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

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

persymmetric jacobi matrices isospectral deformations and orthogonal polynomials abstract persymmetric jacobi matrices are invariant under reflection with respect to the anti diagonal the associated orthogonal polynomials have distinctive properties that are discussed they are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice this is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric jacobi matrices from spectral points isospectral deformations of such matrices are also considered expressions for the associated polynomials and their weights are obtained in terms of the undeformed entities

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

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

efficient analysis of aperture antennas on generally shaped convex multilayered surfaces using a hybrid sd utd method a novel hybrid method is described for analyzing convex multilayered conformal array antennas the hybrid method is based on the spectral domain approach in combination with the ray based uniform theory of diffraction utd method the analysis is divided in two parts first the spectral domain approach is accelerated by using an asymptotic extraction technique where the extracted term of the green s function is calculated using utd it is shown that this new approach results in significant acceleration of the existing spectral domain algorithm without losing accuracy the modified spectral domain method is then used in the second part where generally shaped convex multilayered surfaces are analyzed by using sets of canonically shaped surfaces spheres and or circular cylinders their radii are obtained using the utd formulation which contains important information such as distance and curvature of the generally shaped surface along each geodesic the results obtained using the new algorithm are compared to the available results calculated and measured for different conformal antennas showing very good agreement

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

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

mutual admittance between slots in cylinders of arbitrary shape a numerical formulation involving the fourier transform and the combined field integral equation is presented to calculate the mutual admittance between axial slots or circumferential slots in arbitrarily shaped cylinders two cases are considered coupling between two circumferential slots and coupling between two axial slots the slots are assumed to be waveguide fed apertures excited with the te sub 10 waveguide mode with no higher order modes included in the model the combined field integral equation cfie formulation was used in conjunction with the magnetic field integral equation mfie to solve the resulting two dimensional integral equation formulations for the axial slot case are discussed results concerning the discretization and truncation of the spectrum as well as the accuracy of the procedure are presented for both axial and circumferential cases the accuracy of the procedure is demonstrated by comparison with eigenfunction data for circular cylinders

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 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

a characterization of classical and semiclassical orthogonal polynomials from their dual polynomials we study properties of the dual orthogonal polynomials op introduced by de boor and saff and present a new characterization of the classical and semiclassical op

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

finite sequences of orthogonal polynomials connected by a jacobi matrix abstract if p0 pn is an orthogonal sequence with pj a monic polynomial of exact degree j all j then the sequence q0 qn of monic polynomials defined by the requirement that pj cjqn j 1pn 1 at the zeros of pn with cj suitable constants is also orthogonal with respect to a possibly differently weighted scalar product these two sequences can be shown to be the upper left respectively lower right principal minors of the matrix t j with j a suitable tridiagonal matrix the sequence q occurs in the characterization of the discrete least squares approximation to f from span p0 pj in terms of the n j 1 order divided differences of f p n 1 at the nodes i e the zeros of pn a complete characterization of the pair u v of zero sets of pj 1 and qn j is given and an application is made to the problem of recovering a jacobi matrix from such data

regulator problem for linear discrete time systems with non symmetrical constrained control the regulator problem is studied for linear discrete time systems with non symmetrical constrained control i e systems described by the state equation x k 1 ax k bu k where u k and u k fx k necessary and sufficient conditions allowing us to obtain the largest non symmetrical polyhedral domain of positive invariance and contractivity with respect to motions of the system in the closed loop are established the case of symmetrically constrained control is obtained as a particular case

positively invariant polyhedral sets of discrete time linear systems the problem of the existence of positively invariant polyhedral sets for linear discrete time dynamical systems is studied in the first part of the paper necessary and sufficient conditions for a

constrained stabilization of continuous time linear systems in this paper we consider the stabilization problem for linear continuous time systems under state and control constraints we show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain continuous feedback stabilizing control and we show how to use existing numeric procedures for discrete time systems to solve the continuous time problem we propose a new discontinuous stabilizing control law for scalar input systems which has the advantage of being successfully applicable to systems with quantized control

a utd om technique to design slot arrays on a perfectly conducting paraboloid a procedure that can be employed to design slot arrays on smooth convex conducting surfaces is presented the uniform theory of diffraction utd for the analysis and the orthogonal method om for the synthesis have been used detailed description of the method is given and it is accompanied by a design study of a slot array on a perfectly conducting paraboloid utd has been applied after a proper re construction and re evaluation of the diffraction integral thus a method for treatment of caustics formed on convex surfaces comes out as a by product of our study coupling phenomena between two slots on convex surfaces are also addressed

a class of invariant regulators for the discrete time linear constrained regulation problem abstract stable dynamic systems admit positively invariant domains associated to their lyapunov functions conversely some domains can be made positively invariant for systems with state feedback controllers designed in such a way that some associated non negative definite functions are bound to decrease in particular this approach can be used to establish conditions on the gain matrix for linear constrained regulation problems lcrp we construct fixed and variable regulators easy to compute through linear programming for a class of constrained linear systems

