the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

infinite convolution products and refinable distributions on lie groups sufficient conditions for the convergence in distribution of an infinite convolution product p1 p2 of measures on a connected lie group 5 with respect to left invariant haar measure are derived these conditions are used to construct distributions that satisfy to where t is a refinement operator constructed from a measure t and a dilation automorphism a the existence of a implies 5 is nilpotent and simply connected and the exponential map is an analytic homeomorphism furthermore there exists a unique minimal compact subset ic c 5 such that for any open set u containing ic and for any distribution f on g with compact support there exists an integer n u f such that n n u f implies supp tnf c u if p is supported on an a invariant uniform subgroup r then t is related by an intertwining operator to a transition operator w on c r necessary and sufficient conditions for tnf to converge to e l2 and for the f translates of to be orthogonal or to form a riesz basis are characterized in terms of the spectrum of the restriction of w to functions supported on q k c cn r

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

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

convergence of nonstationary cascade algorithms a nonstationary multiresolution of l 2 mathbb r s is generated by a sequence of scaling functions phi k in l 2 mathbb r s k in mathbb z we consider phi k that is the solution of the nonstationary refinement equations phi k m sum j h k 1 j phi k 1 m cdot j k in mathbb z where h k is finitely supported for each k and m is a dilation matrix we study various forms of convergence in l 2 mathbb r s of the corresponding nonstationary cascade algorithm phi k n m sum j h k 1 j phi k 1 n 1 m cdot j as k or n tends to infty it is assumed that there is a stationary refinement equation at infty with filter sequence h and that sum k h k j h j infty the results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

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

convergence of multidimensional cascade algorithm a necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in l 2 bbb r d of the multidimensional cascade algorithm for any starting function phi 0 whose shifts form a partition of unity

convergence rates of cascade algorithms associated with nonhomogeneous refinement equations this paper is concerned with nonhomogeneous refinement equations of the form x zsa mx g x x rs where the vector of functions 1 r t is unknown g is a given vector of compactly supported functions on rs a is a finitely supported sequence of r r matrices called the refinement mask and m is an s s integer matrix such that limn m n 0 our approach will be to consider the convergence rates of the cascade algorithms associated with nonhomogeneous refinement equations mentioned above the cascade algorithms associated with mask a nonhomogeneous term g and dilation matrix m generates a sequence n n 1 2 by the iterative process n x zsa n 1 mx g x x rs from a starting vector of function 0 in lp rs r 0 p the aim of this paper is to give a characterization of the convergence rates of the cascade algorithms associated with a g 0 and dilation matrix m in lp rs r 0 p in terms of the p norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set e restricted to a certain invariant subspace where the set e is a complete set of representatives of the distinct cosets of the quotient group zs mzs containing 0 some examples are provided to illustrate the method

on asymptotically equivalent shallow water wave equations abstract the integrable third order korteweg de vries kdv equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves however at quadratic order this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other under a group of nonlinear non local normal form transformations introduced by kodama in combination with the application of the helmholtz operator these kodama helmholtz kh transformations are used to present connections between shallow water waves the integrable fifth order kdv equation and a generalization of the camassa holm ch equation that contains an additional integrable case the dispersion relation of the full water wave problem and any equation in this family agree to fifth order the travelling wave solutions of the ch equation are shown to agree to fifth order with the exact solution

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

hankel determinant and orthogonal polynomials for a gaussian weight with a discontinuity at the edge we compute asymptotics for hankel determinants and orthogonal polynomials with respect to a discontinuous gaussian weight in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support their behavior is described in terms of the ablowitz segur family of solutions to the painleve ii equation our results complement the ones in 33 as consequences of our results we conjecture asymptotics for an airy kernel fredholm determinant and total integral identities for painleve ii transcendents and we also prove a new result on the poles of the ablowitz segur solutions to the painleve ii equation we also highlight applications of our results in random matrix theory

the local structure of turbulence in incompressible viscous fluid for very large reynolds numbers 1 we shall denote by u p u x 1 x 2 x 3 t 1 2 3 the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 x 2 x 3 in considering the turbulence it is natural to assume the components of the velocity u p at every point p x 1 x 2 x 3 t of the considered domain g of the four dimensional space x 1 x 2 x 3 t are random variables in the sense of the theory of probabilities cf for this approach to the problem millionshtchikov 1939 denoting by the mathematical expectation of the random variable a we suppose that 2 and d u d x 2 are finite and bounded in every bounded subdomain of the domain g

