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 .
Durrett [ ] DurrettRGW conjectured in 2008 that the random walk should have cutoff on a uniformly chosen @math -regular graph on @math vertices (typically a good expander) with probability tending to 1 as @math ; indeed this is the case, as was verified by the first author and Sly @cite_14 in 2010. Subsequently, expanders without cutoff were constructed in @cite_27 , but these were highly asymmetric. The conjectured behavior of cutoff for all transitive expanders was reiterated in the latter works (see [Conjecture 6.1] LS-gnd and [ ] LSexp ), yet this was not verified nor refuted on any single example to date.
@cite_27 @cite_14
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Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expanderif and only if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties. It also supplies an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNP-complete. 1. Overview 2. Erdos-Renyi random graphs 3. Fixed degree distributions 4. Power laws 5. Small worlds 6. Random walks 7. CHKNS model.