Isoperimetric profiles and random walks on some permutation wreath products

We study the isoperimetric profiles of certain families of finitely generated groups defined via marked Schreier graphs and permutation wreath products. The groups we study are among the "simplest" examples within a much larger class of groups, all defined via marked Schreier graphs and/or action on rooted trees, which includes such examples as the long range group, Grigorchuck group and the basillica group. The highly non-linear structure of these groups make them both interesting and difficult to study. Because of the relative simplicity of the Schreier graphs that define the groups we study here (the key fact is that they contained very large regions that are "one dimensional"), we are able to obtain sharp explicit bounds on the $L^1$ and $L^2$ isoperimetric profiles of these groups. As usual, these sharp isoperimetric profile estimates provide sharp bounds on the probability of return of simple random walk. Nevertheless, within each of the families of groups we study there are also many cases for which the existing techniques appear inadequate and this leads to a variety of open problems.