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  • 1
    Electronic Resource
    Electronic Resource
    Some remarks on AB-percolation in high dimensions (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 1298-1320 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper we consider the AB-percolation model on Z+d and Zd. Let pHalt(Zd) be the critical probability for AB-percolation on Zd. We show that pHalt(Zd)∼1/(2d2). If the probability of a site to be in state A is γ/(2d2) for some fixed γ〉1, then the probability that AB-percolation occurs converges as d→∞ to the unique strictly positive solution y(γ) of the equation y=1−exp(−γy). We also find the limit for the analogous quantities for oriented AB-percolation on Z+d. In particular, pHalt(Z+d)∼2/d2. We further obtain a small extension to the two parameter problem in which even vertices of Zd have probability pA of being in state A and odd vertices have probability pB of being in state B (but without relation between pA and pB). The principal tools in the proofs are a method of Penrose (1993) for asymptotics of percolation on graphs with vertices of high degree and the second moment method. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533188
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  • 2
    Electronic Resource
    Electronic Resource
    The critical probability of bond percolation on the square lattice equals 1/2 (1980)
    Springer
    Communications in mathematical physics 74 (1980), S. 41-59 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We prove the statement in the title of the paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01197577
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  • 3
    Electronic Resource
    Electronic Resource
    Scaling relations for 2D-percolation (1987)
    Springer
    Communications in mathematical physics 109 (1987), S. 109-156 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We prove that the relations 2D-percolation hold for the usual critical exponents for 2D-percolation, provided the exponents δ andv exist. Even without the last assumption various relations (inequalities) are obtained for the singular behavior near the critical point of the correlation length, the percolation probability, and the average cluster size. We show that in our models the above critical exponents have the same value for approach ofp to the critical probability from above and from below.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01205674
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  • 4
    Electronic Resource
    Electronic Resource
    The number of distinguishable alleles according to the Ohta-Kimura model of neutral mutation (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 167-187 
    Details
    ISSN: 1432-1416
    Keywords: Ohta-Kimura model ; Neutral mutations ; Number of distinguishable alleles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We calculate how many alleles one can expect to distinguish in a large sample from a large population which develops according to the Ohta-Kimura model. This number tends to infinity with the sample size, but so slowly that it is bounded for all practical purposes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275840
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  • 5
    Electronic Resource
    Electronic Resource
    Convergence in distribution of products of random matrices (1984)
    Springer
    Probability theory and related fields 67 (1984), S. 363-386 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider a sequence A 2, A 2, ... of i.i.d. nonnegative matrices of size d × d, and investigate convergence in distribution of the product M n: =A 1 ... A n. When d≧2 it is possible for M n to converge in distribution (without normalization) to a distribution not concentrated on the zero matrix. Several equivalent conditions for this to happen are given. These lead to a fairly general family of examples. These conditions can also be used to determine when the a.s. limit of 1/nlog∥M n ∥ equals the logarithm of the largest eigenvalue of E(A 1).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00532045
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  • 6
    Electronic Resource
    Electronic Resource
    Two renewal theorems for general random walks tending to infinity (1996)
    Springer
    Probability theory and related fields 106 (1996), S. 1-38 
    Details
    ISSN: 1432-2064
    Keywords: Mathematics Subject classification (1991): 60K05 ; 60J15 ; 60F15 ; 60G40 ; 60G50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004400050056
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  • 7
    Electronic Resource
    Electronic Resource
    On some growth models with a small parameter (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 435-468 
    Details
    ISSN: 1432-2064
    Keywords: Primary 60K35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models, when a certain parameter becomes small. The basic example treated is the variant of Richardson's growth model on ℤd in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate ɛ≦1 if it has exactly 1 occupied neighbor and, of course, at rate 0 if it has no occupied neighbor. Occupied sites remain occupied forever. Starting from a single occupied site, this model has asymptotic speeds of growth in each direction (as time goes to infinity) and these speeds determine an asymptotic shape in the usual sense. It is proven that as ɛ tends to 0, the asymptotic speeds scale as ɛ1/d and the asymptotic shape, when renormalized by dividing it by ɛ1/d , converges to a cube. Other similar models which are partially oriented are also studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202779
