  We study quotients of the Toeplitz C*-algebra of a random walk, similar to
those studied by the author and Markiewicz for finite stochastic matrices. We
introduce a new Cuntz-type quotient C*-algebra for random walks that have
convergent ratios of transition probabilities. These C*-algebras give rise to
new notions of ratio limit space and boundary for such random walks, which are
computed by appealing to a companion paper by Woess. Our combined results are
leveraged to identify a unique smallest symmetry-equivariant quotient
C*-algebra for any symmetric random walk on a hyperbolic group, shedding light
on a question of Viselter on C*-algebras of subproduct systems.
