Let ∧ be a finite set and let P(∧) denote the set of subsets of ∧. In this chapter we will look at various classes of probability measures on P(∧) that might arise in simple physical and biological models. The points of ∧ can be interpreted as sites, each of which can be either empty or occupied by a particle (or some other entity); the subset A ∈ P(∧) will be regarded as describing the state of the model when the points of A are occupied and the points of ∧ - A are empty. The elements of P(∧) will sometimes be called configurations. Most physical models (even simple ones) involving configurations would be dynamic in nature; the probability measures that we will look at will describe the distribution of the configurations when the model is in some state of dynamic equilibrium.

The set ∧, representing the sites in the model, can be expected to have some additional structure, for example we might know the distances between the sites, or we might know that certain sites are connected. We will consider structures on ∧ of the latter kind, thus we suppose that the points of ∧ are the vertices of some finite graph G = (∧, e), where e is the set of edges of G. We do not allow G to have any multiple edges or loops. The following notation and definitions will be used: If x,y ∈ ∧ and there is an edge of the graph between x and y then we will say that x and y are neighbours.

In this chapter we will examine certain probability measures on P(S), the set of all subsets of S, where S is a countable set. We will be interested in the existence and uniqueness of measures satisfying certain properties. As before we will regard the points of S as sites, each of which can be either empty or occupied by a particle (or some other entity); the subset A ∈ P(S) will be regarded as describing the situation when the points of A are occupied and the points of S - A are empty. The elements of P(S) will sometimes be called configurations and the probability measures on P(S) that we will look at will describe the equilibrium distribution of the configurations of some model.

The basic assumption concerning the models that we will consider is the following: let ∧ be a finite subset of S, A ⊂ ∧ and X ⊂ S - ∧; we will suppose that the model is such that the conditional probability of there being particles on ∧ at exactly the points of A, given that on S - ∧ there are particles at exactly the points of X, is specified. (This says that if we know what is happening outside a finite subset of S then we can compute the distribution of particles inside the finite set.) Let us denote the above conditional probability hy f∧(A,X); our aim in this chapter is firstly to determine what relations the f∧(A,X) must satisfy and secondly, to find out whether the f∧(A,X) uniquely determine a probability measure on P(S).

In the year 1968–1969, Professor Mary Cartwright was a visiting member of the Division of Applied Mathematics at Brown University. This was a year of turmoil at Brown—particularly curricular turmoil—and in the course of one of our division meetings Miss Cartwright remarked that when she was a student all mathematics majors were required to know a proof of the nine-point circle theorem. Since the nine-point circle now has a distinct flavor of beautiful irrelevance, Miss Cartwright seemed to be telling us that we should not be too dogmatic as to what constitutes a proper mathematics curriculum. Fashion is spinach even in mathematics, and time often works to “nine-point circle-ize” many of our most relevant and sophisticated topics that are now insisted upon.

At the time, I was giving a course in Numerical Analysis entitled “Iteration Theory in Banach Spaces”, using notes of L. B. Rall, and I saw that it would be possible—and not too far-fetched—to present several lectures which would trace an unlikely path from the nine-point circle to iteration.

This essay presents such a path. The connecting link is the use of conjugate coordinates and the Schwarz reflection function. The path has been faired, as draftsmen say, to pass in a wide arc near a number of allied topics in complex variable theory that have interested me.

In Chapter Two conjugate coordinates are introduced. In Chapter Three elementary notions of plane analytic geometry are expressed in terms of conjugate coordinates.