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In the paper, we consider the extended Gross-Witten-Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the $(i,j)$-entry being the modified Bessel functions of order $i-j-\nu$, $\nu\in\mathbb{C}$. When the degree $n$ is finite, we show that the Toeplitz determinant is described by the isomonodromy $\tau$-function of the Painlev\'{e} III equation. As a double scaling limit, %In the double scaling limit as the degree $n\to\infty$, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings-McLeod solution of the inhomogeneous Painlev\'{e} II equation with parameter $\nu+\frac{1}{2}$. The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift-Zhou nonlinear steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point $z=-1$, where the $\psi$-function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlev\'{e} II equation is involved., Comment: 41 pages, 14 figures

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with $n$ discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlev\'e V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the $n$ discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes $G$-function., Comment: 36 pages, 5 figures

We study the one-parameter family of Fredholm determinants $\det(I-\rho^2\mathcal{K}_{n,x})$, $\rho\in\mathbb{R}$, where $\mathcal{K}_{n,x}$ stands for the integral operator acting on $L^2(x,+\infty)$ with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy-Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the $n$-th member of the Painlev\'{e} II hierarchy. Using the Riemann-Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlev\'{e} II transcendents as $x\to -\infty$ for $0<|\rho|<1$ and $|\rho|>1$, respectively. In the case of $0<|\rho|<1$, we are able to calculate the constant term in the asymptotic expansion of the determinants, while for $|\rho|>1$, the relevant asymptotics exhibit singular behaviors. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel., Comment: 39 pages, 7 figures

The Clarkson-McLeod solutions of the fourth Painlev\'e equation behave like $\kappa D_{\alpha-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $\kappa$ is some real constant and $D_{\alpha-\frac{1}{2}}(x)$ is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we derive the asymptotic behaviors for this class of solutions as $x\to-\infty$. This completes a proof of Clarkson and McLeod's conjecture on the asymptotics of this family of solutions. The total integrals of the Clarkson-McLeod solutions and the asymptotic approximations of the $\sigma$-form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the $\sigma$-form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel., Comment: 52 pages, 11 figures

This is a survey article on an old topic in classical analysis. We present some new developments in asymptotics in the last fifty years. We start with the classical method of Darboux and its generalizations, including an uniformity treatment which has a direct application to the Heisenberg polynomials. We then present the development of an asymptotic theory for difference equations, which is a major advancement since the work of Birkhoff and Trjitzinsky in 1933. A new method was introduced into this field in the nineteen nineties, which is now known as the nonlinear steepest descent method or the Riemann-Hilbert approach. The advantage of this method is that it can be applied to orthogonal polynomials which do not satisfy any differential or difference equations neither do they have any integral representations. As an example, we mention the case of orthogonal polynomials with respect to the Freud weight. Finally, we show how the Wiener-Hopf technique can be used to derive asymptotic expansions for the solutions of an integral equation on a half line., Comment: 46 pages

We consider the Clarkson-McLeod solutions of the fourth Painlev\'e equation. This family of solutions behave like $\kappa D_{\alpha-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $\kappa $ is an arbitrary real constant and $D_{\alpha-\frac{1}{2}}(x)$ is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we obtain the singular asymptotics of the solutions as $x\to-\infty$ when $\kappa \left( \kappa -\kappa ^*\right )>0$ for some real constant $\kappa ^*$. The connection formulas are also explicitly evaluated. This proves and extends Clarkson and McLeod's conjecture that when the parameter $\kappa >\kappa ^*>0$, the Clarkson-McLeod solutions have infinitely many simple poles on the negative real axis., Comment: 24 pages, 9 figures, 1 table

In this paper, we consider the monodromy and, in particularly, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlev\'{e} equations when the Painlev\'e transcendents go to one of the actual singular points of the linear systems. While the isomonodromy sets of accessory parameters for the Heun class equations are described by the Taylor or Laurent coefficients of the corresponding Painlev\'{e} functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly-confluent Heun equation and the reduced biconfluent Heun equation., Comment: 42 pages

