Asymmetry of near-critical percolation interfaces

Nolin, Pierre¹ ; Werner, Wendelin¹

¹ DMA, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France, and Laboratoire de Mathématiques, Bât. 425, Université Paris-Sud 11, 91405 Orsay Cedex, France

Journal of the American Mathematical Society, Tome 22 (2009) no. 3, pp. 797-819

Résumé

We study the possible scaling limits of percolation interfaces in two dimensions on the triangular lattice. When one lets the percolation parameter $p(N)$ vary with the size $N$ of the box that one is considering, three possibilities arise in the large-scale limit. It is known that when $p(N)$ does not converge to $1/2$ fast enough, then the scaling limits are degenerate, whereas if $p(N) - 1 / 2$ goes to zero quickly, the scaling limits are SLE(6) as when $p=1/2$. We study some properties of the (non-void) intermediate regime where the large scale behavior is neither SLE(6) nor degenerate. We prove that in this case, the law of any scaling limit is singular with respect to that of SLE(6), even if it is still supported on the set of curves with Hausdorff dimension equal to $7/4$.

Détail

DOI : 10.1090/S0894-0347-08-00619-X

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