Cover times for Brownian motion and random walks in two dimensions

Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer

doi:10.4007/annals.2004.160.433

Cited by 161 publications

(218 citation statements)

References 31 publications

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“…Next we state a result regarding the mean hitting time of a subset of cells for a random walk on the √ n × √ n discrete torus. It is a consequence of Lemma 2.1 in [4] and the strong approximation (of random walk by Brownian motion) results as used in the proof of Theorem 1.1 in [4].…”

Section: Optimality Of Schemementioning

confidence: 99%

Optimal throughput-delay scaling in wireless networks - part I: the fluid model

Gamal

Mammen

Prabhakar

et al. 2006

IEEE Trans. Inform. Theory

319

451

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Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per source-destination pair is Θ 1/ √ n log n . Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per source-destination pair.In most applications delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this trade-off for wireless networks in a general framework. Optimal throughput-delay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughput-delay trade-off is given by D(n) = Θ(nT (n)), where T (n) and D(n) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of Θ(1) for the Grossglauser-Tse scheme is given and the associated delay scaling is shown to be Θ(n log n). The optimal throughput-delay trade-off for mobile networks is also established. To capture physical movement in the real world, a random walk model for node mobility is assumed. It is shown that for throughput of O 1/ √ n log n , which can also be achieved in static networks, the throughput-delay trade-off is the same as in static networks, i.e., D(n) = Θ(nT (n)). Surprisingly, for almost any throughput of a higher order, the delay is shown to be Θ(n log n), which is the delay for throughput of Θ(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in real-world networks would necessitate an abrupt and very large increase in delay.

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Section: Optimality Of Schemementioning

confidence: 99%

Optimal throughput-delay scaling in wireless networks - part I: the fluid model

Gamal

Mammen

Prabhakar

et al. 2006

IEEE Trans. Inform. Theory

319

451

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show abstract

“…Relaxation requires that a fraction of the sites of the ball is covered by the active group which is essentially a random walker with rate q. Classic results on random walks [Ald83,DPRZ04] imply that this requires a time (up to log corrections) r d times the inverse of the diffusion rate of the walker, which indeed yields τ ∼ 1/q k+1 .…”

Section: Fa1f and Other Non Cooperative Modelsmentioning

confidence: 99%

Is there a fractional breakdown of the Stokes-Einstein relation in kinetically constrained models at low temperature?

Blondel¹,

Toninelli²

2014

EPL

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We study the motion of a tracer particle injected in facilitated models which are used to model supercooled liquids in the vicinity of the glass transition. We consider the East model, FA1f model and a more general class of non-cooperative models. For East previous works had identified a fractional violation of the Stokes-Einstein relation with a decoupling between diffusion and viscosity of the form D ∼ τ −ξ with ξ ∼ 0.73. We present rigorous results proving that instead log(D) = − log(τ )+O(log(1/q)), which implies at leading order log(D)/ log(τ ) ∼ −1 for very large time-scales. Our results do not exclude the possibility of SE breakdown, albeit non fractional. Indeed extended numerical simulations by other authors show the occurrence of this violation and our result suggests Dτ ∼ 1/q α , where q is the density of excitations. For FA1f we prove fractional Stokes Einstein in dimension 1, and D ∼ τ −1 in dimension 2 and higher, confirming previous works. Our results extend to a larger class of non-cooperative models.

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“…Given κ 6, let G i be the graph with vertex set V ( (13), it is possible to check that there exists a constant c 1 such that μ G (B d (x, r) ∩ B d (0, κi)) c 1 r α (18) for every x ∈ V (G i ), r ∈ [1, 2κi], i 1. Indeed, for balls such that B d (x, r) ⊆ B d (0, κi) (which includes the case r = 1) this is obvious.…”

Section: Infinite Graphsmentioning

confidence: 99%

“…Indeed, recent years have seen the order of growth of the cover time computed for various families of graphs [1,8], and a strong connection has been made between the cover time and the maximum of the Gaussian free field for any graph [20]. Moreover, in some special cases where there is concentration of the cover time about its mean, extremely precise distributional convergence results are known, notably for the two-dimensional discrete torus [18,19]. Partly motivated by providing techniques for studying the cover time in settings where there is not concentration of the cover time, as is the case for many self-similar fractals, in this article we study the continuity properties of local times on graphs.…”

Section: Introductionmentioning

confidence: 99%

Moduli of continuity of local times of random walks on graphs in terms of the resistance metric

Croydon

2015

Transactions of the London Mathematical Society

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In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self-similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.

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Cover times for Brownian motion and random walks in two dimensions

Cited by 161 publications

References 31 publications

Optimal throughput-delay scaling in wireless networks - part I: the fluid model

Optimal throughput-delay scaling in wireless networks - part I: the fluid model

Is there a fractional breakdown of the Stokes-Einstein relation in kinetically constrained models at low temperature?

Moduli of continuity of local times of random walks on graphs in terms of the resistance metric

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