Properties of the solution

In many cases it is possible to determine various properties of the solution to the problem we are interested in without actually obtaining that solution. Asymptotic behaviour is one of the most important. For example, the existence of the growth constant, $\mu$, tells us that the coefficients, $c_n$ must be of the form

$$c_n = \mu^n \theta(n),$$ (4)

for some function $\theta(n)$ that satisfies

$${ \lim_{ n \rightarrow \infty}} \; \theta(n) ^{1/n} = 1.$$

It is believed (but not yet proved) that $c_n$ should actually behave like

$$c_n \sim A \mu^n n^{\gamma-1} n \rightarrow \infty,$$ (5)

where $\gamma$ is called a critical exponent. This asymptotic form is expected to hold for polyominoes and a great number of other models.

In the case of the self-avoiding walk (SAW)[32] the exact value of the exponent $\gamma$ is known in two dimensions (but not rigorously) because of a connection, first observed by de Gennes [11], with a magnet model called the $N$-vector model. In the limit $N \rightarrow 0$ this magnet model (in $d$-dimensions) reduces to the self-avoiding walk (in $d$-dimensions). Nienhuis [33] was able to calculate the exponents of the $N$-vector model in two dimensions for general $N$, and found that $\gamma = \frac{43}{32}$ (when $N=0$). This same technique also gives the SAW length exponent $\nu= 3/4$ (see below), and the growth constant on the honeycomb lattice as $\mu = \sqrt{2+\sqrt{2}}$.

Some knowledge of the analytic structure of the generating function may also be of use; there exist many techniques for ``discovering'' solutions from the first few terms of the generating function (such as NEWGRQD [18], GFUN and MGFUN [23]). These techniques rely on the solution satisfying simple differential or algebraic equations. If we can show that solution has certain properties that mean it cannot satisfy such an equation then we will not be able to ``guess'' the solution using these techniques, no matter how many terms we obtain.