l 2 that are the ranges of projections given by the tracy widom operators for the soft edge of the gaussian unitary ensemble and hard edge of the jacobi ensemble

a variational principle in the dual pair of reproducing kernel hilbert spaces and an application given a positive definite bounded linear operator a on the hilbert space mathcal h 0 l2 e we consider a reproducing kernel hilbert space mathcal h with a reproducing kernel a x y here e is any countable set and a x y x y e is the representation of a w r t the usual basis of mathcal h 0 imposing further conditions on the operator a we also consider another reproducing kernel hilbert space mathcal h with a kernel function b x y which is the representation of the inverse of a in a sense so that mathcal h mathcal h 0 mathcal h becomes a rigged hilbert space we investigate the ratios of determinants of some partial matrices of a and b we also get a variational principle on the limit ratios of these values we apply this relation to show the gibbsianness of the determinantal point process or fermion point process defined by the operator a i a 1 on the set e

dynamics of two mutually coupled slow inhibitory neurons abstract inhibition in oscillatory networks of neurons can have apparently paradoxical effects sometimes creating dispersion of phases sometimes fostering synchrony in the network we analyze a pair of biophysically modeled neurons and show how the rates of onset and decay of inhibition interact with the timescales of the intrinsic oscillators to determine when stable synchrony is possible we show that there are two different regimes in parameter space in which different combinations of the time constants and other parameters regulate whether the synchronous state is stable we also discuss the construction and stability of nonsynchronous solutions and the implications of the analysis for larger networks the analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow flows and fast jumps

correlated attractors from uncorrelated stimuli analytic results for a neural network model proposed to account for neurophysiological experiments in which attractors for uncorrelated patterns are found to be correlated are presented exact expressions for the associative attractors and their correlations are described

determinantal point processes with j hermitian correlation kernels let x be a locally compact polish space and let m be a reference radon measure on x let x denote the configuration space over x that is the space of all locally finite subsets of x a point process on x is a probability measure on x a point process is called determinantal if its correlation functions have the form k n x1 xn det k xi xj i j 1 n the function k x y is called the correlation kernel of the determinantal point process assume that the space x is split into two parts x x1 x2 a kernel k x y is called j hermitian if it is hermitian on x1 x1 and x2 x2 and k x y k y x for x x1 and y x2 we derive a necessary and sufficient condition of existence of a determinantal point process with a j hermitian correlation kernel k x y

morphically primitive words in the present paper we introduce an alternative notion of the primitivity of words that unlike the standard understanding of this term is not based on the power and hence the concatenation of words but on morphisms for any alphabet s we call a word w s morphically imprimitive provided that there are a shorter word v and morphisms h h s s satisfying h v w and h w v and we say that w is morphically primitive otherwise we explain why this is a well chosen terminology we demonstrate that morphic im primitivity of words is a vital attribute in many combinatorial domains based on finite words and morphisms and we study a number of fundamental properties of the concepts under consideration

linear stability analysis of retrieval state in associative memory neural networks of spiking neurons we study associative memory neural networks of the hodgkin huxley type of spiking neurons in which multiple periodic spatiotemporal patterns of spike timing are memorized as limit cycle type attractors in encoding the spatiotemporal patterns we assume the spike timing dependent synaptic plasticity with the asymmetric time window analysis for periodic solution of retrieval state reveals that if the area of the negative part of the time window is equivalent to the positive part then crosstalk among encoded patterns vanishes phase transition due to the loss of the stability of periodic solution is observed when we assume fast function for direct interaction among neurons in order to evaluate the critical point of this phase transition we employ floquet theory in which the stability problem of the infinite number of spiking neurons interacting with function is reduced to the eigenvalue problem with the finite size of matrix numerical integration of the single body dynamics yields the explicit value of the matrix which enables us to determine the critical point of the phase transition with a high degree of precision

