Introduction

Abstract The general approach to lattice models of walks, polygons, trees and animals is rooted in statistical mechanics, and particularly in critical phenomena. It is from this vantage point that questions about these models are asked, and answers are also interpreted within this framework. In this chapter a very brief, and perhaps incomplete, overview is given. The motivation for studying these models comes from the chemistry and physics of macromolecules, and the models are attempts at representing, in a simple manner, the entropic contribution to the free energy made by the conformational degrees of freedom in these molecules. The mathematical description of almost all the models in this monograph is based on a basic theorem of sub-additive functions, which I discuss in Section 1.2.1. The scaling properties and critical exponents are discussed in Section J .3, and a Flory argument for estimating the metric exponent of linear polymers, and of branched polymers, is given in Sections 1.2.3 and 1.3.3. For the most part I shall accept the Flory values of exponents as adequate; while they are known to be accurate in some cases, it is also the case that in some dimensions (three dimensions for walks) the Flory argument gives a value which is incorrect (but not far off the mark).