Thought Toys · Chance & inference · Exhibit 30
Two clouds of possibility, released from opposite corners of the same six-state ring, hop on identical fixed odds. There is one hair's-breadth setting that decides whether they chase each other forever or quietly become indistinguishable — the long run forgetting exactly where each one began.
How far apart are the two clouds?
Press play and watch.
Six states sit on a ring. Each step, every bit of "possibility" at a state either stays put — with probability r, the stickiness dial — or hops one state clockwise, with probability 1−r. That's the whole rule: fixed odds, applied identically everywhere, forever.
Now release two clouds. Cloud A starts entirely at state 0; cloud B starts entirely at the opposite state, 3. Circle size at each state shows how much of that cloud is currently there — a big circle is a near-certainty, a tiny one is almost nothing. Watch what happens as they hop: with the dial above zero, both clouds spread out, wash over the whole ring, and settle down to exactly the same steady mix — one-sixth of the mass on every state, no matter that they began on opposite sides. The long run forgot where each one started.
Drag the dial down to exactly 0 and something different happens: nothing spreads at all. With no chance of staying, every step is a forced, perfect hop — the two clouds stay pinned as single dots, chasing each other around the ring in lockstep, forever the same distance apart. At r=0 the walk has a fixed period (it returns to where it started every 6 steps, on the nose), and a periodic chain — one where returns are only possible at multiples of some fixed number of steps can never settle into one steady distribution. The instant you nudge the dial off zero, even to 0.01, a tiny chance of staying breaks that rigid rhythm, and convergence — slow at first — becomes not just possible but certain.
improve/verify/30-markov.js): rows and
columns of P sum to 1 for every r; two clouds started at any of the 30 distinct opposite
pairs converge to under 1e-5 apart within 300 steps at r=0.2; and mixing time is U-shaped —
slow near both r=0 (periodic) and r→1 (almost frozen), fastest near the middle.
Counter-examples: at r=0 the total-variation distance is pinned at exactly 1 for 500
straight steps — never dipping, confirming this isn't just "slow," it never converges at all. And even with
plenty of stickiness, a reducible chain — one split into separate islands with no path
between them keeps every cloud trapped in its own island forever: a shared stationary distribution needs
the ring to be fully connected (irreducible), not just aperiodic.
Also in Chance & inference: Averages that never settle →
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