We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at $z=1$. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlev\'{e} V system in dimension four. This leads to a Painlev\'e-type representation of the confluent hypergeometric-kernel determinant. Moreover, the large gap asymptotics, including the constant terms, are derived by evaluating the total integral of the Hamiltonian. In particular, we reproduce the large gap asymptotics of the confluent hypergeometric-kernel determinant obtained by Deift, Krasovsky and Vasilevska, and the sine-kernel determinant as a special case, including the constant term conjectured earlier by Dyson., Comment: 52 pages, 6 figures, misprints corrected

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained respectively in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques., Comment: 31 pages, 1 figure (in two separated files)

We consider the Gaussian unitary ensemble perturbed by a Fisher-Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlev\'{e} XXXIV equation and the $\sigma$-form of the Painlev\'{e} II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlev\'{e} XXXIV functions and the $\sigma$-form of the Painlev\'{e} II equation are also studied., Comment: 35 pages, 5 figures

We consider three special cases of the initial value problem of the first Painlev\'e equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained., Comment: 24 pages, 1 figure

We consider a pair of special functions, $u_\beta$ and $v_\beta$, defined respectively as the solutions to the integral equations \begin{equation*} u(x)=1+\int^\infty_0 \frac {K(t) u(t) dt}{t+x} ~~\mbox{and}~~v(x)=1-\int^\infty_0 \frac{ K(t) v(t) dt}{t+x},~~x\in [0, \infty), \end{equation*} where $K(t)= \frac {1} \pi \exp \left (- t^\beta \sin\frac {\pi\beta} 2\right )\sin \left ( t^\beta\cos\frac{\pi\beta} 2 \right )$ for $\beta\in (0, 1)$. In this note, we establish the existence and uniqueness of $u_\beta$ and $v_\beta$ which are bounded and continuous in $[0, +\infty)$. Also, we show that a solution to a model Riemann-Hilbert problem in Kriecherbauer and McLaughlin [Int. Math. Res. Not., 1999] can be constructed explicitly in terms of these functions. A preliminary asymptotic study is carried out on the Stokes phenomena of these functions by making use of their connection formulas. Several open questions are also proposed for a thorough investigation of the analytic and asymptotic properties of the functions $u_\beta$ and $v_\beta$, and a related new special function $G_\beta$., Comment: 14 pages, 1 figure

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex plane. When the parameters $n$, $t$ and the degree $k$ are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy $\tau$-function of the Painlev\'e III equation. When the degree $k=n$, $n$ is large and $t$ is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronqu\'ee solution to the Painlev\'e I equation. Our approach is based on the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems., Comment: More details added. 35 pages, 6 figures

We study the asymptotic expansion for the Landau constants $G_n$, \begin{equation*} \pi G_{n}\sim \ln(16N)+\gamma+\sum^{\infty}_{k=1}\frac{\alpha_k}{N^k} ~~\mbox{as} ~ n\rightarrow\infty, \end{equation*} where $N=n+1$, and $\gamma$ is Euler's constant. We show that the signs of the coefficients $\alpha_{k}$ demonstrate a periodic behavior such that $(-1)^{\frac {l(l+1)} 2} \alpha_{l+1}< 0$ for all $l$. We further prove a conjecture of Granath which states that $(-1)^{\frac {l(l+1)} 2} \varepsilon_l(N)<0$ for $l=0,1,2,\cdots$ and $n=0,1,2,\cdots$, $\varepsilon_l(N)$ being the error due to truncation at the $l$-th order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the form \begin{equation*} \ln(16N)+\gamma+\sum_{k=1}^{p}\frac{\alpha_{k}}{N^{k}}<\pi G_{n}<\ln(16N)+\gamma+\sum_{k=1}^{q}\frac{\alpha_{k}}{N^{k}} \end{equation*} for all $n=0,1,2\cdots$, all $p=4s+1,\; 4s+2$ and $q=4m,\; 4m+3$, with $s=0,1,2,\cdots$ and $m=0, 1, 2,\cdots$., Comment: 17 pages, 1 figure. Proof of Theorem 2 simplified as compared with V1