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

integrals over grassmannians and random permutations testing the independence of two gaussian populations involves the distribution of the sample canonical correlation coefficients given that the actual correlation is zero the laplace transform as a function of x of this distribution is not only an integral over the grassmannian gr p f n of p dimensional planes in real complex or quaternion n space f n but is also related to a generalized hypergeometric function such integrals are solutions of painleve like equations in the complex case they are solutions to genuine painleve equations these integrals over gr p c n have remarkable expansions in x related to random words of length l formed with an alphabet of p letters 1 p the coefficients of these expansions are given by the probability that a word i contains a subsequence of letters p p 1 1 in that order and ii that the maximal length of the disjoint union of p 1 increasing subsequences of the word is less than or equal tok where k refers to the power of x note that if each letter appears in the word then the maximal length of the disjoint union of p increasing subsequences of the word is automatically l and is thus trivial c 2003 elsevier inc all rights reserved

stability analysis of mean field type nonlinear fokker planck equations associated with a generalized entropy and its application to the self gravitating system multidimensional nonlinear fokker planck equations of mean field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions two types of eigenvalue equations are studied the nonlinear fokker planck equations are shown to exhibit an h theorem with a liapunov functional that takes the form of a free energy involving generalized entropies of tsallis the second order variation of the liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean field coupling potential assuming quasiequilibrium for the velocity distribution the reduced eigenvalue equation with space coordinates alone is also obtained the alternative type of eigenvalue equation based on the linearization of the nonlinear fokker planck equations is presented taking the mean field coupling potential to be the gravitational one the nonlinear fokker planck equation in terms of three dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean field model of self gravitating system by solving the eigenvalue equation for the eigenfunction with 0 eigenvalue the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed

determinantal point process models and statistical inference summary statistical models and methods for determinantal point processes dpps seem largely unexplored we demonstrate that dpps provide useful models for the description of spatial point pattern data sets where nearby points repel each other such data are usually modelled by gibbs point processes where the likelihood and moment expressions are intractable and simulations are time consuming we exploit the appealing probabilistic properties of dpps to develop parametric models where the likelihood and moment expressions can be easily evaluated and realizations can be quickly simulated we discuss how statistical inference is conducted by using the likelihood or moment properties of dpp models and we provide freely available software for simulation and statistical inference

extreme gaps between eigenvalues of random matrices this paper studies the extreme gaps between eigenvalues of random matrices we give the joint limiting law of the smallest gaps for haar distributed unitary matrices and matrices from the gaussian unitary ensemble in particular the kth smallest gap normalized by a factor n 4 3 has a limiting density proportional to x3k 1e x3 concerning the largest gaps normalized by n logn they converge in lp to a constant for all p 0 these results are compared with the extreme gaps between zeros of the riemann zeta function

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

stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations abstract we discuss two fundamental aspects of fokker planck equations that are nonlinear with respect to probability densities first we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions second we demonstrate that nonlinear fokker planck equations can be interpreted as linear fokker planck equations describing nonlinear families of markov diffusion processes we exploit this finding in order to derive complete hierarchies of probability densities from nonlinear fokker planck equations

statistical mechanics for neural networks with continuous time dynamics the present paper is intended to summariz our rent knowledge about the long time behaviour of nehwrks of graded response neurons with continuous time dynamics we demonstrate the workings of our previously developed statistical mechanical approach to continuous time dynamics by applying it to networks with various forms of synaptic organization leaming rules and neural composition neuron types as encoded in gain functions as well as to networks varying with respect to the ensemble of stored data unbiased and low activity patterns we present phase diagrams and compute distributions of local fields for a variety of examples local field distributions are found to deviate from the gaussian form obtained for stochastic neumns in the wntexi of the replica approach a solution to the low firing rates problem within the framework of nets of analogue neurons is also briefly discussed finally the statisticalmechanical approach to the analysis of continuowtime dynamics is extended to include effects of fast stochastic noise detailed balance solutions are shown to be unique and of canonical form governed by hamiltonians which exhibit a reciprocity relation between potential dynamics and fuing rate dynamics for the potential dynamics the hamiltonian is given by the lyapounw function of the system expressed in terms of the fuing rates and it generates a gibbs distribution over fuing rate space for the noisy firinerate dynamics the same lyapounov function now expressed in terms of neural potentials generates a gibbs distribution over the space of these potentials as a consequence the firing rate dynamics will freeze in configurations saturating the neural input xtput relations whenever such saturation levels exist both types of stationary distribution are shown to exist only under unrealistic assumptions about the noise in the system

