Abstract
We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.
Highlights
The object of this paper is to prove a result concerning the chemical distance inside large open clusters in critical independent bond percolation on Z2
Distances inside the infinite cluster in supercritical percolation are known to be comparable to the Euclidean distance on Zd, through the work of G
We present our result on the chemical distance in terms of circuits in annuli
Summary
The object of this paper is to prove a result concerning the chemical distance inside large open clusters in critical independent bond percolation on Z2. Chemical distance from the work of Kozma and Nachmias (see [11, Theorem 2.8] for a more general result, which applies to long-range percolation). These estimates presumably hold for any dimension above the critical dimension d = 6, but the current proofs rely on results derived from the lace expansion. Theorem 2.2, is the following: as n → ∞, ESn n2π3(n) This shows that in an averaged sense, Sn is much shorter than the typical size of Ln. The formulation (1.4) in terms of circuits in annuli serves as an illustration of the fractal nature of percolation clusters.
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