## Notation

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 *d*th power,
or *n* to the 1/3 power we will write Z^d or n^{1/3}.

## Lattices and Neighbors

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).

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