stein s method and plancherel measure of the symmetric group we initiate a stein s method approach to the study of the plancherel measure of the symmetric group a new proof of kerov s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained the proof gives an error term the construction of an exchangeable pair needed for applying stein s method arises from the theory of harmonic functions on bratelli diagrams we also find the spectrum of the markov chain on partitions underlying the construction of the exchangeable pair this yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

the mean field theory of a q state neural network model the mean field equations of a q state clock neural network model are derived in the replica symmetric approximation using the replica method these equations are studied for the cases q 2 3 4 and q to infinity it is shown that an infinite number of patterns can be stored in the network even in the limit q to infinity the phase diagram and storage capacity of the network are calculated and the information content considered although the overlap achieved with a nominated pattern is shown to decrease as q increases the information stored in the network is shown to increase

the asep and determinantal point processes we introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line with correlation kernels defined via spectral projections for the associated jacobi matrices for classical weights we show how such ensembles arise as limits of various hypergeometric orthogonal polynomial ensembles we then prove that the q laplace transform of the height function of the asep with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete laguerre ensemble a member of the new family this allows us to obtain the large time asymptotics of the asep in three limit regimes a for finitely many rightmost particles b gue tracy widom asymptotics of the height function c kpz asymptotics of the height function for the asep with weak asymmetry we also give similar results for two instances of the stochastic six vertex model in a quadrant the proofs are based on limit transitions for the corresponding determinantal point processes

replica symmetric theory of nonlinear analogue neural networks on the basis of the theory of the naive mean field model of spin glasses analogue neural networks of the hopfield type are investigated in the saturation limit the saddle point equations for the order parameters describing retrieval and spin glass phases of the networks are obtained by means of a statistical mechanical analysis within the framework of the replica symmetric theory and are shown to undergo a modification to that of ags theory due to the absence of the onsager reaction term in the tap equation based on the equations the memory storage capacity of the networks is analysed as a function of the analogue gain beta a small increase in the critical storage capacity is found for finite values of beta compared with that of the ising model networks with corresponding inverse temperature although the qualitative nature of the phase diagram is unchanged

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

asymptotics of q plancherel measures in this paper we are interested in the asymptotic size of the rows and columns of a random young diagram under a natural deformation of the plancherel measure coming from hecke algebras the first lines of such diagrams are typically of order n so it does not fit in the context of the work of biane int math res notices 4 179 192 2001 and niady probab theory relat fields 136 263 297 2006 using the theory of polynomial functions on young diagrams of kerov and olshanski we are able to compute explicitly the first and second order asymptotics of the length of the first rows our method also works for other measures for example those coming from schur weyl representations

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

oscillatory associative memory network with perfect retrieval abstract inspired by the discovery of possible roles of synchronization of oscillations in the brain networks of coupled phase oscillators have been proposed before as models of associative memory based on the concept of temporal coding of information here we show however that error free retrieval states of such networks turn out to be typically unstable regardless of the network size in contrast to the classical hopfield model we propose a remedy for this undesirable property and provide a systematic study of the improved model in particular we show that the error free capacity of the network is at least 2 e 2 log n patterns per neuron where n is the number of oscillators neurons and e the strength of the second order mode in the coupling function

