We show that on every affine Weyl group natural random walks are noise sensitive in total variation.

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.

We show that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

We show the exact dimensionality of harmonic measures associated with random walks on groups acting on a hyperbolic space under a finite first moment condition, and establish the dimension formula by the entropy over the drift. We also treat the case when a group acts on a non-proper hyperbolic space acylindrically. Applications of this formula include continuity of the Hausdorff dimension with respect to driving measures and Brownian motions on regular coverings of a finite volume Riemannian manifold.

For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson–Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitary versions of dimensional properties of the harmonic measure.