Local Parametrix Research Articles

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by Magnus and extends earlier results by Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at $x = +1$.

On the Riemann–Hilbert approach to asymptotics of tronquée solutions of Painlevé I

In this paper, we revisit large variable asymptotic expansions of tronquée solutions of the Painlevé I equation, obtained via the Riemann–Hilbert approach and the method of steepest descent. The explicit construction of an extra local parametrix around the recessive stationary point of the phase function, in terms of complementary error functions, makes it possible to give detailed information about exponential-type contributions beyond the standard Poincaré expansions for tronquée and tritronquée solutions.

Riemann–Hilbert theory without local parametrix problems: Applications to orthogonal polynomials

We study whether in the setting of the Deift–Zhou nonlinear steepest descent method one can avoid solving local parametrix problems, while still obtaining asymptotic results. We show that this can be done, provided an a priori estimate for the exact solution of the Riemann–Hilbert problem is known. This enables us to derive asymptotic results for orthogonal polynomials on [−1,1] with a new class of weight functions. In these cases, the weight functions are too badly behaved to allow a reformulation of the local parametrix problem to a global one with constant jump matrices. Possible implications for edge universality in random matrix theory are also discussed.

The local universality of Muttalib–Borodin ensembles when the parameter θ is the reciprocal of an integer

The Muttalib–Borodin ensemble is a probability density function for n particles on the positive real axis that depends on a parameter θ and a weight w. We consider a varying exponential weight that depends on an external field V. In a recent article, the large n behavior of the associated correlation kernel at the hard edge was found for , where only few restrictions are imposed on V. In the current article we generalize the techniques and results of this article to obtain analogous results for , where r is a positive integer. The approach is to relate the ensemble to a type II multiple orthogonal polynomial ensemble with r weights, which can then be related to an (r + 1) × (r + 1) Riemann–Hilbert problem. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter

The Muttalib–Borodin biorthogonal ensemble is a probability density function for n particles on the positive real line that depends on a parameter and an external field . For we find the large n behavior of the associated correlation kernel with only few restrictions on . The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a Riemann–Hilbert problem which we then solve with the Deift–Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.

Solvability of interior transmission problem for the diffusion equation by constructing its Green function

Abstract Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients. We prove the unique solvability of the interior transmission problem by constructing its Green function. First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter. Second, by carefully analyzing the analyticity of the local parametrix in the Laplace domain and estimating it there, a local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to generate a global parametrix for the parabolic interior transmission problem and then compensate it to get the Green function by the Levi method. The uniqueness of the Green function is justified by using the duality argument, and then the unique solvability of the interior transmission problem is concluded. We would like to emphasize that the Green function for the parabolic interior transmission problem is constructed for the first time in this paper. It can be applied for active thermography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property.

Asymptotic Behavior and Zero Distribution of Polynomials Orthogonal with Respect to Bessel Functions

We consider polynomials $$P_n$$ orthogonal with respect to the weight $$J_{\nu }$$ on $$[0,\infty )$$ , where $$J_{\nu }$$ is the Bessel function of order $$\nu $$ . Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros of $$P_n$$ are complex and accumulate as $$n \rightarrow \infty $$ near the vertical line $${{\mathrm{{\text {Re}}\,}}}z = \frac{\nu \pi }{2}$$ . We prove this fact for the case $$0 \le \nu \le 1/2$$ from strong asymptotic formulas that we derive for the polynomials $$P_n$$ in the complex plane. Our main tool is the Riemann–Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift–Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for $$\nu \le 1/2$$ .

Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

We consider the double scaling limit for a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t=0$ at the same positive value $x=a$, remain positive, and are conditioned to end at time $t=1$ at $x=0$. After appropriate rescaling, the paths fill a region in the $tx$--plane as $n\to \infty$ that intersects the hard edge at $x=0$ at a critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling limits for the positions of the paths at time $t\neq t^{*}$ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a $3\times 3$ matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix.

