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11
Cutoff phenomena for random walks on random regular graphs
 Duke Math. J
"... Abstract. The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chain ..."
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Abstract. The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on G(n, d), a random dregular graph on n vertices. It is well known that almost every such graph for d ≥ 3 is an expander, and even essentially Ramanujan, implying a mixingtime of O(log n). According to a conjecture of Peres, the simple random walk on G(n, d) for such d should then exhibit cutoff whp. As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3regular graph is whp (6 + o(1)) log2 n. In this work we confirm the above conjectures, and establish cutoff in totalvariation, its location and its optimal window, both for simple and for nonbacktracking random walks on G(n, d). Namely, for any fixed d d ≥ 3, the simple random walk on G(n, d) whp has cutoff at d−2 logd−1 n with window order √ log n. Surprisingly, the nonbacktracking random walk on G(n, d) whp has cutoff already at logd−1 n with constant window order. We further extend these results to G(n, d) for any d = n o(1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.
Expansion of random graphs: New proofs, new results
, 2014
"... We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let Γ be a random dregular graph on n vertices, and let λ be the largest absolut ..."
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Cited by 10 (0 self)
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We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let Γ be a random dregular graph on n vertices, and let λ be the largest absolute value of a nontrivial eigenvalue of its adjacency matrix. It was conjectured by Alon [Alo86] that a random dregular graph is “almost Ramanujan”, in the following sense: for every ε> 0, a.a.s. λ < 2 d − 1 + ε. Friedman famously presented a proof of this conjecture in [Fri08]. Here we suggest a new, substantially simpler proof of a nearlyoptimal result: we show that a random dregular graph satisfies λ < 2 d − 1 + 1 asymptotically almost surely. A main advantage of our approach is that it is applicable to a generalized conjecture: A dregular graph on n vertices is an ncovering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph Ω, we study the spectrum of Γ, a random ncovering of Ω. Let
Zeta Functions of weighted graphs and covering graphs
, 2007
"... We find a condition for weights on the edges of a graph which insures that the Ihara zeta function has a 3term determinant formula. Then we investigate the locations of poles of abelian graph coverings and compare the results with random covers. We discover that the zeta function of the random cove ..."
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Cited by 5 (0 self)
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We find a condition for weights on the edges of a graph which insures that the Ihara zeta function has a 3term determinant formula. Then we investigate the locations of poles of abelian graph coverings and compare the results with random covers. We discover that the zeta function of the random cover satisfies an approximate Riemann hypothesis while that of the abelian cover does not.
On the number of perfect matchings in random lifts
, 2009
"... Let G be a fixed connected multigraph with no loops. A random nlift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the nsets corresponding to the endpoints of the e ..."
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Let G be a fixed connected multigraph with no loops. A random nlift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the nsets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity. Our aim is to prove a concentration result for XG using the small subgraph conditioning method. While we have been unable to prove concentration in general, we present several results including an asymptotic formula for the expectation of XG when G is dregular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Difficulties arise in the calculation of the second moment of XG, where we provide some partial results. Full details are given for two example multigraphs, including the complete graph K4. To assist in our calculations we provide a theorem for estimating summation over multiple dimensions using Laplace’s method. This result is phrased as a summation over lattice points, and may prove useful in future applications. Keywords: random graphs, random multigraphs, random lift, perfect matchings, Laplace’s method.
Zeta Functions and Chaos
, 2009
"... Abstract: The zeta functions of Riemann, Selberg and Ruelle are briefly introduced along with some others. The Ihara zeta function of a finite graph is our main topic. We consider two determinant formulas for the Ihara zeta, the Riemann hypothesis, and connections with random matrix theory and quant ..."
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Abstract: The zeta functions of Riemann, Selberg and Ruelle are briefly introduced along with some others. The Ihara zeta function of a finite graph is our main topic. We consider two determinant formulas for the Ihara zeta, the Riemann hypothesis, and connections with random matrix theory and quantum chaos. 1
Looking into a Graph Theory Mirror of Number Theoretic Zetas
, 2009
"... We survey the Ihara zeta function for irregular graphs, emphasizing connections with number theory zetas and open problems. ..."
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We survey the Ihara zeta function for irregular graphs, emphasizing connections with number theory zetas and open problems.
CUTOFF ON ALL RAMANUJAN GRAPHS
"... Abstract. We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the totalvariation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s logn is asymptotically P(Z> c s) where Z is a standard normal variab ..."
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Abstract. We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the totalvariation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s logn is asymptotically P(Z> c s) where Z is a standard normal variable and c = c(d) is an explicit constant. Furthermore, for all 1 ≤ p ≤ ∞, dregular Ramanujan graphs minimize the asymptotic Lpmixing time for SRW among all dregular graphs. Our proof also shows that, for every vertex x in G as above, its distance from n − o(n) of the vertices is asymptotically logd−1 n. 1.