To write math in HTML we will usually use programming conventions and write our vectors as v[i] and arrays as a[i,j]. If we have to write x subscript i, we will will follow the notation of TeX and write x_i. Likewise if we have to write superscripts such as Z to the dth power, or n to the 1/3 power we will write Z^d or n^{1/3}.
Some of our one dimensional processes will occur on the integers { ... -2, -1, 0, 1, 2, ... } but most will take place on the square lattice, which is the set of all points in the plane both of whose coordinates are integers. Points in our spatial grid are called sites.
The defining features of our models, which we will affectionately call particle systems, are that:
(a) Each site x in the lattice can be in one of a finite number of states, so the state of the process at time t is described by giving state_t[x] for each site x in the grid. As mentioned above the _t indicates a subscript t. However one can ignore this detail and read state_t[x] as "the state of x at time t.
(b) Sites change their state at rates that depend on the states of a finite number of "neighboring" sites. In symbols the rate is a function
f(state_t[x], state_t[x+y_1], ... state_t[x+y_n])
Often in two dimensions x will be influenced only by its nearest neighbors, i.e., y_1 = (1,0), y_2 = (0,1), y_3 = (-1, 0), y_4 = (0,-1). Another of our favorite choices will be the square of radius r, i.e., the y_i are all the points all of whose coordinates are smaller than r.
In simulating processes in the computer we are limited, of course, to a finite set. We will typically take the square grid of points with each coordinate 0, 1, ..., or L-1. To avoid boundary effects we often use periodic boundary conditions, i.e., consider (i,0) to be a neighbor of (i,L-1) and (0,i) to be a neighbor of (L-1,i). However, it is sometimes convenient to use open boundary conditions where it is imagined that the grid is sitting in the infinite plane and sites outside are held in a fixed state (e.g., all sites empty).