Taizo Sadahiro

Local move connectedness of domino tilings with diagonal impurities

We study the perfect matchings in the dual of the square-octagon lattice graph, which can be considered as domino tilings with impurities in some sense. In particular, we show the local move connectedness, that is, if G is a vertex induced finite subgraph which is simply connected, then any perfect matching in G can be transformed into any other perfect matching in G by applying a sequence of local moves each of which involves only two edges.

Multiple points of tilings associated with Pisot numeration systems

This paper deals with a kind of aperiodic tilings associated with Pisot numeration systems, originally due to W.P. Thurston, in the formulation of S. Akiyama. We treat tilings whose generating Pisot units β are cubic and not totally real. Each such tiling gives a numeration system on the complex plane; we can express each complex number z in the following form: z = ck α-k + ck-1 α-k+1 +...+ c1 α-1 + c0 + c-1 α1 + c-2 α2 +..., where α is a conjugate of β, and c-m c-m+1... ck-1 ck is the β-expansion of some real number for any integer m. We determine the set of complex numbers which have three or more representations. This is equivalent to determining the triple points of the tiling, which is shown to be a collection of model sets (or cut-and-project sets). We also determine the set of complex numbers with eventually periodic representations.

A Bijection Theorem for Domino Tilings with Diagonal Impurities

We consider the dimer problem on a planar non-bipartite graph G, where there are two types of dimers one of which we regard as impurities. Computer simulations reveal a reminiscence of the Cheerios effect, that is, impurities are attracted to the boundary, which is the motivation to study this particular graph. Our main theorem is a variant of the Temperley bijection: a bijection between the set of dimer coverings and the set of spanning forests with certain conditions. We further discuss some implications of this theorem: (1) the local move connectedness yielding an ergodic Markov chain on the set of all possible dimer coverings, and (2) a rough bound for the number of dimer coverings and that for the probability of finding an impurity at a given edge, which is an extension of a result in (Nakano and Sadahiro in arXiv:0901.4824).

A -ExpansionAssociated to Sturmian Sequences

Abstract. We consider a -expansion which makes use of the structure of the corresponding Sturmian sequences, and study some basic properties.

Domino Tilings with One Diagonal Impurity

This paper studies the dimer model on the dual graph of the square-octagon lattice, which can be viewed as the domino tilings with impurities in some sense. In particular, under a certain boundary condition where only one impurity is contained, we give an exact formula representing the probability of finding an impurity at a given site in a uniformly random dimer configuration in terms of simple random walks on the square lattice.

A generalization of carries process and riffle shuffles

As a continuation to our previous work (Nakano and Sadahiro, 2014), we consider a generalization of carries process. Our results are: (i) right eigenvectors of the transition probability matrix, (ii) correlation of carries between different steps, and (iii) generalized riffle shuffle whose corresponding descent process has the same distribution as that of the generalized carries process.