Shape curvatures and transversal fluctuations in the first passage percolation model
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چکیده
We consider the first passage percolation model on Z2. In this model, {t(e) : e an edge of Z2} is an independent identically distributed family with a common distribution F . We denote by T (0, v) the passage time from the origin to v for v ∈ R2 and B(t) = {v ∈ Rd : T (0, v) ≤ t}. It is well known that if F (0) < pc, there exists a compact shape BF ⊂ R2 such that for all ǫ > 0, tBF (1 − ǫ) ⊂ B(t) ⊂ tBF (1 + ǫ), eventually with a probability 1. For each shape boundary point u, we denote its rightand leftcurvature exponents by κ+(u) and κ−(u). In addition, for each vector u, we denote the transversal fluctuation exponent by ξ(u). In this paper, we can show that ξ(u) ≤ 1−max{κ−(u)/2, κ+(u)/2} for all shape boundary points u. To pursue a curvature on BF , we consider passage times with a special distribution infsupp(F ) = l and F (l) = p > ~pc, where l is a positive number and ~pc is a critical point for the oriented percolation model. With this distribution, it is known that there is a flat segment on the shape boundary between angles 0 < θ− p < θ + p < 90 ◦. In this paper, we show that the shape are strictly convex at the directions θ± p . Moreover, we also show that for all r > 0, ξ((r, θ± p )) = 0.5 and ξ((r, θ)) = 1 for all θ − p < θ < θ + p and r > 0. Note that this rules out the conjecture that ξ(u) = 2/3 for all u. In addition, if χ(u) is the longitudinal exponent, it is believed that χ(u) = 2ξ(u) − 1. Furthermore, it is estimated that χ(u) ≥ (1 − ξ(u))/2 for F (0) < pc and infsupp(F ) = 0. However, both the equation and the inequality do not hold for our special distribution when θ = θ± p .
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تاریخ انتشار 2007