Strong asymptotics of orthogonal polynomials with respect to exponential weights

The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp(− exp(−x)), the Gumbel distribution, and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density exp(−x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, M = M 0 + √ 2SV, with M 0 diagonal with independent elements and V from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.

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“…Proof (Theorem 1.9) We will use the following asymptotic formula for the Hermite polynomials, [3], valid for…”

Section: The Mns-modelmentioning

confidence: 99%

From Gumbel to Tracy-Widom

Johansson

2006

Probab. Theory Relat. Fields

100

119

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“…This can be easily achieved by setting T(z) = Y(z)z −nσ 3 , but this simple-minded transformation brings the problem at infinity to the origin. What Deift, Zhou and their collaborators have shown (in [8,10,11], etc.) is that normalization can be done by using the logarithmic potential (also called the g-function) defined by…”

Section: Step 1 Y −→ Tmentioning

confidence: 99%

“…A significant development in asymptotic analysis of orthogonal polynomials took place in the 1990s, when Deift, Zhou and their associates introduced the nonlinear steepest descent method; see, e.g., [8][9][10][11]. (The term "nonlinear steepest descent method" appeared in a paper by Deift, Venakides and Zhou [12]; see also Bleher and Its [3], and Its and Kapaev [21].…”

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confidence: 99%

“…Although orthogonal polynomials always satisfy a three-term recurrence relation (equation), the coefficient functions involved in the equation may not have readily available asymptotic expansions, in which case there may be no result in the existing theory of asymptotic analysis that can be applied directly. Since the appearance of the work by Deift et al [10,11], there have been tremendous amount of activities in this area of research; see, e.g., [2-4, 20, 22, 24-31, 38]. All these papers are concerned with the strong asymptotics of orthogonal polynomials; they provide asymptotic approximations (expansions) in domains, which overlap and together cover the entire complex plane.…”

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confidence: 99%

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Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

Qiu

Wong

2008

Numer Algor

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By using the steepest descent method for Riemann-Hilbert problems introduced by Deift-Zhou (Ann Math 137:295-370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial L (α) n (νz) as n → ∞, where ν = 4n + 2α + 2. One expansion holds uniformly in a right halfplane Re z ≥ δ 1 , 0 < δ 1 < 1, which contains the critical point z = 1; the other expansion holds uniformly in a left half-plane Re z ≤ 1 − δ 2 , 0 < δ 2 < 1 − δ 1 , which contains the other critical point z = 0. The two half-planes together cover the entire complex z-plane. The critical points z = 1 and z = 0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by L (α) n (νz).

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“…The purpose of this paper is to study the asymptotic behavior of the polynomials L (α) n (z), when α is given by (1.6). As in [11], our method is also based on the Riemann-Hilbert approach introduced by Deift and Zhou in [4] and further developed in [2,5,6,17]; see also [3]. The starting point of this method is a theorem of Fokas, Its and Kitaev [9], which connects orthogonal polynomials and a Riemann-Hilbert problem.…”

Section: Introductionmentioning

confidence: 97%

Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

Dai

Wong

2008

Ramanujan J

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In this paper, we study the asymptotic behavior of the Laguerre polynomials L (α n ) n (nz) as n → ∞. Here α n is a sequence of negative numbers and −α n /n tends to a limit A > 1 as n → ∞. An asymptotic expansion is obtained, which is uniformly valid in the upper half plane C + = {z : Im z ≥ 0}. A corresponding expansion is also given for the lower half plane C − = {z : Im z ≤ 0}. The two expansions hold, in particular, in regions containing the curve in the complex plane, on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.

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Strong asymptotics of orthogonal polynomials with respect to exponential weights

Cited by 544 publications

References 40 publications

From Gumbel to Tracy-Widom

From Gumbel to Tracy-Widom

Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

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