$L^2$-isoperimetric profile was introduced by Grigor’yan in the form of Faber–Krahn inequalities on manifolds. The theory was further developed in his joint work with Coulhon and Pittet to establish the close relation between $L^2$-isoperimetric profile and decay of heat semigroup on general Riemannian manifolds and infinite graphs. By studying the isoperimetric profiles, we show that random walks exhibit rich exotic behavior on Neumann-Segal type branch groups. I will also discuss a general result that bounds the growth of entropy of random walks in terms of the return probability.