on the correlation measure of a family of commuting hermitian operators with applications to particle densities of the quasi free representations of the car and ccr let x be a locally compact second countable hausdorff topological space we consider a family of commuting hermitian operators a indexed by all measurable relatively compact sets in x a quantum stochastic process over x for such a family we introduce the notion of a correlation measure we prove that if the family of operators possesses a correlation measure which satisfies some condition of growth then there exists a point process over x having the same correlation measure furthermore the operators a can be realized as multiplication operators in the l2 space with respect to this point process in the proof we utilize the notion of positive definiteness proposed in y g kondratiev t kuna harmonic analysis on the configuration space i general theory infin dimens anal quantum probab relat top 5 2002 201 233 in particular our result extends the criterion of existence of a point process from that paper to the case of the topological space x which is a standard underlying space in the theory of point processes as applications we discuss particle densities of the quasi free representations of the car and ccr which lead to fermion boson fermion like and boson like e g para fermions and para bosons of order 2 point processes in particular we prove that any fermion point process corresponding to a hermitian kernel may be derived in this way

conversion of temporal correlations between stimuli to spatial correlations between attractors it is shown that a simple modification of synaptic structures of the hopfield type constructed to produce autoassociative attractors produces neural networks whose attractors are correlated with several learned patterns used in the construction of the matrix the modification stores in the matrix a fixed sequence of uncorrelated patterns the network then has correlated attractors provoked by the uncorrelated stimuli thus the network converts the temporal order or temporal correlation expressed by the sequence of patterns into spatial correlations expressed in the distributions of neural activities in attractors the model captures phenomena observed in single electrode recordings in performing monkeys by miyashita et al the correspondence is as close as to reproduce the fact that given uncorrelated patterns as sequentially learned stimuli the attractors produced are significantly correlated up to a separation of 5 five in the sequence this number 5 is universal in a range of parameters and requires essentially no tuning we then discuss learning scenarios that could lead to this synaptic structure as well as experimental predictions following from it finally we speculate on the cognitive utility of such an arrangement

gibbsianness of fermion random point fields we consider fermion or determinantal random point fields on euclidean space d given a bounded translation invariant and positive definite integral operator j on l2 d we introduce a determinantal interaction for a system of particles moving on d as follows the n points located at x1 x n d have the potential energy given by where j x y is the integral kernel function of the operator j we show that the gibbsian specification for this interaction is well defined when j is of finite range in addition and for d 2 if the intensity is small enough we show that the fermion random point field corresponding to the operator j i j 1 is a gibbs measure admitted to the specification

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

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

dirichlet forms and diffusion processes for fermion random point fields abstract we construct the dirichlet forms and the associated diffusion processes on the configuration space of particles moving on the euclidean space r d for which certain fermion random point fields are invariant

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

restricted ambiguity of erasing morphisms a morphism h is called ambiguous for a string s if there is another morphism that maps s to the same image as h otherwise it is called unambiguous in this paper we examine some fundamental problems on the ambiguity of erasing morphisms we provide a detailed analysis of so called ambiguity partitions and our main result uses this concept to characterise those strings that have a morphism of strongly restricted ambiguity furthermore we demonstrate that there are strings for which the set of unambiguous morphisms depending on the size of the target alphabet of these morphisms is empty finite or infinite finally we show that the problem of the existence of unambiguous erasing morphisms is equivalent to some basic decision problems for nonerasing multi pattern languages

a power series identity and bessel type integrals over unitary groups in 2008 lehner wettig guhr and wei conjectured a power series identity and showed that it implied a determinantal formula for a bessel type integral over the unitary supergroup the integral is the supersymmetric extension of bessel type integrals over the unitary group appearing as partition functions in quantum chromodynamics the identity is proved by interpreting both sides as the same unitary integral which can be computed using the cartan decomposition an equivalent identity of schur functions is also given and interpreted probabilistically in terms of random partitions in 2008 lehner wettig guhr and wei conjectured a power series identity and showed that it implied a determinantal formula for a bessel type integral over the unitary supergroup the integral is the supersymmetric extension of bessel type integrals over the unitary group appearing as partition functions in quantum chromodynamics the identity is proved by interpreting both sides as the same unitary integral which can be computed using the cartan decomposition an equivalent identity of schur functions is also given and interpreted probabilistically in terms of random partitions

