Cutoff phenomena and comparison techniques

For finite irreducible aperiodic kernels Pₙ, let dₙ and tₙ(ε) use the same worst-case TV convention as before. Assume the reference mixing times tₙ(1/4) diverge. Cutoff means that all fixed tolerances strictly between zero and one have asymptotically the same mixing time: the ratio for any two such tolerances tends to one. Equivalently, at times c times a cutoff scale aₙ, the distance tends to one for every c below one and to zero for every c above one. Integer times are rounded for discrete chains.

A window is a smaller scale wₙ=o(aₙ) over which the transition occurs, requiring matching limiting bounds on both sides. A single finite-chain graph cannot establish these quantifiers, nor does divergence of its family’s mixing times guarantee cutoff. The clock also matters: one update, one sweep of many coordinates and one unit of continuous time are different units. Multiplying all times in one family by a known factor changes the numerical location, so that factor must be recorded before comparing formulas across models.

For a concrete continuous-time example, take n independent bits, each refreshed to a fair bit by its own rate-one Poisson clock. Starting from all zeros, each sign has mean e⁻ᵗ; symmetry makes all deterministic starts equivalent in distance from the uniform product law. The sum of signs has mean ne⁻ᵗ and variance at most n, whereas its stationary mean is zero and variance n. At t=(1/2−η)log n for fixed positive η below one half, the squared mean divided by n diverges. Chebyshev bounds then distinguish the two distributions with probability difference tending to one.

For the upper bound, the product-law chi-squared discrepancy relative to uniform is exactly (1+e⁻²ᵗ)ⁿ−1. Total variation is at most one half its square root. At t=(1/2+η)log n this tends to zero, proving cutoff at one half log n in the specified per-coordinate clock. These are proof bounds, not simulated observations. They also indicate a bounded-width window when constant offsets are used. A discrete chain updating one uniformly chosen coordinate per step uses a different clock and needs its own conversion or argument.

For two finite irreducible reversible kernels sharing π, their Rayleigh quotients have the same variance denominator. An inequality between Dirichlet forms therefore transfers directly to their ordinary gaps in the displayed direction. A common way to establish it replaces each transition of Q by a path of P-edges, bounds the squared endpoint difference by path length times edge differences, and controls the accumulated congestion on each edge. The original comparison theorem makes this geometric mechanism precise. If stationary laws differ, additional density-ratio bounds enter and cannot be silently omitted.

Ordinary-gap comparison still leaves negative eigenvalues unless both chains are lazy or their negative spectra are separately controlled. Even for lazy chains, constant-factor bounds on relaxation or mixing scales do not force cutoff, whose tolerance ratios must approach one. A sequence of increasingly slow two-state chains already demonstrates the issue: its profiles remain exponential under rescaling rather than becoming step functions. Comparison is a powerful way to obtain estimates, while cutoff requires enough additional concentration or profile information to locate the transition more precisely.