A general bridge theorem for self-avoiding walks

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Abstract

Let X be an infinite, locally finite, connected, quasi-transitive graph without loops or multiple edges. A graph height function on X is a map adapted to the graph structure, assigning to every vertex an integer, called height. Bridges are self-avoiding walks such that heights of interior vertices are bounded by the heights of the start- and end-vertex. The number of self-avoiding walks and the number of bridges of length n starting at a vertex o of X grow exponentially in n and the bases of these growth rates are called connective constant and bridge constant, respectively. We show that for any graph height function h the connective constant of the graph is equal to the maximum of the two bridge constants given by increasing and decreasing bridges with respect to h. As a concrete example, we apply this result to calculate the connective constant of the Grandparent graph.

Section snippets

Introduction and results

Let (X,) be a simple, locally finite, connected, infinite graph consisting of a countable set of vertices X and a symmetric neighbourhood relation . We consider a walk on the graph as a sequence of vertices, where consecutive vertices in the walk must be adjacent in the graph and call it self-avoiding (or a SAW), if no vertex of the graph is contained twice in this sequence. The length of a walk is the length of the sequence reduced by one, so the number of “steps” in the walk.

We shall

Terminology and preliminaries

We consider a graph (X,) as a countable set of vertices X together with a symmetric neighbourhood relation on X. We usually write X for the graph and omit . Two vertices u and v are called adjacent if uv. The degree degX(v) of a vertex v in X is the number of vertices of X which are adjacent to v. We call X locally finite, if degX(v)< for every vX.

A walk π on a graph X is a sequence of vertices (v0,v1,,vn) of X such that any two consecutive vertices of the sequence are adjacent in X. We

The bridge theorem

One of the main results of [6] is the bridge theorem.

Theorem 3.1

Thm. 4.3 in [6]

Let XX possess an unimodular graph height function (h,Γ). Then μ(X)=β(X,h).

Note that as a consequence, the bridge constant β(X,h) does not depend on the choice of the unimodular graph height function h. However, there are simple examples showing that unimodularity is required in this theorem, one of them being the Grandparent graph, which will be discussed in Section 4.

The main result in this paper is the following extension of this bridge

Bridges in the Grandparent graph

In this section we provide an example of a graph which does not possess a unimodular graph height function. We calculate the bridge constants and use the bridge theorem to obtain the connective constant.

An end of a tree is an equivalence class of one-way infinite SAWs, where two walks are equivalent if they share all but finitely many initial vertices. Fix some end ω of the infinite 3-regular tree T3 and let the graph “hang down” from the end ω. Then the graph can be seen as the union of

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (13)

  • GrimmettGeoffrey R. et al.

    Locality of connective constants

    Discrete Math.

    (2018)
  • BauerschmidtRoland et al.

    Lectures on self-avoiding walks

  • Duminil-CopinHugo et al.

    The connective constant of the honeycomb lattice equals 2+2

    Ann. of Math. (2)

    (2012)
  • FloryPaul J.

    Principles of Polymer Chemistry

    (1953)
  • GrimmettGeoffrey R. et al.

    Connective constants and height functions for Cayley graphs

    Trans. Amer. Math. Soc.

    (2017)
  • GrimmettGeoffrey R. et al.

    Self-avoiding walks and amenability

    Electron. J. Combin.

    (2017)
There are more references available in the full text version of this article.

This work was partially supported by the Austrian Science Fund : FWF P31889-N35 and FWF W1230.

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