Locked Polyomino Tilings

A locked $t$-omino tiling is a grid tiling by $t$-ominoes such that, if you remove any pair of tiles, the only way to fill in the remaining $2t$ grid cells with $t$-ominoes is to use the same two tiles in the exact same configuration as before. We exclude degenerate cases where there is only one tiling overall due to small dimensions. It is a classic (and straightforward) result that finite grids do not admit locked 2-omino tilings. In this paper, we construct explicit locked $t$-omino tilings for $t \geq 3$ on grids of various dimensions. Most notably, we show that locked 3- and 4-omino tilings exist on finite square grids of arbitrarily large size, and locked $t$-omino tilings of the infinite grid exist for arbitrarily large $t$. The result for 4-omino tilings in particular is remarkable because they are so rare and difficult to construct: Only a single tiling is known to exist on any grid up to size $40 \times 40$. Locked $t$-omino tilings arise as obstructions to widely used political redistricting algorithms in a model of redistricting where the underlying census geography is a grid graph. Most prominent is the ReCom Markov chain, which takes a random walk on the space of redistricting plans by iteratively merging and splitting pairs of districts (tiles) at a time. Locked $t$-omino tilings are isolated states in the state space of ReCom. The constructions in this paper are counterexamples to the meta-conjecture that ReCom is irreducible on graphs of practical interest

Tiling a simply connected figure with bars of length 2 or 3

AbstractLet F be a simply connected figure formed from a finite set of cells of the planar square lattice. We first prove that if F has no peak (a peak is a cell of F which has three of its edges in the contour of F), then F can be tiled with rectangular bars formed from 2 or 3 cells. Afterwards, we devise a linear-time algorithm for finding a tiling of F with those bars when such a tiling exists

Tiling with Bars and Satisfaction of Boolean Formulas

AbstractLetFbe a figure formed from a finite set of cells of the planar square lattice. We first prove that the problem of tiling such a figure with bars formed from 2 or 3 cells can be reduced to the logic problem 2-SAT. Afterwards, we deduce a linear-time algorithm of tiling with these bars

Local dimer dynamics in higher dimensions

We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes $[n]^d$. These consist of successively switching the dimers along alternating cycles of prescribed (small) lengths. We study the connectivity properties of the dimer configuration space equipped with these transitions. Answering a question of Freire, Klivans, Milet and Saldanha, we show that in three dimensions any configuration admits an alternating cycle of length at most 6. We further establish that any configuration on $[n]^d$ features order $n^{d-2}$ alternating cycles of length at most $4d-2$. We also prove that the dynamics of dimer configurations on the unit hypercube of dimension $d$ is ergodic when switching alternating cycles of length at most $4d-4$. Finally, in the planar but non-bipartite case, we show that parallelogram-shaped boxes in the triangular lattice are ergodic for switching alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon and R\'emila, which also uses 8-cycles. None of our proofs make reference to height functions.Comment: 14 pages, 4 figure

Ergodic Archimedean dimers

We study perfect matchings, or close-packed dimer coverings, of finite sections of the eleven Archimedean lattices and give a constructive proof showing that any two perfect matchings can be transformed into each other using small sets of local ring-exchange moves. This result has direct consequences for formulating quantum dimer models with a resonating valence bond ground state, i.e., a superposition of all dimer coverings compatible with the boundary conditions. On five of the composite Archimedean lattices we supplement the sufficiency proof with translationally invariant reference configurations that prove the strict necessity of the sufficient terms with respect to ergodicity. We provide examples of and discuss frustration-free deformations of the quantum dimer models on two tripartite lattices.Comment: Submission to SciPost; 21 pages, 17 figure

Some tiling moves explored

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 135).by David Gupta.Ph.D