ABSTRACT:In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density 1, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area A c for percolation in this model. The proof is in two parts: First we give a rigorous reduction of the problem to a finite problem, and then we solve this problem using Monte-Carlo methods. We prove that, with 99.99% confidence, the critical area lies between 4.508 and 4.515. For the corresponding problem with the disc replaced by the square we prove, again with 99.99% confidence, that the critical area lies between 4.392 and 4.398.
We prove that there exist natural generalizations of the classical bootstrap percolation model on Z 2 that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property.Van Enter (1987) (in the case d = r = 2) and Schonmann (1992) (for all d r 2) proved that r-neighbour bootstrap percolation models have trivial critical probabilities on Z d for every choice of the parameters d r 2: that is, an initial set of density p almost surely percolates Z d for every p > 0. These results effectively ended the study of bootstrap percolation on infinite lattices.Recently Bollobás, Smith and Uzzell introduced a broad class of percolation models called U-bootstrap percolation, which includes r-neighbour bootstrap percolation as a special case. They divided two-dimensional U-bootstrap percolation models into three classes -subcritical, critical and supercritical -and they proved that, like classical 2-neighbour bootstrap percolation, critical and supercritical U-bootstrap percolation models have trivial critical probabilities on Z 2 . They left open the question as to what happens in the case of subcritical families. In this paper we answer that question: we show that every subcritical U-bootstrap percolation model has a non-trivial critical probability on Z 2 . This is new except for a certain 'degenerate' subclass of symmetric models that can be coupled from below with oriented site percolation. Our results re-open the study of critical probabilities in bootstrap percolation on infinite lattices, and they allow one to ask many questions of subcritical bootstrap percolation models that are typically asked of site or bond percolation.
Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.
-We model the dynamical behavior of the neuropil, the densely interconnected neural tissue in the cortex, using neuropercolation approach.Neuropercolation generalizes phase transitions modeled by percolation theory of random graphs, motivated by properties of neurons and neural populations.The generalization includes (i) a noisy component in the percolation rule, (ii) a novel depression function in addition to the usual arousal function, (iii) non-local interactions among nodes arranged on a multi-dimensional lattice. This paper investigates the role of non-local (axonal) connections in generating and modulating phase transitions of collective activity in the neuropil. We derive a relationship between critical values of the noise level and non-locality parameter to control the onset of phase transitions. Finally, we propose a potential interpretation of ontogenetic development of the neuropil maintaining a dynamical state at the edge of criticality.
An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χ a (G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χ a (G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χ a (G) ≤ Δ(G) + 2 for bipartite graphs. These bounds are tight. For k-chromatic graphs G without isolated edges we prove a weaker result of the form χ a (G) = Δ(G) + O(log k).
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