Descent Method For Riemann-Hilbert Problems Research Articles

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|2αe−(x4+tx2), x ∈ R, where α is a constant larger than −1/2 and t is any real number. They consider this problem in three separate cases: (i) c > −2, (ii) c = −2, and (iii) c < −2, where c:= tN−1/2 is a constant, N = n + α and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μt is a single interval, whereas in the third case the support of μt consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993).

The nonlinear steepest descent method for Riemann-Hilbert problems of low regularity

We prove a nonlinear steepest descent theorem for Riemann-Hilbert problems with Carleson jump contours and jump matrices of low regularity and slow decay. We illustrate the theorem by deriving the ...

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

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.

Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis

We consider the uniform asymptotics of polynomials orthogonal on (0, ∞) with respect to the exponential weight w(x) = x α e −Q(x) ,w here α> − 1a ndQ(x) is a polynomial with positive leading coefficient. In this paper, we have obtained two types of asymptotic expansions in terms of Laguerre polynomials and elementary functions for z in different overlapping regions, respectively. These two regions together cover the whole complex plane. Our approach is based on the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295-368).

On the distribution of the length of the longest increasing subsequence of random permutations

The authors consider the length, $l_N$, of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and scaled, converges to the Tracy-Widom distribution of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of $l_N$.

New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems

New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems