Asymmetry of near-critical percolation interfaces

Nolin, Pierre; Werner, Wendelin

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

Asymmetry of near-critical percolation interfaces
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by Pierre Nolin and Wendelin Werner

J. Amer. Math. Soc. 22 (2009), 797-819

DOI: https://doi.org/10.1090/S0894-0347-08-00619-X

Published electronically: September 16, 2008

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Abstract:

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 $7/4$.

References

Michael Aizenman, Scaling limit for the incipient spanning clusters, Mathematics of multiscale materials (Minneapolis, MN, 1995–1996) IMA Vol. Math. Appl., vol. 99, Springer, New York, 1998, pp. 1–24. MR 1635999, DOI 10.1007/978-1-4612-1728-2_{1}
M. Aizenman and A. Burchard, Hölder regularity and dimension bounds for random curves, Duke Math. J. 99 (1999), no. 3, 419–453. MR 1712629, DOI 10.1215/S0012-7094-99-09914-3
C. Borgs, J. T. Chayes, H. Kesten, and J. Spencer, The birth of the infinite cluster: finite-size scaling in percolation, Comm. Math. Phys. 224 (2001), no. 1, 153–204. Dedicated to Joel L. Lebowitz. MR 1868996, DOI 10.1007/s002200100521
Federico Camia and Charles M. Newman, Two-dimensional critical percolation: the full scaling limit, Comm. Math. Phys. 268 (2006), no. 1, 1–38. MR 2249794, DOI 10.1007/s00220-006-0086-1
Federico Camia and Charles M. Newman, Critical percolation exploration path and $\textrm {SLE}_6$: a proof of convergence, Probab. Theory Related Fields 139 (2007), no. 3-4, 473–519. MR 2322705, DOI 10.1007/s00440-006-0049-7
Federico Camia, Luiz Renato G. Fontes, and Charles M. Newman, The scaling limit geometry of near-critical 2D percolation, J. Stat. Phys. 125 (2006), no. 5-6, 1159–1175. MR 2282484, DOI 10.1007/s10955-005-9014-6
Federico Camia, Luiz Renato G. Fontes, and Charles M. Newman, Two-dimensional scaling limits via marked nonsimple loops, Bull. Braz. Math. Soc. (N.S.) 37 (2006), no. 4, 537–559. MR 2284886, DOI 10.1007/s00574-006-0026-x
J. T. Chayes, Finite-size scaling in percolation, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 113–122. MR 1648146
J. T. Chayes, L. Chayes, and J. Fröhlich, The low-temperature behavior of disordered magnets, Comm. Math. Phys. 100 (1985), no. 3, 399–437. MR 802552
J. T. Chayes, L. Chayes, Daniel S. Fisher, and T. Spencer, Finite-size scaling and correlation lengths for disordered systems, Phys. Rev. Lett. 57 (1986), no. 24, 2999–3002. MR 925751, DOI 10.1103/PhysRevLett.57.2999
Geoffrey Grimmett, Percolation, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. MR 1707339, DOI 10.1007/978-3-662-03981-6
Harry Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston, Mass., 1982. MR 692943
Harry Kesten, Scaling relations for $2$D-percolation, Comm. Math. Phys. 109 (1987), no. 1, 109–156. MR 879034
Gregory F. Lawler, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. MR 2129588, DOI 10.1090/surv/114
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237–273. MR 1879850, DOI 10.1007/BF02392618
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237–273. MR 1879850, DOI 10.1007/BF02392618
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, One-arm exponent for critical 2D percolation, Electron. J. Probab. 7 (2002), no. 2, 13. MR 1887622, DOI 10.1214/EJP.v7-101
Gregory Lawler, Oded Schramm, and Wendelin Werner, Conformal restriction: the chordal case, J. Amer. Math. Soc. 16 (2003), no. 4, 917–955. MR 1992830, DOI 10.1090/S0894-0347-03-00430-2

46

Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288. MR 1776084, DOI 10.1007/BF02803524
Stanislav Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244 (English, with English and French summaries). MR 1851632, DOI 10.1016/S0764-4442(01)01991-7
Stanislav Smirnov, Towards conformal invariance of 2D lattice models, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421–1451. MR 2275653
Stanislav Smirnov and Wendelin Werner, Critical exponents for two-dimensional percolation, Math. Res. Lett. 8 (2001), no. 5-6, 729–744. MR 1879816, DOI 10.4310/MRL.2001.v8.n6.a4
Wendelin Werner, Random planar curves and Schramm-Loewner evolutions, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1840, Springer, Berlin, 2004, pp. 107–195. MR 2079672, DOI 10.1007/978-3-540-39982-7_{2}
Wendelin Werner, The conformally invariant measure on self-avoiding loops, J. Amer. Math. Soc. 21 (2008), no. 1, 137–169. MR 2350053, DOI 10.1090/S0894-0347-07-00557-7