Run coupling from the past on a live Ising grid, watch two monotone bounds certify every possible start at once, then compute a three-state counterexample showing exactly why forward first-meeting can be biased.
One world begins all cold, one all hot. Both receive the same site and the same uniform number at every negative time. Ferromagnetic heat-bath updates preserve their order, so every other starting grid is trapped between them.
P(spin i = +1 | neighbors) = 1 / (1 + exp(-2 beta sum of four neighbor spins)); J = 1, external field = 0.
Press the gold button. The complete calculation is local to this page.
The maps run forward from a negative index. Nothing here inverts a Markov transition. When the horizon doubles, the old events keep their times and only a new prefix is added.
The certificate can look like an initialization diagnostic. The small machine below removes that escape: it solves the target law, enumerates every short history, and computes the law selected by each stopping rule.
The dismissal: if all starts meet eventually, why bother indexing randomness in the past? Just run them forward and stop at the first meeting.
D = (0, 0, 1) and U = (1, 2, 2). Each is chosen with probability 1/2. A tuple lists the image of states 0, 1, 2.
Whose chain this is. The chain is Propp and Wilson's own. Their 1996 paper introduces it in these words:
The curious reader may verify this by considering the Markov chain whose states are 0, 1, and 2, and in which transitions are implemented using a fair coin, by the rule that one moves from state i to state min(i+1,2) if the coin comes up heads and to state max(i-1,0) otherwise.
Two honest qualifications, because the distinction matters. In that paper the chain is raised against a different mistake, drawing fresh random bits on each restart; their argument against stopping a forward run is the general observation that a state with a unique predecessor can never be the one you land on at the instant every history first agrees. And the worked numbers below, the exact law (1/2, 0, 1/2) and its total variation distance 1/3, appear in neither Propp and Wilson nor Häggström: they are computed here, and the check panel recomputes them. The published forward counterexample, Häggström's Example 10.1, is a two-state chain with stationary law (2/3, 1/3), not this one.
| from \ to | 0 | 1 | 2 |
|---|
Computing the exact law...
The correctness question here is closed: Propp and Wilson proved in 1996 that a valid CFTP output has the stationary law. The live edge is whether an all-state certificate can be represented and reached at useful cost for the model and coupling at hand. This page does not claim a universal runtime bound.
Chanal and Krauth reported one frustrated Ising spin-glass realization at the displayed size and parameters. Random starts supplied only a lower bound. Their patch representation retained all compatible states and supplied the upper certificate.
The opened 2008 paper contrasts a sampled set of starts with the full sublattice state space. The reported coupling time belongs to one coupling, bond realization, and random history. It is not a universal constant and not the chain's intrinsic mixing time.
These values are recomputed by the shipped engine, not copied into the boxes. The dependency-free verifier separately derives them, executes this page's own engine in a Node VM, and mutation-tests the checks.