Phase Transitions in Random Dyadic Tilings and Rectangular Dissections

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

“…However, their techniques did not extend to the critical point, and they left as an open problem bounding the mixing time when λ = 1; it is notoriously often quite difficult to bound mixing times at or near critical points. Our main result, Theorem 1.1, uses a different, non-local approach to finally answer the question of [14] and [2] by showing that the mixing time of the edge-flip Markov chain at critical point λ = 1 is at most polynomial in n, substantially less than the mixing time when λ > 1. Furthermore, our Theorem 1.2 combined with the result for the weighted case in [2] shows that the behaviour at the (unweighted) critical point λ = 1 is also substantially different than when λ < 1.…”

“…Our main result, Theorem 1.1, uses a different, non-local approach to finally answer the question of [14] and [2] by showing that the mixing time of the edge-flip Markov chain at critical point λ = 1 is at most polynomial in n, substantially less than the mixing time when λ > 1. Furthermore, our Theorem 1.2 combined with the result for the weighted case in [2] shows that the behaviour at the (unweighted) critical point λ = 1 is also substantially different than when λ < 1. While it follows from the path coupling analysis in [2] that the relaxation time is O(n) for all fixed λ < 1, Theorem 1.2 establishes a super-linear lower bound on the relaxation time when λ = 1.…”

A Lyapunov function for Glauber dynamics on lattice triangulations

“…Since ground state edges of distinct midpoints are all compatible with one another, we have thatσ = {σ x : x ∈ Λ} is a ground state triangulation. In the lemma below we use the partial order on the set E ξ x , which is defined in (4), and the set of midpoints of constraint edges Λ bc = ξ ∩ Λ. Proposition 4.1.…”

“…Concurrently to [6], a similar model of random lattice triangulations has independently appeared in the statistical physics literature [13,14]. Following [6], a similar model has been introduced to study the mixing time of random rectangular dissections and dyadic tilings [4].…”

Polynomial mixing time of edge flips on quadrangulations

“…On the other hand, for many natural chains on Catalan structures, triangulations and related objects not even a polynomial upper bound for the mixing time is known. One such example is that of lattice triangulations, where polynomial bounds are only known for biased versions of the chain [8,9,26]; see also works on rectangular dissections, for which polynomial bounds were obtained very recently [7,6].…”