We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time -the expected time required to visit every node in a graph at least once -and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probablistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t-connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds. *
Abstract. We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d > 6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension, that is, pt(x, x) = t −2/3+o(1) . This establishes a conjecture of Alexander and Orbach [4]. En route we calculate the one-arm exponent with respect to the intrinsic distance.
We consider the nearest-neighbor simple random walk on Z d , d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy > 0 exceeds the threshold for bond percolation on Z d . For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2n-step return probability P 2n ω (0, 0). We prove that P 2n ω (0, 0) is bounded by a random constant times n −d/2 in d = 2, 3, while it is o(n −2 ) in d ≥ 5 and O(n −2 log n) in d = 4. By producing examples with anomalous heat-kernel decay approaching 1/n 2 we prove that the o(n −2 ) bound in d ≥ 5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d = 4.
We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.
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