For circular beta ensembles with beta = 1, 2, 4 and even beta, bulk-scaled correlations and spacing distributions expand in even powers of 1/N, with leading corrections given by a second derivative of the limiting form.
Bornemann,A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge, Ann
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Finite size corrections in the bulk for circular $\beta$ ensembles
For circular beta ensembles with beta = 1, 2, 4 and even beta, bulk-scaled correlations and spacing distributions expand in even powers of 1/N, with leading corrections given by a second derivative of the limiting form.