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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Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process. For many random walks on sufficiently large finite groups the so-called cut-off phenomenon occurs: roughly stated, there exists a number k 0 , depending on the size of the group, such that k 0 steps are necessary and sufficient for the random walk to closely approximate uniformity. As a first example on a continuous group, Rosenthal recently proved the occurrence of this cut-off phenomenon for a specific random walk on SO(N). Here we present and [for the case of O(N)] prove results for random walks on O(N), U(N) and Sp(N), where the one-step distribution is a suitable probability measure concentrated on reflections. In all three cases the cut-off phenomenon occurs at k 0 = 1 2N log N. We show that the total variation mixing time of the simple random walk on the giant component of supercritical Gn,p and Gn,m is i¾?log2n. This statement was proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are "decorated expanders" - an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 383-407, 2014 This paper gives a new way of showing that certain constant degree graphs are graph expanders. This is done by giving new proofs of expansion for three permutations of the Gabber-Galil expander. Our results give an expansion factor of @math for subgraphs of these three-regular graphs with @math inputs for @math prime. The proofs are not based on eigenvalue methods or higher algebra. The same methods show the expected number of probes for unsuccessful search in double hashing is bounded by @math , where a is the load factor. This assumes a double hashing scheme in which two hash functions are randomly and independently chosen from a specified uniform distribution. The result is valid regardless of the distribution of the inputs. This is analogous to Carter and Wegman’s result for hashing with chaining. This paper concludes by elaborating on how any sufficiently sized subset of inputs in any distribution expands in the subgraph of the Gabber-Galil graph expander of focus. This is related to any key distribution having expected @math probes for unsuccessful search for double hashing given the initial random, independent, and uniform choice of two universal hash functions.