We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $\theta $-functions.

We consider the probability that no points lie on [Formula: see text] large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order [Formula: see text], and we prove this conjecture for [Formula: see text].

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$ H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and $$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$ H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operators arise when one studies the interior problem of tomography. The diagonalization of $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L has been previously obtained, but only asymptotically when $$b_L\ne -b_R$$ b L ≠ - b R . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

We consider the probability that no points lie on g large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order 1, and we prove this conjecture for g = 1.

The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability P there are no points in the Bessel process on (0, x1)where 0 < x1 < • • • < x2g+1 and g ≥ 0 is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order 1. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a g-dimensional torus, whose integrand involves ratios of Riemann θ-functions associated to a genus g Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain "good Diophantine properties", we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has "good Diophantine properties" (which is always the case for g = 1, and "almost always" the case for g ≥ 2), these results can be combined, yielding particularly precise and explicit large gap asymptotics.