Local move connectedness of domino tilings with diagonal impurities

Nakano, Fuminiko; Ono, Hiroyuki; Sadahiro, Taizo

doi:10.1016/j.disc.2010.02.015

Cited by 6 publications

(3 citation statements)

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“…We will prove that for four-cycle free graphs H, X H has the pivot property (Theorem 4.1). Many properties similar to the pivot property have appeared in the literature (often by the name local-move connectedness) [4,20,22,34]. For instance consider the following problem: Let G be a finite undirected graph without multiple edges and self-loops.…”

Section: Introductionmentioning

confidence: 99%

Four-cycle free graphs, height functions, the pivot property and entropy minimality

Chandgotia

2016

Ergod. Th. Dynam. Sys.

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Abstract. Fix d ≥ 2. Given a finite undirected graph H without self-loops and multiple edges, consider the corresponding 'vertex' shift, Hom(Z d , H) denoted by XH. In this paper we focus on H which is 'four-cycle free'. The two main results of this paper are: XH has the pivot property, meaning that for all distinct configurations x, y ∈ XH which differ only at finitely many sites there is a sequence of configurations x = x 1 , x 2 , . . . , x n = y ∈ XH for which the successive configurations x i , x i+1 differ exactly at a single site. Further if H is connected then XH is entropy minimal, meaning that every shift space strictly contained in XH has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the 'lifts' of the configurations in XH to the universal cover of H and the introduction of 'height functions' in this context. IntroductionBy H we will always denote an undirected graph without multiple edges (and by abuse of notation also denote its set of vertices). In this paper we focus on H which is four-cycle free, that is, it is finite, it has no self-loops and the four-cycle, denoted by C 4 is not a subgraph of H. Fix d ≥ 2. The basic object of study is X H , the space of graph homomorphisms from Z d to H. Here by Z d we will mean both the group and its standard Cayley graph.Such a space of configurations X H , referred to as a hom-shift, can be obtained by forbidding certain patterns on edges of H Z d . If H is a finite graph then X H is a nearest neighbour shift of finite type. In addition it is also 'isotropic' and 'symmetric', that is, given vertices a, b ∈ H if a is not allowed to sit next to b in X H for some coordinate direction, then a is not allowed to sit next to b in all coordinate directions. Most of the concepts related to shift spaces are introduced in Section 2.Related to a shift space X is its topological entropy denoted by h top (X) which measures the growth rate of the number of patterns allowed in X with the size of the underlying shape (usually rectangular). For a given shift space, its computation is a very difficult task (look for instance in [23] and the references within). We will focus on a different aspect: as in [8], a shift space X is called entropy minimal if for all shift spaces Y X, h top (Y ) < h top (X). Thus if a shift space has zero entropy and is entropy minimal then it is a topologically minimal system. For d = 1, it is well known that all irreducible nearest neighbour shifts of finite type are entropy minimal [15]. However not much is known about it in higher dimensions: Shift spaces with a strong mixing property called uniform filling are entropy minimal [32], however nearest neighbour shifts of finite type with weaker mixing properties like block-gluing may not be entropy minimal [2]. If H is a four-cycle free graph then X H is not even block-gluing (we do not prove this but is implied by our results). There has been some recent work [33] which describes some conditions which are equivalent to entropy minimality for s...

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Section: Introductionmentioning

confidence: 99%

Four-cycle free graphs, height functions, the pivot property and entropy minimality

Chandgotia

2016

Ergod. Th. Dynam. Sys.

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Section: Introductionmentioning

confidence: 99%

Four-Cycle Free Graphs, Height Functions, the Pivot Property and Entropy Minimality

Chandgotia

2014

Preprint

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Fix d ≥ 2. Given a finite undirected graph H without self-loops and multiple edges, consider the corresponding 'vertex' shift, Hom(Z d , H) denoted by XH. In this paper we focus on H which is 'four-cycle free'. The two main results of this paper are: XH has the pivot property, meaning that for all distinct configurations x, y ∈ XH which differ only at finitely many sites there is a sequence of configurations x = x 1 , x 2 , . . . , x n = y ∈ XH for which the successive configurations x i , x i+1 differ exactly at a single site. Further if H is connected then XH is entropy minimal, meaning that every shift space strictly contained in XH has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the 'lifts' of the configurations in XH to the universal cover of H and the introduction of 'height functions' in this context.

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“…We shall review the known results on the perfect matching problem on G (k) . In [8], they consider two types of elementary moves which transforms a perfect matching on G (k) to a different one, and showed that any two perfect matchings are connected via a sequence of elementary moves. Then we can construct a Markov chain on M(G (k) ) with the uniform stationary distribution.…”

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confidence: 99%

Hitting Matrix and Domino Tiling with Diagonal Impurities

Nakano

Sadahiro

2013

J Stat Phys

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As a continuation to our previous work [9, 10], we consider the domino tiling problem with impurities. (1) if we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the hitting matrix method[4]. (2) we have an alternative proof of the main result in [9] and result in (1) above using the formula by Kenyon-Wilson [6, 7] of counting the number of groves on the circular planar graph. (3) we study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity, as well as the scaling limit of those.Comment: 29 pages, 27 figure

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Local move connectedness of domino tilings with diagonal impurities

Cited by 6 publications

References 6 publications

Four-cycle free graphs, height functions, the pivot property and entropy minimality

Four-cycle free graphs, height functions, the pivot property and entropy minimality

Four-Cycle Free Graphs, Height Functions, the Pivot Property and Entropy Minimality

Hitting Matrix and Domino Tiling with Diagonal Impurities

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