The asymptotics of a Bessel-kernel determinant which arises in Random Matrix Theory

Abstract

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 α ( x , y ) = x y J α ( x ) y J α ′ ( y ) − J α ( y ) x J α ′ ( x ) x 2 − y 2 , x , y > 0 , α > − 1 . In particular, the so-called hard edge gap probabilities P ( α ) ( R ) can be expressed as the Fredholm determinants of the corresponding integral operator B α restricted to the finite interval [ 0 , R ] . Using operator theoretic methods we are going to compute their asymptotics as R → ∞ , i.e., we show that P ( α ) ( R ) : = det ( I − B α ) | L 2 [ 0 , R ] ∼ exp ( − R 2 4 + α R − α 2 2 log R ) G ( 1 + α ) ( 2 π ) α / 2 , where G stands for the Barnes G-function. In fact, this asymptotic formula will be proved for all complex parameters α satisfying | Re α | < 1 .