Partial and random lattice covering times in two dimensions

Coutinho, Kaline; Coutinho‐Filho, M. D.; Gomes, M. A. F.; Nemirovsky, A. M.

doi:10.1103/physrevlett.72.3745

Cited by 27 publications

(32 citation statements)

References 15 publications

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“…These authors investigated the evolution of the number of islands, I (t), and several of their properties on finite lattices of up to 1200 2 sites. In a short report [5] we showed that an analytic calculation is possible, both on finite and infinite lattices, for some (not all) of the quantities considered by Coutinho et al [4], and that good agreement between theory and computer simulation is obtained. In this work we present a full account of most of the results announced in [5] and consider a great many related questions.…”

Section: Introductionmentioning

confidence: 84%

“…of E(t) and I (t) being black-and-white observables. In section 4.5 we shall come back to I (t) and also make a comparison with the numerical simulations by Coutinho et al [4].…”

Section: Examplesmentioning

confidence: 97%

“…It is now possible to make a comparison with the simulations by Coutinho et al [4], carried out on finite square lattices of up to N = 1200 2 sites with periodic boundary conditions. These authors were interested in the 'fragmentation' of the finite lattice into islands and the way the average number and size of the islands eventually tend to zero.…”

Section: Examplesmentioning

confidence: 99%

“…Questions about the statistical properties of the islands are natural and our interest in them was raised by a simulation study by Coutinho et al [4]. These authors investigated the evolution of the number of islands, I (t), and several of their properties on finite lattices of up to 1200 2 sites.…”

Section: Introductionmentioning

confidence: 99%

“…Several of the observables, M(t), of interest just mentioned are treated as examples. A comparison is made, where possible, with the simulations by Coutinho et al [4] (ii) Average behaviour and fluctuations on the infinite lattice. On an infinite lattice the fluctuating properties of the support manifest a universality that can be described as follows.…”

Section: Introductionmentioning

confidence: 99%

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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: 84%

“…of E(t) and I (t) being black-and-white observables. In section 4.5 we shall come back to I (t) and also make a comparison with the numerical simulations by Coutinho et al [4].…”

Section: Examplesmentioning

confidence: 97%

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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