Polynomial mixing of the edge-flip Markov chain for unbiased dyadic tilings [article]

Sarah Cannon, David Levin, Alexandre Stauffer
2016 arXiv   pre-print
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] × [b2^-t, (b+1)2^-t] for non-negative integers a,b,s,t. The edge-flip Markov chain
more » ... s 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 Ω(n^1.38), improving upon the previously best lower bound of Ω(n n) coming from the diameter of the chain.
arXiv:1611.03636v1 fatcat:gssfcjs5lrhadobw7tjngdlqwm