subdivisions with infinitely supported mask in this paper we investigate the convergence of subdivision schemes associated with masks being polynomially decay sequences two scale vector refinement equations are the email protected x email protected email protected za a f email protected email protected r where the vector of functions f f 1 f r t is in l 2 r r and email protected a a a z is polynomially decay sequence of rxr matrices called refinement mask associated with the mask a is a linear operator on l 2 r r given byq af x email protected email protected za a f email protected email protected r f f 1 f r email protected l 2 r r by using same methods in b han r q jia characterization of riesz bases of wavelets generated from multiresolution analysis manuscript b han refinable functions and cascade algorithms in weighted spaces with infinitely supported masks manuscript r q jia q t jiang z w shen convergence of cascade algorithms associated with nonhomogeneous refinement equations proc amer math soc 129 2001 415 427 r q jia convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets in advances in wavelet hong kong 1997 springer singapore 1998 pp 199 227 a characterization of convergence of the sequences q a nf n 1 2 in the l 2 norm is given which extends the main results in r q jia s d riemenschneider d x zhou vector subdivision schemes and multiple wavelets math comp 67 1998 1533 1563 on convergence of the subdivision schemes associated with a finitely supported mask to the case in which mask a is polynomially decay sequence as an application we also obtain a characterization of smoothness of solutions of the refinement equation mentioned above for the case r 1

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

wave group dynamics in weakly nonlinear long wave models abstract the dynamics of wave groups is studied for long waves using the framework of the extended korteweg de vries equation it is shown that the dynamics is much richer than the corresponding results obtained just from the korteweg de vries equation first a reduction to a nonlinear schrodinger equation is obtained for weakly nonlinear wave packets and it is demonstrated that either the focussing or the defocussing case can be obtained this is in contrast to the corresponding reduction for the korteweg de vries equation where only the defocussing case is obtained next the condition for modulational instability is obtained it is shown that wave packets are unstable only for a positive sign of the coefficient of the cubic nonlinear term in the extended korteweg de vries equation and for a high carrier frequency at the boundary of this parameter space a modified nonlinear schrodinger equation is derived and its steady state solutions including an algebraic soliton are found the exact breather solution of the extended korteweg de vries equation is analysed it is shown that in the limit of weak nonlinearity it transforms to a wave group with an envelope described by soliton solutions of the nonlinear schrodinger equation and its modification as described above numerical simulations demonstrate the main features of wave group evolution and show some differences in the behaviour of the solutions of the extended korteweg de vries equation compared with those of the nonlinear schrodinger equation

determinants of airy operators and applications to random matrices the purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of wiener hopf operators the determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum

large gap asymptotics for airy kernel determinants with discontinuities we obtain large gap asymptotics for airy kernel fredholm determinants with any number m of discontinuities these m point determinants are generating functions for the airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles our main result is that the m point determinants can be expressed asymptotically as the product of m 1 point determinants multiplied by an explicit constant pre factor which can be interpreted in terms of the covariance of the counting function of the process

vector subdivision schemes and multiple wavelets we consider solutions of a system of refinement equations written in the form formula math where the vector of functions 1 r t is in l p r r and a is a finitely supported sequence of r r matrices called the refinement mask associated with the mask a is a linear operator q a defined on l p r r by q a f z a f 2 this paper is concerned with the convergence of the subdivision scheme associated with a i e the convergence of the sequence q a n f n 1 2 in the l p norm our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask along the way properties of the joint spectral radius and its relation to the subdivision scheme are discussed in particular the l 2 convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace we analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations finally the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions this leads us to construct a class of continuous orthogonal double wavelets with symmetry

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

on the spectra of periodic waves for infinite dimensional hamiltonian systems abstract we consider the problem of determining the spectrum for the linearization of an infinite dimensional hamiltonian system about a spatially periodic traveling wave by using a bloch wave decomposition we recast the problem as determining the point spectra for a family of operators j l where j is skew symmetric with bounded inverse and l is symmetric with compact inverse our main result relates the number of unstable eigenvalues of the operator j l to the number of negative eigenvalues of the symmetric operator l the compactness of the resolvent operators allows us to greatly simplify the proofs as compared to those where similar results are obtained for linearizations about localized waves the theoretical results are general and apply to a larger class of problems than those considered herein the theory is applied to a study of the spectra associated with periodic and quasi periodic solutions to the nonlinear schrodinger equation as well as periodic solutions to the generalized korteweg de vries equation with power nonlinearity

convergence of cascade algorithms by frequency approach abstract starting with an initial function 0 the cascade algorithm generates a sequence q a n 0 n 1 by cascade operator q a defined by q a f z d a f m a function is refinable if it satisfies q a the refinable functions play an important role in wavelet analysis and computer graphics the cascade algorithm is the main approach to approximate the refinable functions and to study their properties this note establishes a sufficient condition in terms of fourier transforms of the initial function 0 and the refinable function for the convergence of cascade algorithm our results apply to the case where neither the initial function is compactly supported nor the refinement mask is finitely supported as a byproduct we prove that any compactly supported refinable function has a positive sobolev regularity exponent provided it is in l 2

convergence of cascade algorithms in sobolev spaces and integrals of wavelets the cascade algorithm with mask a and dilation m generates a sequence phi n n 1 2 ldots by the iterative process

