Glauber Dynamics of Lattice Triangulations on Thin Rectangles

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Proves concrete mixing bounds for lattice triangulations on rectangles with fixed height m: O(n^2) mixing for λ<1 using path coupling with an exponential metric and a Lyapunov function to force a n^2 “burn‑in” into a good set; shows exp(Ω(n^2)) mixing for λ>1 (rate weakens as m grows) by exhibiting an exponential bottleneck (herringbone variant); obtains polynomial relaxation on polylog scales via improved canonical paths so the restricted chain mixes faster; key numbers: burn‑in ~n^2, stay time ~n^10, final polylog decoupling.

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" chapters (12)

1. 0:00Intro
2. 2:36Spin Model
3. 5:24Phase Transition
4. 6:11Rapid Mixing
5. 9:49Standard Approach
6. 10:35herringbone bottleneck
7. 20:53Constraint edges
8. 24:06Special mixing
9. 30:42Bounds
10. 33:31Improved Canonical Argument
11. 36:34Good Set
12. 38:39Construction