Bond animals

Figure 13: The bond animal kingdom -- the only solved models (other than polygon models) are spiral walks (on the square lattice, and one of the three possible models on the triangular lattice), partially directed walks and directed walks.

Like polyominoes and site-animals, bond animals remain unsolved in two dimensions and higher, it therefore makes sense to consider bond animals with simpler topologies (see figure 13):

Definition 6

• Directed bond animals are bond animals in which all bonds can be reached from the vertex at the origin by a path (remaining inside the animal) that takes only north and east steps.
• Lattice trees are bond animals that contain no closed loops.
• Self-avoiding walks (SAWs) are bond animals with a linear topology. Alternatively they are bond animals, such that every vertex visited by the animal is of degree $2$, excepting two vertices (the endpoints) which are of degree $1$.

Unfortunately these restrictions have not been sufficient to lead to a solution in any dimension higher than one. It is curious that directed bond animals are yet to be solved when directed site animals have been solved for almost two decades; we discuss possible reasons for this in chapter 3.

Self-avoiding walks (SAWs) are of considerable importance as a model of linear polymers (see below) and have been intensively studied for many years. Even though no exact solution has been forthcoming, a great deal is known about them, by both rigorous and numerical methods (see [32] and references within) -- for example, the connection between SAWs and the $N$-vector model [11] gives (non-rigorously) both the critical exponents $\nu$ and $\gamma$ in two dimensions, and the growth constant on the honeycomb lattice [33].

Many modified walk models have been introduced to mimic various physical situations and to allow for easier analysis. These modifications can lead to changes in the asymptotic behaviour of the model. A large family of modified SAW models are two-step-restricted walks (TSRWs). A TSRW is constructed by starting at the origin and adding bonds so that the walk is self-avoiding and so that the possible directions of the $n^{th}$ bond (or step) depend upon the direction of the $(n-1)^{th}$ bond according to some rule (consequently the bonds of a TSRW are ordered). In two dimensions, a wide ranging study of TSRW [21] found (using computer enumeration and numerical methods) that the asymptotic behaviour of these TSRWs can be linked to the symmetries of the restricting rule. We describe this survey in chapter 9. Some of the more interesting²¹ restricted walks are:

Definition 7

• $3$-choice walks are TSRW for which clockwise turns are forbidden after steps in the $\pm x$ direction; i.e. after an east step it is not possible to step south, and after a west step it is not possible to step north.
• $2$-choice walks are $3$-choice walks that are forbidden to make more than one successive step in the $\pm y$ direction.
• Spiral walks are oriented SAWs in which all clockwise turns are forbidden; i.e. after an east step it is not possible to step south, after a north step it is not possible to step east, etc.
• Partially directed walks are SAWs whose intersection with any vertical line is connected. Equivalently the walk is forbidden to step west.
• Directed walks are SAWs that consist entirely of north and east steps.

These walks are discussed in more detail in chapter 9. The only ``non-trivial'' walk models to have been solved are partially directed walks, directed walks [37] and spiral walks. Two different spiral walk models have been solved; spiral walks on the square lattice [22,27,42,5], and one model²² of spiral walks on the triangular lattice [28] (although the asymptotics of the other two is known [39]). The growth constants of $3$-choice and $2$-choice walks are known [43].

In chapter 9 we examine the behaviour of three dimensional TSRW. There are $2^{30} = 1073741824$ TSRW models in three dimensions, and so we have taken only a very small subset of these, chosen so as to examine the relationship between asymptotic behaviour of the walk and the symmetries of the restricting rule. Given the extreme difficulty of finding rigorous results for any bond animal models, we have used computer enumeration and numerical methods in this study.