calculating the mutual coupling between apertures on a convex circular cylinder using a hybrid utd mom method the problem of calculating the mutual coupling between apertures on a perfectly conducting circular cylinder is considered the mutual coupling is calculated by a combination of the uniform theory of diffraction utd and the method of moments mom this hybrid method makes it possible to include higher order aperture modes in order to improve the accuracy it is found that the utd mom solution gives accurate results even for small cylinders the results are verified against results found in the literature as well as against measurements performed at ericsson microwave system in molndal sweden for this purpose a cylindrical test antenna with 54 apertures was built using a single mode approximation good agreement down to about 60 db with comparisons to measured data has been obtained if additional modes are added the results improve significantly both in amplitude and phase in several cases the agreement is good down to coupling levels of about 80 db this requires that about six modes are included however the most important mode besides the fundamental te sub 10 mode is the tm sub 11 mode

analysis of nearly cylindrical antennas and scattering problems using a spectrum of two dimensional solutions this paper presents a powerful method for analysing antennas which can be considered principally two dimensional 2 d or cylindrical except for some three dimensional 3 d physical or equivalent sources e g dipoles or slots it is shown by fourier transform techniques that such antennas can be analyzed as 2 d problems with harmonic longitudinal field variation the radiation pattern can often be determined directly from a finite set of such 2 d solutions each one obtained by any method e g the moment method the mutual interaction between the cylindrical scatterer and the sources must be calculated to determine the exact current distribution on the sources and their impedances or admittances this is facilitated by performing an inverse fourier transform of an infinite spectrum of the numerical 2 d solutions followed by a moment method solution in the spatial domain to satisfy the boundary conditions on the 3 d equivalent sources themselves the inverse fourier transform is simplified by the use of asymptote extraction the method is in itself a hybrid technique as one method is used to solve the harmonic 2 d problem and the other to solve for the source currents

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

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

investigation of the mutual coupling between apertures on doubly curved convex surfaces theory and measurements a hybrid method of high frequency approximation utd and method of moments is used to calculate the mutual coupling between circular apertures on a doubly curved surface this requires that the geodesics on doubly curved surfaces be found to represent the aperture fields a single waveguide mode approximation is used in order to verify the theoretical results a general paraboloid of revolution with circular waveguide fed apertures was built at ericsson microwave systems ab molndal sweden this has given a possibility to verify the theoretical model for a general doubly curved surface against measurements to the authors knowledge no such verification is available in the literature it was found that measured and theoretical results in terms of mutual coupling agree very well for copolar coupling as expected the mutual coupling results are very dependent on the polarization in fact the definition of polarization directions and the need for polarization control are important issues for doubly curved conformal array antennas

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

eigenstructure assignment for state constrained linear continuous time systems abstract linear state constraints define a polyhedron in the state space of a dynamical system if this polyhedron can be made positively invariant by state feedback the constraints are satisfied all along the trajectory for any initial point in this polyhedron in the context of continuous time linear systems some algebraic positive invariance conditions are established for unbounded symmetrical polyhedra the existence of a solution to the corresponding set of equations and inequalities depends on the structural properties of an associated input output system in particular the number of zeros of this system should be appropriate and they should be stable if these conditions are satisfied the problem can easily be solved by eigenvalue and eigenvector assignment

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

constrained stabilization via smooth lyapunov functions abstract we consider the problem of stabilizing a dynamic system by means of bounded controls we show that the largest domain of attraction can be arbitrarily closely approximated by a smooth domain of attraction for which we provide an analytic expression such an expression allows for the determination of a non linear control law in explicit form

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

the regulator problem for class of linear systems with constrained control abstract this paper deals with the regulator problem for a class of linear discrete time systems with nonsymmetrical constrained controls for such a class a nonsymmetrical set of positive invariance and global attractivity exists this allows a globally asymptotically stable regulator to be obtained despite the existence of the constrained controls

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

polyhedral cones associated to m matrices and stability of time varying discrete time systems abstract this paper establishes sufficient conditions for asymptotic stability of linear time varying discrete time systems a new approach is proposed related to polyhedral cones associated to m matrices thus stability conditions are transformed in geometrical conditions between the dynamical matrix parameters of the time varying system

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

admittance of rectangular waveguide radiating from a conducting cylinder theory and experiment are compared for the admittance presented to a rectangular waveguide which terminates at an arbitrary angle in the surface of a conducting cylinder two methods are described for calculating the admittance a single mode approximation from the modal solution and an alternative asymptotic solution using these methods the effect of aperture orientation on admittance and the results obtained are compared with the experiment the single mode approximation is in good agreement with experimental results for the case when the input waveguide supports the te sub 10 mode only the accuracy of the approximate formula for waveguide admittance was also verified the admittance is shown to be weakly dependent on aperture orientation for moderately large cylinders

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

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