Wiener sausage volume moments

Berezhkovskii, Alexander M.; Makhnovskii, Yu. A.; Suris, R. A.

doi:10.1007/bf01023647

Cited by 85 publications

(76 citation statements)

References 6 publications

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“…Historically this continuum problem precedes its lattice counterpart. It was considered as early as 1933 by Leontovitsh and Kolmogorov [16] and is still today an active subject of investigation in probability theory ( [17]; see [18] for a recent overwiew) and in mathematical physics [19].…”

Section: Introductionmentioning

confidence: 99%

Statistical properties of the set of sites visited by the two-dimensional random walk

Wijland¹,

Caser²,

Hilhorst³

1997

J. Phys. A: Math. Gen.

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Abstract. We study the support (i.e. the set of visited sites) of a t-step random walk on a two-dimensional square lattice in the large t limit. A broad class of global properties, M(t), of the support is considered, including for example the number, S(t), of its sites; the length of its boundary; the number of islands of unvisited sites that it encloses; the number of such islands of given shape, size, and orientation; and the number of occurrences in space of specific local patterns of visited and unvisited sites. On a finite lattice we determine the scaling functions that describe the averages, M(t), on appropriate lattice size-dependent time scales. On an infinite lattice we first observe that the M(t) all increase with t as ∼ t/ log k t, where k is an M-dependent positive integer. We then consider the class of random processes constituted by the fluctuations around average M(t). We show that, to leading order as t gets large, these fluctuations are all proportional to a single universal random process, η(t), normalized to η 2 (t) = 1. For t → ∞ the probability law of η(t) tends to that of Varadhan's renormalized local time of self-intersections. An implication is that in the long time limit all M(t) are proportional to S(t).

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Section: Introductionmentioning

confidence: 99%

Statistical properties of the set of sites visited by the two-dimensional random walk

Wijland¹,

Caser²,

Hilhorst³

1997

J. Phys. A: Math. Gen.

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“…In this regard, the realization-averaged probability P mon (t) can be thought off as the moment generating function of the volume of the Wiener sausage (see [28] for more details). We also hasten to remark that Eq.…”

Section: A Reminder On the Monomer Trapping Problem Inmentioning

confidence: 99%

Kinetics of anchoring of polymer chains on substrates with chemically active sites

et al. 1998

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We consider dynamics of an isolated polymer chain with a chemically active end-bead on a 2D solid substrate containing immobile, randomly placed chemically active sites (traps). For a particular situation when the end-bead can be irreversibly trapped by any of these sites, which results in a complete anchoring of the whole chain, we calculate the time evolution of the probability P ch (t) that the initially nonanchored chain remains mobile until time t. We find that for relatively short chains P ch (t) follows at intermediate times a standard-form 2D Smoluchowski-type decay law lnP ch (t) ∼ −t/ln(t), which crosses over at very large times to the fluctuationinduced dependence lnP ch (t) ∼ −t 1/2 , associated with fluctuations in the spatial distribution of traps. We show next that for long chains the kinetic behavior is quite different; here the intermediate-time decay is of the form lnP ch (t) ∼ −t 1/2 , which is the Smoluchowski-type law associated with subdiffusive motion of the endbead, while the long-time fluctuation-induced decay is described by the dependence lnP ch (t) ∼ −t 1/4 , stemming out of the interplay between fluctuations in traps distribution and internal relaxations of the chain.

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“…This volume in turn is known quite precisely [33,4]. The coverage problem was suggested as an open problem by Konstantopoulos [24].…”

Section: Motivation and Related Workmentioning

confidence: 99%

“…So, finally, using the asymptotic expression for the expected volume of the Wiener sausage in two dimensions [4,33]…”

Section: R) Be a Wiener Sausage With Drift G Up To Time Tmentioning

confidence: 99%

Mobile Geometric Graphs: Detection, Coverage and Percolation

Peres¹,

Sinclair²,

Sousi³

et al. 2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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Static wireless networks are by now quite well understood mathematically through the random geometric graph model. By contrast, there are relatively few rigorous results on the practically important case of mobile networks. In this paper we consider a natural extension of the random geometric graph model to the mobile setting by allowing nodes to move in space according to Brownian motion. We study three fundamental questions in this model: detection (the time until a given target point-which may be either fixed or moving-is detected by the network), coverage (the time until all points inside a finite box are detected by the network), and percolation (the time until a given node is able to communicate with the giant component of the network). We derive precise asymptotics for these problems by combining ideas from stochastic geometry, coupling and multi-scale analysis. We also give an application of our results to analyze the time to broadcast a message in a mobile network.

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Wiener sausage volume moments

Cited by 85 publications

References 6 publications

Statistical properties of the set of sites visited by the two-dimensional random walk

Statistical properties of the set of sites visited by the two-dimensional random walk

Kinetics of anchoring of polymer chains on substrates with chemically active sites

Mobile Geometric Graphs: Detection, Coverage and Percolation

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