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

convergence of cascade algorithms in sobolev spaces associated with multivariate refinement equations abstract this paper is concerned with multivariate inhomogeneous refinement equations written in the form x z s a mx g x x r s where is the unknown function defined on the s dimentional euclidean space r s g is a given compactly supported function on r s a is a finitely supported sequence on z s and m is an s s dilation matrix with m det m let 0 be an initial function in the sobolev space wk2 r s for n 1 2 define n x z s a n 1 mx g x x r s in this paper we give a characterization for the strong convergence in the sobolev space wk2 r s k n of the cascade sequence n n n for the case in which m is isotropic

computing nearly singular solutions using pseudo spectral methods in this paper we investigate the performance of pseudo spectral methods in computing nearly singular solutions of fluid dynamics equations we consider two different ways of removing the aliasing errors in a pseudo spectral method the first one is the traditional 2 3 dealiasing rule the second one is a high 36th order fourier smoothing which keeps a significant portion of the fourier modes beyond the 2 3 cut off point in the fourier spectrum for the 2 3 dealiasing method both the 1d burgers equation and the 3d incompressible euler equations are considered we demonstrate that the pseudo spectral method with the high order fourier smoothing gives a much better performance than the pseudo spectral method with the 2 3 dealiasing rule moreover we show that the high order fourier smoothing method captures about 12 15 more effective fourier modes in each dimension than the 2 3 dealiasing method for the 3d euler equations the gain in the effective fourier codes for the high order fourier smoothing method can be as large as 20 over the 2 3 dealiasing method another interesting observation is that the error produced by the high order fourier smoothing method is highly localized near the region where the solution is most singular while the 2 3 dealiasing method tends to produce oscillations in the entire domain the high order fourier smoothing method is also found be very stable dynamically no high frequency instability has been observed in the case of the 3d euler equations the energy is conserved up to at least six digits of accuracy throughout the computations 2007 elsevier inc all rights reserved

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

subcritical multiplicative chaos for regularized counting statistics from random matrix theory for an n times n haar distributed random unitary matrix u n we consider the random field defined by counting the number of eigenvalues of u n in a mesoscopic arc centered at the point u on the unit circle we prove that after regularizing at a small scale epsilon n 0 the renormalized exponential of this field converges as n to infty to a gaussian multiplicative chaos measure in the whole subcritical phase we discuss implications of this result for obtaining a lower bound on the maximum of the field we also show that the moments of the total mass converge to a selberg like integral and by taking a further limit as the size of the arc diverges we establish part of the conjectures in ostrovsky nonlinearity 29 2 426 464 2016 by an analogous construction we prove that the multiplicative chaos measure coming from the sine process has the same distribution which strongly suggests that this limiting object should be universal our approach to the l1 phase is based on a generalization of the construction in berestycki electron commun probab 22 27 12 2017 to random fields which are only asymptotically gaussian in particular our method could have applications to other random fields coming from either random matrix theory or a different context

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

direct dynamical energy cascade in the modified kdv equation in this study we examine the energy transfer mechanism during the nonlinear stage of the modulational instability mi in the modified korteweg de vries equation the particularity of this study consists in considering the problem essentially in the fourier space a dynamical energy cascade model of this process originally proposed for the focusing nls type equations is transposed to the mkdv setting using the existing connections between the kdv type and nls type equations the main predictions of the d cascade model are outlined and thoroughly discussed finally the obtained theoretical results are validated by direct numerical simulations of the mkdv equation using the pseudo spectral methods a general good agreement is reported in this study the nonlinear stages of the mi evolution are also investigated for the mkdv equation

nonuniform cascade algorithms abstract for a sequence of bounded linear operators hk k 0 1 on a banach space s the algorithm k n hk k 1 n 1 generates a family of sequences k n n 0 k 0 1 from an initial family of vectors k 0 s k 0 1 we study the convergence of k n as n and give an application on the convergence of cascade algorithms for nonuniform splines when s is the space of all sequences i i z with norm sup i z i and i i z belong to the banach space x l 2 r

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

a fredholm determinant identity and the convergence of moments for random young tableaux we obtain an identity between fredholm determinants of two kinds of operators one acting on functions on the unit circle and the other acting on functions on a subset of the integers this identity is a generalization of an identity between a toeplitz determinant and a fredholm determinant that has appeared in the random permutation context using this identity we prove in particular convergence of moments for arbitrary rows of a random young diagram under plancherel measure

randomly incomplete spectra and intermediate statistics by randomly removing a fraction of levels from a given spectrum a model is constructed that describes a crossover from this spectrum to a poisson spectrum the formalism is applied to the transitions towards poisson from random matrix theory rmt spectra and picket fence spectra it is shown that the fredholm determinant formalism of rmt extends naturally to describe incomplete rmt spectra

an index theorem for the stability of periodic travelling waves of korteweg de vries type we consider the stability of periodic travelling wave solutions to a generalized korteweg de vries gkdv equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling wave ordinary differential equation we illustrate this result with several examples including the integrable kdv and modified kdv equations the l 2 critical kdv 4 equation that arises in the study of blow up and the kdv equation which is an idealized model for plasmas

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