Agoston Pisztora

Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh PA 15213, USA

Scaling Inequalities for Shortest Paths in Regular and Invasion Percolation

Abstract: For $x \in \boldmath {Z}^d$ , we consider the length S(0,nx) of a shortest open path (chemical distance) in supercritical (Bernoulli) percolation between 0 and nx. We show that if the subcritical correlation length satisfies $\xi(p)\le C(p-p_c)^{-{\nu}}$ for some ${\nu}< 1$ then, wi th probability tending to 1 exponentially fast, for every p>p_c sufficiently close to p_c, $S(0,nx)/\Vert nx \Vert _2 \ge (p-p_c)^{-(1-{\nu})}/\phi((p-p_c)^{-1})$ , where $\phi$ has logarithmic growth. The estimate is uniform in $x \in \boldmath {Z}^d$ and our lower bound $1-{\nu}$ for the ``true'' exponent is presumably optimal in the mean-field regime. The assumption on the correlation length is known to hold in high dimensions [Hara, 1990] and there is numerical evidence that it holds for $d\ge 3$ (but it can not hold in d=2, [Kesten, 1987].) Using a partly different approach we prove a similar theorem in two dimensions. In this case no extra assumptions are required. As an essential ingredient of our proofs, finite-size scaling estimates for the chemical distance are derived by perturbing corresponding estimates for critical systems, [Kesten & Zhang, 1993] and [Aizenman & Burchard, 1999]. In the second part of the paper we consider invasion percolation in $d\ge 2$ and establish power-law lower bounds on the length of a shortest path $S_{\rm inv}(0,x)$ between the the origin and an invaded site in distance n within the invasion cluster. Under similar assumptions as before, we show that with high probability, $S_{\rm inv}(0,x)$ is at least of order $\Vert\hspace{0.1em} x \Vert ^{1+\delta}$ for some $\delta>0$ , indicating thereby the fractal nature (geometry) of the invasion cluster. In the course of the proof we derive bounds on the probability that the invasion cluster intersects $C_\infty(p)$ before leaving the box of size n centered at 0 for some appropriate p=p(n).

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