We study for each fixed integer $g \ge 2$, for all primes $\ell $ and p with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell $-marked principally polarized superspecial abelian varieties of dimension g in characteristic p, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of p. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles–Goren–Lauter-type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$. As a byproduct, we also show that the finite regular directed graphs constructed by Jordan and Zaytman also has the same property.

A geometric mass concerning supersingular abelian varieties with real multiplications is formulated and related to an arithmetic mass. We determine the exact geometric mass formula for superspecial abelian varieties of Hubert-Blumenthal type. As an application, we compute the number of the irreducible components of the supersingular locus of some Hubert-Blumenthal varieties in terms of special values of the zeta function.