13 research outputs found
Characterizations of Flip-Accessibility for Domino Tilings of the Whole Plane
International audienceIt is known that any two domino tilings of a polygon are flip-accessible, \emph{i.e.}, linked by a finite sequence of local transformations, called flips. This paper considers flip-accessibility for domino tilings of the \emph{whole plane}, asking whether two of them are linked by a \emph{possibly infinite} sequence of flips. The answer turning out to depend on tilings, we provide three equivalent \emph{characterizations} of flip-accessibility
Distances on Lozenge Tilings
International audienceIn this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the flipdistance (i.e. the number of necessary local transformations involving three lozenges) between two given tilings. It is here proven that, for n5, We show that there is some deficient pairs of tilings for which the flip connection needs more flips than the combinatorial lower bound indicates
Stochastic Flips on Dimer Tilings
International audienceThis paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called , which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a bound, where is the number of tiles of the tiling. We prove a upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case
Fast domino tileability
Domino tileability is a classical problem in Discrete Geometry, famously
solved by Thurston for simply connected regions in nearly linear time in the
area. In this paper, we improve upon Thurston's height function approach to a
nearly linear time in the perimeter.Comment: Appeared in Discrete Comput. Geom. 56 (2016), 377-39
The -invariant massive Laplacian on isoradial graphs
We introduce a one-parameter family of massive Laplacian operators
defined on isoradial graphs, involving elliptic
functions. We prove an explicit formula for the inverse of , the
massive Green function, which has the remarkable property of only depending on
the local geometry of the graph, and compute its asymptotics. We study the
corresponding statistical mechanics model of random rooted spanning forests. We
prove an explicit local formula for an infinite volume Boltzmann measure, and
for the free energy of the model. We show that the model undergoes a second
order phase transition at , thus proving that spanning trees corresponding
to the Laplacian introduced by Kenyon are critical. We prove that the massive
Laplacian operators provide a one-parameter
family of -invariant rooted spanning forest models. When the isoradial graph
is moreover -periodic, we consider the spectral curve of the
characteristic polynomial of the massive Laplacian. We provide an explicit
parametrization of the curve and prove that it is Harnack and has genus . We
further show that every Harnack curve of genus with
symmetry arises from such a massive
Laplacian.Comment: 71 pages, 13 figures, to appear in Inventiones mathematica
Martin boundary of killed random walks on isoradial graphs
Avec un appendice d'Alin BostanWe consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially non-homogeneous. Despite these crucial differences, we compute the asymptotics of the Martin kernel, deduce the Martin boundary and show that it is minimal. Similar results on the grid are derived in a celebrated work of Ney and Spitzer
A Characterization of Flip-accessibility for Rhombus Tilings of the Whole Plane
International audienceIt is known that any two rhombus tilings of a polygon are flip-accessible, that is, linked by a finite sequence of local transformations called flips. This paper considers flip-accessibility for rhombus tilings of the whole plane, asking whether any two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on tilings, a characterization of flip-accessibility is provided. This yields, for example, that any tiling by Penrose tiles is flip-accessible from a Penrose tiling