Arm exponents in high dimensional percolation

Kozma, Gady¹ ; Nachmias, Asaf²

¹ The Weizmann Institute of Science, Rehovot POB 76100, Israel
² Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Journal of the American Mathematical Society, Tome 24 (2011) no. 2, pp. 375-409

Résumé

We study the probability that the origin is connected to the sphere of radius $r$ (an arm event) in critical percolation in high dimensions, namely when the dimension $d$ is large enough or when $d>6$ and the lattice is sufficiently spread out. We prove that this probability decays like $r^{-2}$. Furthermore, we show that the probability of having $\ell$ disjoint arms to distance $r$ emanating from the vicinity of the origin is $r^{-2\ell }$.

Détail

DOI : 10.1090/S0894-0347-2010-00684-4

Bibliographie

[1] Aizenman, Michael On the number of incipient spanning clusters Nuclear Phys. B 1997 551 582

[2] Aizenman, Michael, Barsky, David J. Sharpness of the phase transition in percolation models Comm. Math. Phys. 1987 489 526

[3] Aizenman, Michael, Newman, Charles M. Tree graph inequalities and critical behavior in percolation models J. Statist. Phys. 1984 107 143

[4] Athreya, Krishna B., Ney, Peter E. Branching processes 1972

[5] Barsky, D. J., Aizenman, M. Percolation critical exponents under the triangle condition Ann. Probab. 1991 1520 1536

[6] Van Den Berg, J., Fiebig, U. On a combinatorial conjecture concerning disjoint occurrences of events Ann. Probab. 1987 354 374

[7] Van Den Berg, J., Kesten, H. Inequalities with applications to percolation and reliability J. Appl. Probab. 1985 556 569

[8] Bollobã¡S, Bã©La, Riordan, Oliver Percolation 2006

[9] Borgs, C., Chayes, J. T., Randall, D. The van den Berg-Kesten-Reimer inequality: a review 1999 159 173

[10] Brydges, David, Spencer, Thomas Self-avoiding walk in 5 or more dimensions Comm. Math. Phys. 1985 125 148

[11] Burton, R. M., Keane, M. Density and uniqueness in percolation Comm. Math. Phys. 1989 501 505

[12] Chayes, J. T., Chayes, L. On the upper critical dimension of Bernoulli percolation Comm. Math. Phys. 1987 27 48

[13] Durrett, Richard Probability: theory and examples 1996

[14] Erdå‘S, P., Rã©Nyi, A. On the evolution of random graphs Magyar Tud. Akad. Mat. Kutató Int. Közl. 1960 17 61

[15] Grimmett, Geoffrey Percolation 1999

[16] Hara, Takashi Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals Ann. Probab. 2008 530 593

[17] Hara, Takashi, Van Der Hofstad, Remco, Slade, Gordon Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models Ann. Probab. 2003 349 408

[18] Hara, Takashi, Slade, Gordon Mean-field critical behaviour for percolation in high dimensions Comm. Math. Phys. 1990 333 391

[19] Hara, Takashi, Slade, Gordon The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents J. Statist. Phys. 2000 1075 1168

[20] Harris, T. E. A lower bound for the critical probability in a certain percolation process Proc. Cambridge Philos. Soc. 1960 13 20

[21] Heydenreich, Markus, Van Der Hofstad, Remco, Sakai, Akira Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk J. Stat. Phys. 2008 1001 1049

[22] Van Der Hofstad, Remco, Den Hollander, Frank, Slade, Gordon Construction of the incipient infinite cluster for spread-out oriented percolation above 4+1 dimensions Comm. Math. Phys. 2002 435 461

[23] Van Der Hofstad, Remco, Den Hollander, Frank, Slade, Gordon The survival probability for critical spread-out oriented percolation above 4+1 dimensions. I. Induction Probab. Theory Related Fields 2007 363 389

[24] Van Der Hofstad, Remco, Den Hollander, Frank, Slade, Gordon The survival probability for critical spread-out oriented percolation above 4+1 dimensions. II. Expansion Ann. Inst. H. Poincaré Probab. Statist. 2007 509 570

[25] Kesten, Harry The critical probability of bond percolation on the square lattice equals 1\over2 Comm. Math. Phys. 1980 41 59

[26] Kesten, Harry Percolation theory for mathematicians 1982

[27] Kozma, Gady, Nachmias, Asaf The Alexander-Orbach conjecture holds in high dimensions Invent. Math. 2009 635 654

[28] Kozã¡Kovã¡, Iva Critical percolation of free product of groups Internat. J. Algebra Comput. 2008 683 704

[29] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin One-arm exponent for critical 2D percolation Electron. J. Probab. 2002

[30] Men′Shikov, M. V. Coincidence of critical points in percolation problems Dokl. Akad. Nauk SSSR 1986 1308 1311

[31] Nguyen, Bao Gia Gap exponents for percolation processes with triangle condition J. Statist. Phys. 1987 235 243

[32] Reimer, David Proof of the van den Berg-Kesten conjecture Combin. Probab. Comput. 2000 27 32

[33] Sakai, Akira Mean-field behavior for the survival probability and the percolation point-to-surface connectivity J. Statist. Phys. 2004 111 130

[34] Schonmann, Roberto H. Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs Comm. Math. Phys. 2001 271 322

[35] Slade, G. The lace expansion and its applications 2006

[36] Smirnov, Stanislav, Werner, Wendelin Critical exponents for two-dimensional percolation Math. Res. Lett. 2001 729 744