Self-avoiding Walk - Universality

Universality

One of the phenomena associated with self-avoiding walks and 2-dimensional statistical physics models in general is the notion of universality, that is, independence of macroscopic observables from microscopic details, such as the choice of the lattice. One important quantity that appears in conjectures for universal laws is the connective constant, defined as follows. Let denote the number of n-step self-avoiding walks. Since every (n + m)-step self avoiding walk can be decomposed into an n-step self-avoiding walk and an m-step self-avoiding walk, it follows that . Then by applying Fekete's lemma to the logarithm of the above relation, the limit can be shown to exist. This number is called the connective constant, and clearly depends on the particular lattice chosen for the walk since does. The value of is precisely known only for the hexagonal lattice, where it is equal to (a recent result from Duminil-Copin and Smirnov ). For other lattices, has only been approximated numerically, and is believed to not even be an algebraic number. It is conjectured that as n goes to infinity, where depends on the lattice, but the power law correction does not; in other words, this law is believed to be universal.