Inequalities for the Time Constant in First-Passage Percolation
نویسندگان
چکیده
منابع مشابه
Variational Formula for the Time - Constant of First - Passage Percolation
We consider first-passage percolation with positive, stationary-ergodic weights on the square lattice Z. Let T (x) be the first-passage time from the origin to a point x in Z. The convergence of the scaled first-passage time T ([nx])/n to the time-constant as n → ∞ can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation. We derive an exact variational ...
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We consider the directed first passage percolation model on Z2. In this model, we assign independently to each edge e a passage time t(e) with a common distribution F . We denote by ~ T (0, (r, θ)) the passage time from the origin to (r, θ) by a northeast path for (r, θ) ∈ R+ × [0, π/2]. It is known that ~ T (0, (r, θ))/r converges to a time constant ~μF (θ). Let ~ pc denote the critical probab...
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We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in rst{passage percolation, as a functional on the space of distribution functions. The present counterexample only works for rst{passage percolation on Z Z d for d large.
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We present improved lower and upper bounds for the time constant of first-passage percolation on the square lattice. For the case of lower bounds, a new method, using the idea of a transition matrix, has been used. Numerical results for the exponential and uniform distributions are presented. A simulation study is included, which results in new estimates and improved upper confidence limits of ...
متن کاملOn a Lower Bound for the Time Constant of First-passage Percolation
We consider the Bernoulli first-passage percolation on Z (d ≥ 2). That is, the edge passage time is taken independently to be 1 with probability 1− p and 0 otherwise. Let μ(p) be the time constant. We prove in this paper that μ(p1)− μ(p2) ≥ μ(p2) 1− p2 (p2 − p1) for all 0 ≤ p1 < p2 < 1 by using Russo’s formula. AMS classification: 60K 35. 82B 43.
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ژورنال
عنوان ژورنال: The Annals of Applied Probability
سال: 1993
ISSN: 1050-5164
DOI: 10.1214/aoap/1177005507