期刊
ADVANCES IN MATHEMATICS
卷 225, 期 6, 页码 3088-3133出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.aim.2010.05.020
关键词
Random matrix; Gap probability; Determinant asymptotics; Wiener-Hopf operator; Toeplitz and Hankel operator
类别
资金
- NSF [DMS-0901434]
- Direct For Mathematical & Physical Scien [0901434] Funding Source: National Science Foundation
- Division Of Mathematical Sciences [0901434] Funding Source: National Science Foundation
In Random Matrix Theory the local correlations of the Laguerre and Jacobi Unitary Ensemble in the hard edge scaling limit can be described in terms of the Bessel kernel B-alpha(x, y) = root xy J(alpha)(x)yJ'(alpha)(y) - J(alpha)(y)xJ'(alpha)(x)/x(2)-y(2), x, y > 0, alpha > -1. In particular, the so-called hard edge gap probabilities P-(alpha) (R) can be expressed as the Fredholm determinants of the corresponding integral operator B-alpha restricted to the finite interval [0, R]. Using operator theoretic methods we are going to compute their asymptotics as R -> infinity, i.e., we show that P-(alpha) (R) :=det(I - B-alpha)vertical bar(L2[0, R]) similar to exp(- R-2/4 + alpha R - alpha(2)/2 log R) G(1+alpha)/(2 pi)(alpha/2), where G stands for the Barnes G-function. In fact, this asymptotic formula will be proved for all complex parameters alpha satisfying vertical bar Re alpha vertical bar < 1. (C) 2010 Elsevier Inc. All rights reserved.
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