Universality in the lattice-covering time problem

Nemirovsky, A. M.; Mártin, H. O.; Coutinho‐Filho, M. D.

doi:10.1103/physreva.41.761

Cited by 38 publications

(30 citation statements)

References 11 publications

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“…(1) depends on lattice type and space dimension. It is the inverse of the mean number B d of visits of a random walker to its initial position [34][35][36] (see also chapter 3 in Ref. [37]).…”

Section: Statistics Of Sites Not Visited By a Random Walkmentioning

confidence: 99%

Percolation of sites not removed by a random walker inddimensions

Kantor

Kardar

2019

Phys. Rev. E

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How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of N = uL d steps on a d-dimensional hypercubic lattice of size L d (with periodic boundaries). We systematically explore dependence of the probability Π d (L, u) of percolation (existence of a spanning cluster) of sites not removed by the RW on L and u. The concentration of unvisited sites decays exponentially with increasing u, while the visited sites are highly correlated -their correlations decaying with the distance r as 1/r d−2 (in d > 2). On increasing L, the percolation probability Π d (L, u) approaches a step function, jumping from 1 to 0 when u crosses a percolation threshold uc that is close to 3 for all 3 ≤ d ≤ 6. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with ν = 2/(d − 2). There is no percolation threshold at the lower critical dimension of d = 2, with the percolation probability approaching a smooth function Π2(∞, u) > 0. PACS numbers: 05.70.Jk 05.40.Fb 68.35.Rh 36.20.Ey

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Section: Statistics Of Sites Not Visited By a Random Walkmentioning

confidence: 99%

Percolation of sites not removed by a random walker inddimensions

Kantor

Kardar

2019

Phys. Rev. E

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“…This indicates that possible corrections to τ * (G N ) must go in the direction of decreasing its value. Since previous results in the literature suggest just the opposite, i.e., that, if anything, τ * (G N ) may be missing a (1 + c/ ln N ) correction with c > 0 [5,6,9], we are led to believe that the constant C d with d = d max is overshooting. Supplementary evidence comes from the behavior of the Union Jack lattice with respect to d. Although the bound (1) requires d d max (G N ), our data suggest that this requirement is probably not optimal.…”

Section: Monte Carlo Datamentioning

confidence: 80%

“…We choose the beta distribution because it has finite support and can take several different shapes; moreover, beta-like PDFs for the cover times of some special graphs were found in previous investigations [5,6,21]. The empirical data together with the adjusted beta PDF appear in Fig.…”

Section: The Empirical Probability Distribution Function Of the mentioning

confidence: 99%

“…The functional form τ (G N ) ∼ N (ln N ) 2 for the cover time of the square lattice had been guessed earlier on the basis of Monte Carlo simulations and scaling analysis, where a multiplicative correction (1 + c/ ln N ) to this form was detected, with c (the magnitude of the leading scaling correction) a constant depending on the boundary conditions of the finite graphs [5,6,9]. Following a sophisticated probabilistic-geometric analysis of the cover time of the square * jricardo@if.usp.br lattice, Dembo et al [19] conjectured that for the honeycomb, square, and triangular lattices, corresponding respectively to d = d max = 3, 4, and 6, the constants C d appearing in Eq.…”

Section: Introductionmentioning

confidence: 99%

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Numerical evidence against a conjecture on the cover time of planar graphs

Mendonça

2011

Phys. Rev. E

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We investigate a conjecture on the cover times of planar graphs by means of large Monte Carlo simulations. The conjecture states that the cover time τ(G(N)) of a planar graph G(N) of N vertices and maximal degree d is lower bounded by τ(G(N))≥C(d)N(lnN)2 with C(d)=(d/4π)tan(π/d), with equality holding for some geometries. We tested this conjecture on the regular honeycomb (d=3), regular square (d=4), regular elongated triangular (d=5), and regular triangular (d=6) lattices, as well as on the nonregular Union Jack lattice (dmin=4, dmax=8). Indeed, the Monte Carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices, with an interesting issue in the case of the Union Jack lattice. The data for the honeycomb lattice, however, violate the bound with the conjectured constant. The empirical probability distribution function of the cover time for the square lattice is also briefly presented, since very little is known about cover time probability distribution functions in general.

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“…The constant for bounded 2-d lattices is A R = 0.44 according to [22]. By investigating the point process of placing M ants (points) on a √ N × √ N lattice we obtain the expected maximal distance between an ant and a lattice site.…”

Section: A Stoßzahlansatzmentioning

confidence: 99%

Thermodynamics of emergence: Langton's ant meets Boltzmann

Hamann

Schmickl

2011

2011 IEEE Symposium on Artificial Life (ALIFE)

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Abstract-The current definitions of emergence have no effects in the context of artificial life that could convincingly be called 'constructive'. They are rather descriptive labels or tests. In order to get towards recipes of generating emergence we need to know systemic characteristics that help during the design phase of artificial life systems and worlds. In this paper, we develop and discuss five hypotheses that are not meant to be irrevocable but rather thought-provoking. We introduce two modeling approaches for Langton's ant to clarify these hypotheses. Then we discuss general properties of systems, such as (ir-)reversibility, dependence on initial states, computation, discreetness, and undecidable properties of system states.

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Universality in the lattice-covering time problem

Cited by 38 publications

References 11 publications

Percolation of sites not removed by a random walker inddimensions

Percolation of sites not removed by a random walker inddimensions

Numerical evidence against a conjecture on the cover time of planar graphs

Thermodynamics of emergence: Langton's ant meets Boltzmann

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