Random walks on compact manifolds — the torus, the sphere, projective spaces — converge to the uniform distribution. The mixing time measures how long this takes. The cutoff phenomenon is sharper: not just convergence, but a sharp transition from “far from uniform” to “close to uniform” happening in a narrow time window. Before the cutoff, the walk remembers where it started. After, it doesn't.
The authors (arXiv:2603.22997) prove cutoff for random walks on flat tori, spheres, and projective spaces in the separation distance metric. Separation distance is the strongest notion of mixing — it measures the worst-case ratio between the walk's distribution and the uniform distribution, rather than the total variation or the chi-squared distance.
The cutoff window — the time interval during which the transition happens — is asymptotically narrow relative to the mixing time. The walk is far from mixed at time t_mix - o(t_mix) and close to mixed at time t_mix + o(t_mix). The geometry of the manifold enters through the eigenvalues of the Laplacian, which determine the rate at which different spatial modes decay.
The through-claim: mixing on symmetric spaces is discontinuous in a precise sense. The walk doesn't gradually approach uniformity — it snaps to it. The geometry determines when the snap happens (the mixing time) and how sharp it is (the cutoff window), but the snap itself is universal. Any walk on any of these spaces forgets its origin at a specific moment, not over a gradual process.