Superspecial points on Humbert surfaces and the probability of (N,N)-splitting

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Let $p\equiv3\pmod4$ be a prime and let $N\geq2$ be an integer. We determine, to first order in $1/p$ and with an explicit error bound, the proportion of superspecial principally polarized abelian surfaces over $\overline{\mathbb{F}}_p$ that are optimally $(N,N)$-split. The surfaces are weighted by the reciprocals of their automorphism orders, which is the stationary measure of the Richelot random walk. The main theorem states that for $p\nmid N$ the expected number of optimal $(N,N)$-splittings