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 .
For an infinite family of @math -regular LPS Ramanujan graphs, we show that the diameter of these graphs is greater than or equal to @math , where @math is an odd prime number and @math is the number of vertices. On the other hand, for any @math -regular Ramanujan graph we show that the distance of only tiny fraction of all pairs of vertices is greater than @math . We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically @math and @math , respectively. These are consistent with Sarnak's expectation on the covering exponent of universal quantum gates and our conjecture for the optimal strong approximation for quadratic forms in 4 variables. The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.