Editing Isoperimetric dimension (section)

==Consequences of isoperimetry==<!-- This section is linked from [[Random walk]] -->

A simple integration over ''r'' (or sum in the case of graphs) shows that a ''d''-dimensional isoperimetric inequality implies a ''d''-dimensional [[Growth rate (group theory)|volume growth]], namely

:<math>\operatorname{vol} B(x,r)\geq Cr^d</math>

where ''B''(''x'',''r'') denotes the ball of radius ''r'' around the point ''x'' in the [[Riemannian manifold|Riemannian distance]] or in the [[Glossary of graph theory#Distance|graph distance]]. In general, the opposite is not true, i.e. even uniformly exponential volume growth does not imply any kind of isoperimetric inequality. A simple example can be had by taking the graph '''Z''' (i.e. all the integers with edges between ''n'' and ''n''&nbsp;+&nbsp;1) and connecting to the vertex ''n'' a complete [[binary tree]] of height |''n''|. Both properties (exponential growth and 0 isoperimetric dimension) are easy to verify.

An interesting exception is the case of [[Group (mathematics)|groups]]. It turns out that a group with polynomial growth of order ''d'' has isoperimetric dimension ''d''. This holds both for the case of [[Lie group]]s and for the [[Cayley graph]] of a [[finitely generated group]].

A theorem of [[Nicholas Varopoulos|Varopoulos]] connects the isoperimetric dimension of a graph to the rate of escape of [[random walk]] on the graph. The result states

''Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then''

:<math>p_n(x,y)\leq Cn^{-d/2} </math>

''where'' <math display="inline"> p_n(x,y)</math> ''is the probability that a random walk on'' ''G'' ''starting from'' ''x'' ''will be in'' ''y'' ''after'' ''n'' ''steps, and'' ''C'' ''is some constant.''