Two-dimensional scaling limits via marked nonsimple loops

Bulletin of the Brazilian Mathematical Society

Volumn 37, Issue 4, 2006, Pages 537-559

  Camia, Federico ^a   Fontes, Luiz Renato G ^b   Newman, Charles M ^c  

^a VU UNIVERSITY AMSTERDAM   (Netherlands)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

^c NEW YORK UNIVERSITY   (United States)

Author keywords

Conformal covariance; Finite size scaling; Minimal spanning tree; Near critical; Off critical; Percolation; Scaling limits

Indexed keywords

EID: 33845800266     PISSN: 16787544     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00574-006-0026-x     Document Type: Article

References (26)

1. M. Aizenman, Scaling limit for the incipient spanning clusters, in Mathematics of Multiscale Materials; the IMA Volumes in Mathematics and its Applications (K. Golden, G. Grimmett, R. James, G. Milton and P. Sen, eds.), Springer (1998).
2. M. Aizenman, Continuum limits for critical percolation and other stochastic geometric models, in XII Int. Cong. Math. Phys. (ICMP '97, Brisbane), Internat. Press, Cambridge, MA (1999).
3. M. Aizenman and A. Burchard, Hölder regularity and dimension bounds for random curves, Duke Math. J., 99 (1999), 419-453.
4. M. Aizenman, A. Burchard, C.M. Newman, D. Wilson, Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions, Random Struct. Alg., 15 (1999), 319-367.
5. K.S. Alexander, Percolation and Minimal Spanning Forests in Infinite Graphs, Ann. Probab., 23 (1995), 87-104.
6. C. Borgs, J. Chayes, H. Kesten, J. Spencer, The Birth of the Infinite Cluster: Finite-Size Scaling in Percolation, Comm. Math. Phys., 224 (2001), 153-204.
7. F. Camia, L.R.G. Fontes, C.M. Newman, The Scaling Limit Geometry of NearCritical 2D Percolation, J. Stat. Phys., to appear (2006).
8. F. Camia and C.M. Newman, Continuum Nonsimple Loops and 2D Critical Percolation, J. Stat. Phys., 116 (2004), 157-173.
9. F. Camia and C.M. Newman, The full scaling limit of two-dimensional critical percolation, available at arXiv:math.PR/0504036 (2005), including a main part corresponding to [10] and along appendix roughly corresponding to [11].
10. F. Camia and C.M. Newman, Two-Dimensional Critical Percolation: The Full Scaling Limit, Comm. Math. Phys., 268 (2006), 1-38.
11. 6: a Proof of Convergence, Probab. Theory Related Fields, to appear, available at arXiv:math.PR/0604487.
12. J.L. Cardy, Critical percolation in finite geometries, J. Phys. A, 25 (1992), L201-L206.
13. J.T. Chayes, L. Chayes and C.M. Newman, The Stochastic Geometry of Invasion Percolation, Comm. Math. Phys., 101 (1985), 383-407.
14. L.R.G. Fontes, M. Isopi, C.M. Newman and K. Ravishankar, Coarsening, Nucleation, and the Marked Brownian Web, Ann. Inst. H. Poincaré (B), Prob, and Stat., 42 (2006), 37-60.
15. O. Häggström, Y. Peres and J.E. Steif, Dynamical percolation, Ann. Inst. H. Poincaré Probab. Statist. 33, (1997), 497-528.
16. G.J. Morrow and Y. Zhang, The sizes of the pioneering, lowest crossing and pivotal sites in critical percolation on the triangular lattice, Ann. Appl. Probab., 15 (2005), 1832-1886.
17. C.M. Newman and D.L. Stein, Ground State Structure in a Highly Disordered Spin Glass Model, J. Stat. Phys., 82 (1996), 1113-1132.
18. Y. Peres and J.E. Steif, The number of infinite clusters in dynamical percolation, Probab. Theory Related Fields, 111 (1998), 141-165.
19. N. Read, Minimal spanning trees and random resistor networks in d dimensions, Phys. Rev. E, 72 (2005), 036114-1-036114-17.
20. O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math., 118 (2000), 221-288.
21. O. Schramm, Conformally invariant scaling limits: an overview and a collection of problems, preprint math.PR/0602151 (2006).
22. O. Schramm and J.E. Steif, Quantitative noise sensitivity and exceptional times for percolation, preprint math.PR/0504586 (2005).
23. S. Smirnov, Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits, C.R. Acad. Sci. Paris, 333 (2001), 239-244.
24. S. Smirnov and W. Werner, Critical exponents for two-dimensional percolation, Math. Rev. Lett., 8 (2001), 729-744.
25. D. Wilkinson and J.F. Willemsen, Invasion percolation: A new form of percolation theory, J. Phys. A 16 (1983), 3365-3376.
26. D. Wilson, On the red-green-blue model, Phys. Rev. E 69 (2004), 037105-1-037105-4.