We apply the uniform asymptotics method proposed by Bassom, Clarkson, Law and McLeod to a special Painlev\'{e} V equation, and we provide a simpler and more rigorous proof of the connection formulas for a special solution of the equation, which have been established earlier by McCoy and Tang via the isomonodromy and WKB methods., Comment: 18 pages

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The Plancherel-Rotach type asymptotics of the sieved Pollazek Polynomials are obtained in the whole complex plane. The Riemann-Hilbert method is applied to derive the results. A main feature of the treatment is the appearance of a new band consisting of two adjacent intervals, one of which is a portion of the support of the absolutely continuous measure, the other is the discrete band., Comment: 31 pages, 5 figures

We study the Hankel determinants associated with the weight $$w(x;t)=(1-x^2)^{\beta}(t^2-x^2)^\alpha h(x),~x\in(-1,1),$$ where $\beta>-1$, $\alpha+\beta>-1$, $t>1$, $h(x)$ is analytic in a domain containing $[-1,1]$ and $h(x)>0$ for $x\in[-1,1]$. In this paper, based on the Deift-Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as $n\to \infty$ and $t\to 1$. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo-Miwa-Okamoto $\sigma$-function for the Painlev\'{e} III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained., Comment: 28 pages. Modifications made to the previous version arXiv:1412.8586v1

In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials $f_n^{(\alpha)}(z)$ where $\alpha$ is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and [{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}]., Comment: 24 pages, 7 figures

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^\alpha e^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $\alpha>0$. When the matrix size $n\to\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0, d]$, $d>0$ fixed. A particular Painlev\'{e} III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method., Comment: 21 pages, 3 figures, minor revision, references updated. arXiv admin note: text overlap with arXiv:1309.4354

We study the Jacobi unitary ensemble perturbed by an algebraic singularity at $t>1$. For fixed $t$, this is the modified Jacobi ensemble studied by Kuijlaars {\it{et al.}} The main focus here, however, is the case when the algebraic singularity approaches the hard edge, namely $t\to 1^+$. In the double scaling limit case when $t- 1$ is of the order of magnitude of $1/n^2$, $n$ being the size of the matrix, the eigenvalue correlation kernel is shown to have a new limiting kernel at the hard edge $1$, described by the $\psi$-functions for a certain second-order nonlinear equation. The equation is related to the Painlev\'e III equation by a M\"obius transformation. It also furnishes a generalization of the Painlev\'e V equation, and can be reduced to a particular Painlev\'e V equation via the B\"acklund transformations in special cases. The transitions of the limiting kernel to Bessel kernels are also investigated, with $n^2(t-1)$ being large or small. In the present paper, the approach is based on the Deift-Zhou nonlinear steepest descent analysis for Riemann-Hilbert problems., Comment: 48 pages, 6 figures. Minor modifications made to the previous version arXiv:1404.5105v2

In this paper, we study the singularly perturbed Laguerre unitary ensemble $$ \frac{1}{Z_n} (\det M)^\alpha e^{- \textrm{tr}\, V_t(M)}dM, \qquad \alpha >0, $$ with $V_t(x) = x + t/x$, $x\in (0,+\infty)$ and $t>0$. Due to the effect of $t/x$ for varying $t$, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves $\psi$-functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown equivalent to a particular Painlev\'e III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter $t$ changes in a finite interval $(0, d]$. Our approach is based on Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems., Comment: 41 pages, 6 figures. Modifications are made in Introduction and Sec. 2.2. Typos corrected and references added

We show that the set of Heisenberg polynomials furnishes a simple non-trivial example in the uniform treatment of Darboux's method., Comment: 9 pages