Level-spacing distributions and the Airy kernel

Level-spacing distributions and the Airy kernel. Tracy, C. A & Widom, H. Communications in Mathematical Physics, 159(1):151–174, January, 1994. arXiv: hep-th/9211141 ISBN: 0370-2693

Paper doi abstract bibtex

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 sin π (x - y)/π(x - y). Similarly a double scaling limit at the "edge of the spectrum" leads to the Airy kernel [Ai(x)Ai′(y) - Ai′(x)Ai(y)]/(x-y). We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDE's found by Jimbo, Miwa, Môri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general n, of the probability that an interval contains precisely n eigenvalues. © 1993.

@article{Tracy1994,
	title = {Level-spacing distributions and the {Airy} kernel},
	volume = {159},
	issn = {0010-3616},
	url = {http://link.springer.com/10.1007/BF02100489},
	doi = {10.1007/BF02100489},
	abstract = {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 sin π (x - y)/π(x - y). Similarly a double scaling limit at the "edge of the spectrum" leads to the Airy kernel [Ai(x)Ai′(y) - Ai′(x)Ai(y)]/(x-y). We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDE's found by Jimbo, Miwa, Môri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general n, of the probability that an interval contains precisely n eigenvalues. © 1993.},
	number = {1},
	journal = {Communications in Mathematical Physics},
	author = {Tracy, Craig A and Widom, Harold},
	month = jan,
	year = {1994},
	note = {arXiv: hep-th/9211141
ISBN: 0370-2693},
	keywords = {\#nosource},
	pages = {151--174},
}

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