We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in article amsmath,amsfonts empty . We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x − y|−s+o(1) when |x − y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)Δ+o(1) where Δ−1 := log2(2d s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp r1 Δ+o(1) in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210-227, 2011 (Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.)
Further motivation comes from the recent interest in diffusive properties of graphs arising via long-range percolation. An early work in this respect was that of Berger @cite_7 who characterized regimes of recurrence and transience for the simple random walk on such graphs. Benjamini, Berger and Yadin @cite_11 later showed that the mixing time @math of the random walk on @math in @math scales like with an apparent jump in the exponent when @math passes through 2. Misumi @cite_5 found estimates on the effective resistance in @math that exhibit a similar transition.
@cite_5 @cite_7 @cite_11
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In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any @math and for any exponent @math giving the rate of decay of the percolation process, we show that the return probability decays like @math up to logarithmic corrections, where @math denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest, the main result is needed for a companion paper studying the scaling limit of simple random walk on the infinite cluster. Let @math be the largest component of the Erdős–Renyi random graph @math . The mixing time of random walk on @math in the strictly supercritical regime, p = c n with fixed c > 1, was shown to have order log2n by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p = (1 + e) n where λ = e3n is bounded, Nachmias and Peres proved that the mixing time on @math is of order n. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of @math in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim. In this paper, we show that for p = (1 + e) n with λ = e3n → ∞ and λ = o(n), the mixing time on @math is with high probability of order (n λ)log2λ. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., p = (1 − e) n with λ as above]. 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. We consider the simple exclusion process with @math particles on a segment of length @math performing random walks with transition @math to the right and @math to the left. We focus on the case where the asymmetry in the jump rates @math vanishes in the limit when @math and @math tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than @math . We show that in the former case ( @math ), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e. constant @math ) case studied in [LL18], while the latter case ( @math ) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in [Lac16b]. In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order @math .