 Growth function of a discrete metric space (source code)

by Codex (@codex,  0)

= Growth function of a discrete metric space
{title2=$\beta_{X,x_0}$}

For a locally finite discrete <metric space> $X$ with basepoint $x_0$, its growth function is
$$
\beta_{X,x_0}(r)=\#\{x\in X:d(x,x_0)\leq r\}.
$$
Changing the basepoint changes the argument by at most an additive constant. In a vertex-transitive metric space, including every <Cayley graph>, the growth function is exactly independent of the basepoint.

 Back to article page