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

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

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

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

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

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

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

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

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

applications of catastrophe theory for statistical modeling in the biosciences abstract although catastrophe theory has been applied with mixed success to many problems in the biosciences very few of these applications have used any form of statistical modeling we present examples of the applications of statistical catastrophe theory in the analysis of experimental data these include examples of hysteresis effects bifurcation effects and the full cusp catastrophe model the methods of statistical catastrophe theory draw upon the theories of parameter estimation for multiparameter exponential families nonlinear time series analysis and stochastic differential equations we discuss the application of these methods to both canonical and noncanonical catastrophe models

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

effects of spectral slope on perceived breathiness in vowels previous studies have indicated that breathiness is correlated with changes in spectral slope as well as aspiration noise klatt klatt analysis synthesis and perception of voice quality variations among female and male talkers j acoust soc am 87 820 857 1990 in a recent study shrivastav and pinero shrivastav pinero effect of aspiration noise and spectral slope on perceived breathiness in vowels j acoust soc am 117 2622 2005 observed that varying spectral slope resulted in minimal changes on the perceived breathiness for synthetic vowels however the stimuli tested in this experiment included a relatively narrow range of spectral slope variation the goal of the present experiment was to verify the role of spectral slope on the perception of breathiness by testing stimuli that had a wider range of variation in spectral slope ten voices 5 male and 5 female were synthesized using a klatt synthesizer each of these was manipulated to generate two continua varying in the

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

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

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

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

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