a probabilistic interpretation of the macdonald polynomials the two parameter macdonald polynomials are a central object of algebraic combinatorics and representation theory we give a markov chain on partitions of k with eigenfunctions the coefficients of the macdonald polynomials when expanded in the power sum polynomials the markov chain has stationary distribution a new two parameter family of measures on partitions the inverse of the macdonald weight rescaled the uniform distribution on cycles of permutations and the ewens sampling formula are special cases the markov chain is a version of the auxiliary variables algorithm of statistical physics properties of the macdonald polynomials allow a sharp analysis of the running time in natural cases a bounded number of steps suffice for arbitrarily large k

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

on the distributions of the lengths of the longest monotone subsequences in random words we consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length n from an alphabet of k letters in the limit as k these become the corresponding distributions for permutations on n letters we find toeplitz determinant representations for the exponential generating functions on n of these distribution functions and show that they are expressible in terms of solutions of painleve v equations we show further that in the weakly increasing case the generating unction gives the distribution of the smallest eigenvalue in the k k laguerre random matrix ensemble and that the distribution itself has after centering and normalizing an n limit which is equal to the distribution function for the largest eigenvalue in the gaussian unitary ensemble of k k hermitian matrices of trace zero

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

duality of orthogonal polynomials on a finite set we prove a certain duality relation for orthogonal polynomials defined on a finite set the result is used in a direct proof of the equivalence of two different ways using particles or holes of computing the correlation functions of a discrete orthogonal polynomial ensemble

time structure of the activity in neural network models several neural network models in continuous time are reconsidered in the framework of a general mean field theory which is exact in the limit of a large and fully connected network the theory assumes pointlike spikes which are generated by a renewal process the effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak coupling expansion it is shown that the resulting spike response model is the most general renewal model with linear inputs the standard integrate and fire model forms a special case in a network structure with several pools of identical spiking neurons the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools we derive explicit stability criteria for stationary incoherent and oscillatory coherent solutions it is shown that the stationary state of noiseless systems is almost always unstable noise suppresses fast oscillations and stabilizes the system furthermore collective oscillations are stable only if the firing occurs while the synaptic potential is increasing in particular collective oscillations in a network with delayless excitatory interaction are at most semistable inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations our general results allow a straightforward application to different network models with spiking neurons furthermore the theory allows an estimation of the errors introduced in firing rate or graded response models

weakly unambiguous morphisms a nonerasing morphism s is said to be weakly unambiguous with respect to a word s if s is the only nonerasing morphism that can map s to s s i e there does not exist any other nonerasing morphism t satisfying t s s s in the present paper we wish to characterise those words with respect to which there exists such a morphism this question is nontrivial if we consider so called length increasing morphisms which map a word to an image that is strictly longer than the word our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets we also comprehensively describe those words that have a weakly unambiguous length increasing morphism with a unary target alphabet but we have to leave the problem open for binary alphabets where we can merely give some non characteristic conditions

random sorting networks local statistics via random matrix laws this paper finds the bulk local limit of the swap process of uniformly random sorting networks the limit object is defined through a deterministic procedure a local version of the edelman greene algorithm applied to a two dimensional determinantal point process with explicit kernel the latter describes the asymptotic joint law near 0 of the eigenvalues of the corners in the antisymmetric gaussian unitary ensemble in particular the limiting law of the first time a given swap appears in a random sorting network is identified with the limiting distribution of the closest to 0 eigenvalue in the antisymmetric gue moreover the asymptotic gap in the bulk between appearances of a given swap is the gaudin mehta law the limiting universal distribution for gaps between eigenvalues of real symmetric random matrices the proofs rely on the determinantal structure and a double contour integral representation for the kernel of random poissonized young tableaux of arbitrary shape

