We show that on every Ramanujan graph @math , the simple random walk exhibits cutoff: when @math has @math vertices and degree @math , the total-variation distance of the walk from the uniform distribution at time @math is asymptotically @math where @math is a standard normal variable and @math is an explicit constant. Furthermore, for all @math , @math -regular Ramanujan graphs minimize the asymptotic @math -mixing time for SRW among all @math -regular graphs. Our proof also shows that, for every vertex @math in @math as above, its distance from @math of the vertices is asymptotically @math .
Abstract A general method for obtaining asymptotic isoperimetric inequalities for families of graphs is developed. Some of these inequalities have been applied to functional analysis. This method uses the second smallest eigenvalue of a certain matrix associated with the graph and it is the discrete version of a method used before for Riemannian manifolds. Also some results are obtained on spectra of graphs that show how this eigenvalue is related to the structure of the graph. Combining these results with some known results on group representations many new examples are constructed explicitly of linear sized expanders and superconcentrators.