Finite size corrections in the bulk for circular β $\beta $ beta ensembles

Abstract The circular β $\beta $ beta ensemble for β = 1 , 2 $\beta =1,2$ beta equals 1 comma 2 and 4 corresponds to circular orthogonal, unitary and symplectic ensemble respectively as introduced by Dyson. The statistical state of the eigenvalues is then a determinantal point process ( β = 2 $\beta = 2$ beta equals 2 ) and Pfaffian point process ( β = 1 , 4 $\beta = 1,4$ beta equals 1 comma 4 ). The explicit functional forms of the correlation kernels then imply that the general n -point correlation functions exhibit an asymptotic expansion in 1 / N 2 $1/N^2$ 1 divided by upper N squared , which moreover can be lifted to an asymptotic expansion in 1 / N 2 $1/N^2$ 1 divided by upper N squared for the spacing distributions and their generating function. We use σ $\sigma $ sigma -Painlevé characterisations to show that the functional form of the first correction is related to the leading term via a second derivative. In the case β = 2 $\beta = 2$