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

painleve formulas of the limiting distributions for nonnull complex sample covariance matrices in a recent study of large non null sample covariance matrices a new sequence of functions gener alizing the gue tracy widom distribution of random matrix theory was obtained this paper derives painleve formulas of these functions and use them to prove that they are indeed distribution functions applications of these new distribution functions to last passage percolation queues in tandem and to tally asymmetric simple exclusion process are also discussed as a part of the proof a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented

convergence condition of the tap equation for the infinite ranged ising spin glass model it is shown that the power expansion of the gibbs potential of the sk model up to second order in the exchange couplings leads to the tap equation this result remains valid for the general including a ferromagnetic exchange sk model theorems of power expansions and resolvent techniques are employed to solve the convergence problem the convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations

the problem of harmonic analysis on the infinite dimensional unitary group abstract the goal of harmonic analysis on a noncommutative group is to decompose the most natural unitary representations of this group like the regular representation on irreducible ones the infinite dimensional unitary group u is one of the basic examples of big groups whose irreducible representations depend on infinitely many parameters our aim is to explain what the harmonic analysis on u consists of we deal with unitary representations of a reasonable class which are in 1 1 correspondence with characters central positive definite normalized functions on u the decomposition of any representation of this class is described by a probability measure called spectral measure on the space of indecomposable characters the indecomposable characters were found by dan voiculescu in 1976 the main result of the present paper consists in explicitly constructing a 4 parameter family of natural representations and computing their characters we view these representations as a substitute of the nonexisting regular representation of u we state the problem of harmonic analysis on u as the problem of computing the spectral measures for these natural representations a solution to this problem is given in the next paper harmonic analysis on the infinite dimensional unitary group and determinantal point processes math 0109194 to appear in ann math joint with alexei borodin we also prove a few auxiliary general results in particular it is proved that the spectral measure of any character of u can be approximated by a sequence of discrete spectral measures for the restrictions of the character to the compact unitary groups u n this fact is a starting point for computing spectral measures

asymptotics of plancherel type random partitions abstract we present a solution to a problem suggested by philippe biane we prove that a certain plancherel type probability distribution on partitions converges as partitions get large to a new determinantal random point process on the set z of nonnegative integers this can be viewed as an edge limit transition the limit process is determined by a correlation kernel on z which is expressed through the hermite polynomials we call it the discrete hermite kernel the proof is based on a simple argument which derives convergence of correlation kernels from convergence of unbounded self adjoint difference operators our approach can also be applied to a number of other probabilistic models as an example we discuss a bulk limit for one more plancherel type model of random partitions

quasi symmetries of determinantal point processes the main result of this paper is that determinantal point processes on mathbb r corresponding to projection operators with integrable kernels are quasi invariant in the continuous case under the group of diffeomorphisms with compact support theorem 1 4 in the discrete case under the group of all finite permutations of the phase space theorem 1 6 the radon nikodym derivative is computed explicitly and is given by a regularized multiplicative functional theorem 1 4 applies in particular to the sine process as well as to determinantal point processes with the bessel and the airy kernels theorem 1 6 to the discrete sine process and the gamma kernel process the paper answers a question of grigori olshanski

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

quasi invariance and integration by parts for determinantal and permanental processes abstract determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent their atoms exhibit mutual attraction of repulsion thus these processes are very far from the uncorrelated situation encountered in poisson models we establish a quasi invariance result we show that if atom locations are perturbed along a vector field the resulting process is still a determinantal respectively permanental process the law of which is absolutely continuous with respect to the original distribution based on this formula following bismut approach of malliavin calculus we then give an integration by parts formula

the quasi invariance property for the gamma kernel determinantal measure abstract the gamma kernel is a projection kernel of the form a x b y b x a y x y where a and b are certain functions on the one dimensional lattice expressed through euler s function the gamma kernel depends on two continuous parameters its principal minors serve as the correlation functions of a determinantal probability measure p defined on the space of infinite point configurations on the lattice as was shown earlier a borodin g olshanski adv math 194 2005 141 202 arxiv math ph 0305043 p describes the asymptotics of certain ensembles of random partitions in a limit regime theorem the determinantal measure p is quasi invariant with respect to finitary permutations of the nodes of the lattice this result is motivated by an application to a model of infinite particle stochastic dynamics