Painlevé XXXIV Asymptotics of Orthogonal Polynomials for the Gaussian Weight with a Jump at the Edge

We study the uniform asymptotics of the polynomials orthogonal with respect to analytic weights with jump discontinuities on the real axis, and the influence of the discontinuities on the asymptotic behavior of the recurrence coefficients. The Riemann–Hilbert approach, also termed the Deift–Zhou steepest descent method, is used to derive the asymptotic results. We take as an example the perturbed Gaussian weight , where θ(x) takes the value of 1 for x < 0, and a nonnegative complex constant ω elsewhere, and as . That is, the jump occurs at the edge of the support of the equilibrium measure. The derivation is carried out in the sense of a double scaling limit, namely, and . A crucial local parametrix at the edge point where the jump occurs is constructed out of a special solution of the Painlevé XXXIV equation. As a main result, we prove asymptotic formulas of the recurrence coefficients in terms of a special Painlevé XXXIV transcendent under the double scaling limit. The special thirty-fourth Painlevé transcendent is shown free of poles on the real axis. A consistency check is made with the reduced case when ω= 1, namely the Gaussian weight: the polynomials in this case are the classical Hermite polynomials. A comparison is also made of the asymptotic results for the recurrence coefficients between the case when the jump happens at the edge and the case with jump inside the support of the equilibrium measure. The comparison provides a formal asymptotic approximation of the Painlevé XXXIV transcendent at positive infinity.

Asymptotics for a special solution of the thirty fourth Painlevé equation

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form with α > −1/2. The factor | det M|2α induces critical eigenvalue behaviour near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N → ∞ such that n2/3(n/N − 1) = O(1) by using the Deift–Zhou steepest descent method for the Riemann–Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|2αe−NV(x). Our main attention was on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution uα of the Painlevé XXXIV equation. This solution is related to a particular solution of the Painlevé II equation, which, however, is different from the usual Hastings–McLeod solution. In this paper we compute the asymptotic behaviour of uα(s) as s → ±∞. We conjecture that this asymptotics characterizes uα and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painlevé XXXIV equation which includes uα. We identify this family as the family of tronquée solutions of the thirty fourth Painlevé equation.

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight z−n+νe−Nz(z− 1)2b, in the limit where n, N→∞ with N/n→ 1 and ν is a fixed number in . With the effect of the factor (z− 1)2b, the local parametrix near the critical point z= 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.

Multi-critical unitary random matrix ensembles and the general Painlevé II equation

We study unitary random matrix ensembles of the form Z-'N\detM\2«e-NTrVWdM, where a > -1/2 and V is such that the limiting mean eigenvalue density for n, N - > oc and n/N - ► 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight \x\2ae~NV^x Here the main focus is on the construction of a local parametrix near the origin with ^-functions associated with a special solution qa of the Painleve II equation q = sq + 2q3 - a. We show that qa has no real poles for a > -1/2, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit.

Critical Edge Behavior in Unitary Random Matrix Ensembles and the Thirty-Fourth Painlevé Transcendent

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles Z(n,N)(-1)vertical bar det M vertical bar (2 alpha)e(-NTrV(M)) dM, with alpha > -1/2, defined on n x n Hermitian matrices M. Assuming that the limiting mean eigenvalue density is regular and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n, N -> infinity such that n(2/3)(n/N-1) = O(1). We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials orthogonal with respect to the weight vertical bar x vertical bar(2 alpha)e(-NV(x)). Our main attention is on the construction of a local parametrix near the origin by means of the psi-functions associated with a distinguished solution of the Painleve XXXIV equation.

Large n Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit

We consider the double scaling limit in the random matrix ensemble with an external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.

Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight

We study polynomials that are orthogonal with respect to a varying quartic weight exp(− N(x2/2 + tx4/4)) for t < 0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its and Kitaev showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painlevé I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann–Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric combinations of contours. Special features in the steepest descent analysis are a modified equililbrium problem and the use of Ψ-functions for the Painlevé I equation in the construction of the local parametrix.

Universality of the double scaling limit in random matrix models

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann-Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift-Zhou steepest-descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. (c) 2005 Wiley Periodicals, Inc.

On an algebra of a certain class of operators in a slab domain in $R^2$

In this paper a Dirichlet problem for the Laplacian in a domain with a corner in R2 is treated and a new method of construction of a local parametrix for that Dirichlet problem at a corner point is given. In order to construct a parametrix a certain class of operators in a slab domain is introduced and it is shown that that operator class is an algebra.

ON LOCAL SOLVABILITY OF EQUATIONS OF QUASI-PRINCIPAL TYPE

The question of local solvability for a certain class of partial differential equations is considered. For this class a local parametrix is constructed , the points of singularity of the corresponding kernel of the integral operator are indicated , and the boundedness of the operator realizing the parametrix is demonstrated. Local solvability is then established on the basis of these results.Bibliography: 12 titles.