Asymptotics of the Airy-Kernel Determinant

“…The constant χ (k) has a representation, as in the Airy case k = 0, in terms of a solution of a Painlevé equation and also in terms of a limit of multiple integrals (see Section 6). For k = 0, the relevant multiple integral can be reduced to an explicitly computable Selberg integral, which allowed the authors in [17] to obtain the simple expression (1.26).…”

Section: Remark 16mentioning

confidence: 99%

“…This would appear to be a final formula for χ (1) . Indeed, note that in [17], for χ (0) , the corresponding D n (+∞) was a Selberg integral and therefore the corresponding formulas simplified to (1.26). In the present case, however, there is no known expression for D n (+∞) in terms of elementary functions (or a fixed number of integrals thereof).…”

Section: The Constant Problemmentioning

confidence: 99%

“…To prove the convergence of the determinants (1.15), one would need to obtain global estimates for the polynomials p k (x) (cf. [17,16]), which should be possible in view of [12]. Note that (1.15) for our V (x) = V (x) is equivalent to the following:…”

mentioning

confidence: 99%

“…Let us now try to generalize the argument in [17] which led to the simple expression (1.26) for χ (0) . Consider the kernel (1.2) with V (x) given by…”

mentioning

confidence: 99%

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Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy

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2009

Comm Pure Appl Math

132

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We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the TracyWidom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition we compute large gap asymptotics for the Fredholm determinants.

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Rank 1 real Wishart spiked model

2012

Comm Pure Appl Math

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In this paper, we consider N -dimensional real Wishart matrices Y in the class W R (Σ, M ) in which all but one eigenvalues of Σ is 1. Let the non-trivial eigenvalue of Σ be 1 + τ , then as N , M → ∞, with M/N = γ 2 finite and non-zero, the eigenvalue distribution of Y will converge into the Marchenko-Pastur distribution inside a bulk region. When τ increases from zero, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko-Pastur density. As the this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix. In this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occur. We will first establish the results that are valid for all N and M and will use them to carry out the asymptotic analysis. In particular, we have derived a contour integral formula for the Harish-Chandra Itzykson-Zuber integral O(N ) e tr(XgY g T ) g T dg when X, Y are real symmetric and Y is a rank 1 matrix. This allows us to write down a Fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques. As a result, we obtain an integral formula for the largest eigenvalue distribution in the large N limit characterized by Painlevé transcendents. The approach used in this paper is very different from a recent paper [23], in which the largest eigenvalue distribution was obtained using stochastic operator method. In particular, the Painlevé formula for the largest eigenvalue distribution obtained in this paper is new.

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Asymptotics of the Airy-Kernel Determinant

Abstract: 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.

Cited by 108 publications

References 35 publications

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy

Rank 1 real Wishart spiked model

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