منابع مشابه
On Asymptotics for the Airy Process
The Airy process t → A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s1, s2 and t for the probability Pr (A(0) ≤ s1, A(t) ≤ s2). Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion fo...
متن کاملAiry Kernel and Painlevé II
We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlevé II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the resu...
متن کاملAsymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals
The sine kernel K(x, y) := sin(x− y) π(x− y) arises in many areas of mathematics and mathematical physics. There is an extensive literature on the asymptotics of the eigenvalues of Ks, the operator with this kernel on an interval of length s, as s→ ∞, for example [4, 6, 8, 12], and asymptotic fomulas of various kinds were obtained. Some of these derivations were rigorous, others were more heuri...
متن کاملLevel-Spacing Distributions and the Airy Kernel
Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models of N × N hermitian matrices and then going to the limit N → ∞, leads to the Fredholm determinant of the sine kernel sinπ(x− y)/π(x− y). Similarly a scaling limit at the “edge of the spectrum” leads to the Airy kernel [Ai(x)Ai′(y)−Ai′(x)Ai(y)] /(x − y). In this paper we derive analogues for this Ai...
متن کاملAsymptotics for the Covariance of the Airy 2 Process
In this paper we compute some of the higher order terms in the asymptotic behavior of the two point function P(A2(0)≤ s1,A2(t)≤ s2), extending the previous work of Adler and van Moerbeke (arXiv:math.PR/0302329; Ann. Probab. 33, 1326–1361, 2005) and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We prove that it is possible to represent any order asymptotic approximation as a polynomial and integr...
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ژورنال
عنوان ژورنال: Communications in Mathematical Physics
سال: 2008
ISSN: 0010-3616,1432-0916
DOI: 10.1007/s00220-007-0409-x