Renormalization group

The notion of studying systems by seeing the effect of changing the scale on which one looks at them has been widely used in physics since about 1970, and there is some analogy between this and what I do here with cellular automata. In the lattice version in physics one typically considers what happens to averages over all possible configurations of a system if one does a so-called blocking transformation that replaces blocks of elements by individual elements. And what one finds is that in certain cases—notably in connection with nesting at critical points associated with phase transitions (see page 981)—certain averages turn out to be the same as one would get if one did no blocking but just changed parameters ("coupling constants") in the underlying rules that specify the weighting of different configurations. How such effective parameters change with scale is then governed by so-called renormalization group differential equations. And when one looks at large scales the versions of these equations that arise in practice essentially always show fixed points, whose properties do not depend much on details of the equations—leading to certain universal results across many different underlying systems (see page 983).

What I do in the main text can be thought of as carrying out blocking transformations on cellular automata. But only rarely do such transformations yield cellular automata whose rules are of the same type one started from. And in most cases such rules will not suffice even if one takes averages. And indeed, so far as I can tell, only in those cases where there is fairly simple nested behavior is any direct analog of renormalization group methods useful. (See page 989.)