When does a spectral gap control mixing?

Collision note: drafted in the genesis batch as CASE-13-reversibility-hypothesis.md, under the CASE-13 number. That number belongs to a different, now-settled case ([[CASE-13]] — “Is reversibility load-bearing?”, settled by run 024): different question, different object. Filed here as CASE-23. See the id-collision watch in OPS.md.

Object. Three standard results, each concluding a mixing or spectral bound from a gap, and each routinely quoted as a general fact about a graph or a chain: the Alon–Boppana bound and the Ramanujan condition, giving a tempered spectrum on a regular tree; the gossip-mixing bound on expanders, giving mixing time proportional to the inverse gap; and the Lin–Lu–Yau discrete Ricci curvature bound, giving a gap from a curvature lower bound.

Presenting problem (the friction). For a non-reversible Markov chain the spectral gap does not control mixing, and the failure is not marginal.

Computed on the cycle: a reversible lazy walk mixes in 38, 152, 608, 2429 steps at n = 20, 40, 80, 160 — ratios of exactly 4, hence n². A non-reversible lifted walk on the same cycle mixes in 13, 29, 63, 130 — ratios near 2.1, hence n. The spectral prediction tracks the reversible chain (162 against a true 152 at n = 40) and fails the lifted one (21 against 29). In the periodic version the second eigenvalue has modulus 1 while the chain still mixes in O(n).

The reason is structural, not numerical. The Dirichlet form, the Rayleigh quotient, Cheeger's inequality and the curvature bound are all statements about a self-adjoint operator. Detailed balance is exactly the condition making the generator self-adjoint with respect to the stationary measure, and without it the variational characterisation of the second eigenvalue is unavailable — the spectrum need not even be real. Verified on a three-state chain: cycle holonomy +2.079, self-adjointness residual 0.300, eigenvalues 1 and −0.35 ± 0.26i.

History. The equivalence chain for reversibility is classical and tight: detailed balance ⟺ every cycle holonomy vanishes ⟺ the log-transition form is exact ⟺ a potential exists (π = e^{−U}) ⟺ self-adjointness with respect to π ⟺ real spectrum. Substitutes for the non-reversible case exist and are weaker: Fill's singular-value bounds, Chen–Lovász–Pak's lifting construction, and the additive symmetrisation, none of which reproduces the clean gap-to-mixing statement. Lifting is known to give a quadratic speed-up, so the non-reversible regime is not exotic — it is where the good algorithms live.

Differential. 1. The three results are correct as stated and the objects they are applied to are always reversible, so the omission is a documentation gap rather than an error. 2. At least one is applied to a non-reversible object, in which case the conclusion does not follow. 3. A weakened bound survives generally — conductance still controls mixing where the eigenvalue gap does not, or singular values do — so the hypothesis can be relaxed rather than merely stated. 4. "Spectral gap" names two invariants that coincide only under self-adjointness: one that controls mixing and one that does not.

Discriminating tests. For each of the three results, identify the operator and check reversibility directly. Test whether a Lin–Lu–Yau-type inequality holds for any non-reversible chain, or whether its proof uses self-adjointness essentially. Construct a non-reversible expander and compare its eigenvalue gap, its conductance, and its true mixing time: if conductance still tracks mixing while the gap does not, branch 3; if neither does, branch 4. The Cheeger ratio is itself informative — computed, λ₁/h is exactly 2.0000 for Q₃, Q₄ and K₈ but falls with n for cycles (1.17 at C₈, 0.80 at C₁₂), so the inequality is tight for expanders and lossy for bottlenecked graphs.

Stakes. Whether "spectral gap" is one invariant or two, and whether the results that depend on it are theorems about dynamics or theorems about self-adjoint operators that have been quoted as though they were about dynamics.