capacity of networks with correlated attractors we analyse the extensive loading of the neural network model proposed to describe neurophysiological experiments in which correlated attractors associated to uncorrelated patterns are found the phase diagram is obtained and discussed some generalizations of the original model are also considered in all the cases we demonstrate the existence of a region in the phase diagram with correlated attractors results from numerical simulations which confirm the mean field theory results are also presented

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

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

phase locking in weakly heterogeneous neuronal networks abstract we examine analytically the existence and stability of phase locked states in a weakly heterogeneous neuronal network we consider a model of n neurons with all to all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons we consider both inhibitory and excitatory coupling we derive the conditions under which stable phase locking is possible in homogeneous networks many different periodic phase locked states are possible their stability depends on the dynamics of the neuron and the coupling for weak heterogeneity the phase locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable for enough heterogeneity phase locked solutions either lose stability or are destroyed completely we analyze the possible states the network can take when phase locking is broken

random point fields associated with certain fredholm determinants ii fermion shifts and their ergodic and gibbs properties we construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants under a mild condition they turn out unique gibbs measures also some ergodic properties including the entropy positivity are discussed in the lattice case

nonlinear fokker planck equation exhibiting bifurcation phenomena and generalized thermostatistics a nonlinear fokker planck equation exhibiting bifurcation phenomena is proposed within the framework of generalized thermostatistics the nonlinearity responsible for the occurrence of bifurcation of solutions is assumed to be of the form appearing in the standard mean field model a liapunov function is defined that takes the form of free energy involving generalized entropies of tsallis and an h theorem is proved to show that the free energy which is bounded below continues to decrease until the system approaches one of the equilibrium distributions the h theorem ensures instead of uniqueness of the equilibrium distribution global stability of the system in that either one of multisolutions must be approached for large times local stability analysis is conducted and the second order variation of the liapunov function is computed to find its relevant part whose sign governs stability of the equilibrium distribution of the system the case with a bistable potential is investigated as an example of confirming the theory to give the bifurcation diagram displaying the order parameter as a function of the coefficient of the nonlinear diffusion term

stochastic higher spin six vertex model and macdonald measures we prove an identity that relates the q laplace transform of the height function of a higher spin inhomogeneous stochastic six vertex model in a quadrant on one side and a multiplicative functional of a macdonald measure on the other the identity is used to prove the gue tracy widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding schur measures

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

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

low firing rates an effective hamiltonian for excitatory neurons the authors analyse the behaviour of an attractor neural network which exhibits low mean temporal activity levels despite the fact that the intrinsic neuronal cycle time is very short 2 3 ms information and computation are represented on the excitatory neurons only the influence of inhibitory neurons which are assumed to react on a shorter timescale than the excitatory ones is expressed as an effective interaction of the excitatory neurons this leads to an effective model which describes the interplay of excitation and inhibition acting on excitatory neurons in terms of the excitatory neural variables alone the network operates in the presence of fast noise which is large relative to the frozen randomness induced by the stored patterns the overall fraction of active neurons is controlled by a single free parameter which expresses the relative strength of the effective inhibition associative retrieval is identified as usual with the breakdown of ergodicity in the dynamics of the network in particular with the presence of dynamical attractors corresponding to the retrieval of a given pattern in such an attractor the activity of neurons corresponding to active sites in the stored patterns increases at the expense of other neurons yet only a small fraction of the neurons active in the pattern are in the active state in each elementary time cycle and they vary from cycle to cycle in an uncorrelated fashion due to the noise hence the observed mean activity rate of any individual neuron is kept low this scenario is demonstrated by an analytical study based on the replica method and the results are tested by numerical simulations