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

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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 (containing a parameter α). In particular, the so-called hard edge gap probabilities can be expressed as the Fredholm determinants of the corresponding integral operator restricted to the finite interval [0, R]. Using operator theoretic methods we are going to compute their asymptotics as R goes to infinity under certain assumption on the parameter α.