2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/225661We show a strict hierarchy among various edge and vertex expansion properties of Markov chains. This gives easy proofs of a range of bounds, both classical and new, on chi-square distance, spectral gap and mixing time. The 2-gradient is then used to give an isoperimetric proof that a random walk on the grid [k]^n mixes in time O*(k^2 n).Probability60J10text