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

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain

Approximately Sampling Elements with Fixed Rank in Graded Posets

Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call $\textit{balanced bias}$. A noteworthy application of our method is sampling restricted classes of integer partitions of $n$. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of $n$ from general restricted classes. Several observations allow us to improve the efficiency of this chain to require $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions, expected $O(n^{9/4})$ time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.Comment: 23 pages, 12 figure

Distances on Rhombus Tilings

The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by the study of a quasicrystal growth model, we are here interested in better understanding how "tight" rhombus tiling spaces are flip-connected. We introduce a lower bound (Hamming-distance) on the minimal number of flips to link two tilings (flip-distance), and we investigate whether it is sharp. The answer depends on the number n of different edge directions in the tiling: positive for n=3 (dimer tilings) or n=4 (octogonal tilings), but possibly negative for n=5 (decagonal tilings) or greater values of n. A standard proof is provided for the n=3 and n=4 cases, while the complexity of the n=5 case led to a computer-assisted proof (whose main result can however be easily checked by hand).Comment: 18 pages, 9 figures, submitted to Theoretical Computer Science (special issue of DGCI'09

Domino tilings and related models: space of configurations of domains with holes

We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called {\em flips}. This study allows us to formulate an exhaustive generation algorithm and a uniform random sampling algorithm. We finally extend these results to other types of tilings (calisson tilings, tilings with bicolored Wang tiles).Comment: 17 pages, 11 figure

Quivers, Tilings, Branes and Rhombi

We describe a simple algorithm that computes the recently discovered brane tilings for a given generic toric singular Calabi-Yau threefold. This therefore gives AdS/CFT dual quiver gauge theories for D3-branes probing the given non-compact manifold. The algorithm solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. We study the parameter space of a-maximization; this study is made possible by identifying the R-charges of bifundamental fields as angles in the brane tiling. We also study Seiberg duality from a new perspective.Comment: 36 pages, 40 figures, JHEP