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  • 8
    Electronic Resource
    Electronic Resource
    A central limit theorem for “critical” first-passage percolation in two dimensions (1997)
    Springer
    Probability theory and related fields 107 (1997), S. 137-160 
    Details
    ISSN: 1432-2064
    Keywords: Mathematics Subject Classification (1991): 60K35 ; 60F05 ; 82B43
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Consider (independent) first-passage percolation on the edges of ℤ 2 . Denote the passage time of the edge e in ℤ 2 by t(e), and assume that P{t(e) = 0} = 1/2, P{0〈t(e)〈C 0 } = 0 for some constant C 0 〉0 and that E[t δ (e)]〈∞ for some δ〉4. Denote by b 0,n the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T( 0 ,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0〈C 1 , C 2 〈∞ and γ n such that C 1 ( log n) 1/2 ≦γ n ≦ C 2 ( log n) 1/2 and such that γ n −1 [b 0,n −Eb 0,n ] and (√ 2γ n ) −1 [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed). A similar result holds for the site version of first-passage percolation on ℤ 2 , when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C 0 } = p c (ℤ 2 , site ) := critical probability of site percolation on ℤ 2 , and E[t δ (u)]〈∞ for some δ〉4.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004400050080
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  • 9
    Electronic Resource
    Electronic Resource
    Asymptotic behavior of the critical probability for ρ-percolation in high dimensions (2000)
    Springer
    Probability theory and related fields 117 (2000), S. 419-447 
    Details
    ISSN: 1432-2064
    Keywords: Mathematics Subject Classification (1991): Primary 60K35; Secondary 82B43 ; Key words and phrases:ρ-percolation – Critical probability – Second moment method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. We consider oriented bond or site percolation on ℤ d +. In the case of bond percolation we denote by P p the probability measure on configurations of open and closed bonds which makes all bonds of ℤ d + independent, and for which P p {e is open} = 1 −P p e {is closed} = p for each fixed edge e of ℤ d +. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 〈 ρ≤ 1, if there exists an oriented infinite path v 0 = 0, v 1, v 2, …, starting at the origin, such that lim inf n →∞ (1/n) ∑ i=1 n X(e i ) ≥ρ, where e i is the edge {v i−1 , v i }. [MZ92] showed that there exists a critical probability p c = p c (ρ, d) = p c (ρ, d, bond) such that there is a.s. no ρ-percolation for p 〈 p c and that P p {ρ-percolation occurs} 〉 0 for p 〉 p c . Here we find lim d →∞ d 1/ρ p c (ρd, bond) = D 1 , say. We also find the limit for the analogous quantity for site percolation, that is D 2 = lim d →∞ d 1/ρ p c (ρ, d, site). It turns out that for ρ 〈 1, D 1 〈 D 2 , and neither of these limits equals the analogous limit for the regular d-ary trees.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004400050013
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  • 10
    Electronic Resource
    Electronic Resource
    First-passage percolation, network flows and electrical resistances (1984)
    Springer
    Probability theory and related fields 66 (1984), S. 335-366 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We show that the first-passage times of first-passage percolation on ℤ2 are such that P(θ 0n〈n(μ−ɛ)) and P(θ 0n〉n(μ+ɛ)) decay geometrically as n→∞, where θ may represent any of the four usual first-passage-time processes. The former estimate requires no moment condition on the time coordinates, but there exists a geometrically-decaying estimate for the latter quantity if and only if the time coordinate distribution has finite moment generating function near the origin. Here, μ is the time constant and ɛ〉0. We study the line-to-line first-passage times and describe applications to the maximum network flow through a randomly-capacitated subsection of ℤ2, and to the asymptotic behaviour of the electrical resistance of a subsection of ℤ2 when the edges of the subsection are wires in an electrical network with random resistances. In the latter case we show, for example, that if each edge-resistance equals 1 or ∞ ohms with probabilities p and 1−p respectively, then the effective resistance R n across opposite faces of an n by n box satisfies the following: (a) if p〈1/2 then P(R n=∞)→1 as n→∞, (b) if p〉1/2 then there exists ν(p)〈∞ such that $$P(p^{ - 1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\lim \inf R_n }\limits_{n \to \infty } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\lim \sup R_n }\limits_{n \to \infty } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } v(p)) = 1$$ . There are some corresponding results for certain other two-dimensional lattices, and for higher dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00533701
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