Hölder regularity and dimension bounds for random curves

Duke Mathematical Journal

Volumn 99, Issue 3, 1999, Pages 419-453

  Aizenman, M ^a   Burchard, A ^b  

^a PRINCETON UNIVERSITY   (United States)

^b University of Virginia   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001056718     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-99-09914-3     Document Type: Article

References (41)

1. M. Aizenman, On the number of incipient spanning clusters, Nuclear Phys. B 485 (1997), 551-582.
2. M. Aizenman, “Scaling limit for the incipient spanning clusters” in Mathematics of Multiscale Materials: Percolation and Composites (Minneapolis, Minn., 1995-1996), IMA Vol. Math. Appl. 99, Springer-Verlag, New York, 1998, 1-24.
3. M. Aizenman, A. Burchard, C. Newman, and D. Wilson, Scaling limits for minimal and random spanning trees in two dimensions, preprint, http://xxx.lanl.gov/ps/math.PR/9809145; to appear in Random Structures and Algorithms.
4. K. Alexander, Percolation and minimal spanning forests in infinite graphs, Ann. Probab. 23 (1995), 87-104.
5. K. Alexander, The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees, Ann. Appl. Probab. 6 (1996), 466-494.
6. P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: An explanation of 1 /f noise, Phys. Rev. Lett. 59 (1987), 381-384.
7. ∼rdlyons/#papers.
8. I. Benjamini and O. Schramm, Conformal invariance and Voronoipercolation, Comm. Math. Phys. 197 (1998), 75-107.
9. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
10. C. J. Bishop, P. W. Jones, R. Pemantle, and Y. Peres, The dimension of the Brownian frontier is greater than 1, J. Funct. Anal. 143 (1997), 309-336.
11. C. Borgs, J. Chayes, H. Kesten, and J. Spencer, Birth of the infinite cluster: Finite size scaling in percolation, to appear in Random Structures and Algorithms.
12. K. Burdzy and G. Lawler, Nonintersection exponents for Brownian paths, II: Estimates and applications to a random fractal, Ann. Probab. 18 (1990), 981-1009.
13. J. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992), L201-L206.
14. J. Cardy, The number of incipient spanning clusters in two-dimensional percolation, preprint, http://xxx.lanl.gov/list/cond-mat/9705137.
15. J. Chayes, L. Chayes, and C. M. Newman, The stochastic geometry of invasion percolation, Comm. Math. Phys. 101 (1985), 383-407.
16. M. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc. 1951, no. 6.
17. S. Dunbar, R. Douglass, and W. Camp, The divider dimension of the graph of a function, J. Math. Anal. Appl. 167 (1992), 403-413.
18. P. Erdős and J. Gillis, Note on the transfinite diameter, J. London Math. Soc. 12 (1937), 185-192.
19. K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math. 85, Cambridge Univ. Press, Cambridge, 1985.
20. P. Flory, The configuration of real polymer chains, J. Chem. Phys. 17 (1949), 303-310.
21. G. Grimmett, Percolation, Springer-Verlag, New York, 1989.
22. O. Häggström, Random-cluster measures and uniform spanning trees, Stochastic Process Appl. 59 (1995), 267-275.
23. H. Kesten, Scaling relations for 2D-percolation, Comm. Math. Phys. 109 (1987), 109-156.
24. H. Kesten and Y. Zhang, The tortuosity of occupied crossings of a box in critical percolation, J. Statist. Phys. 70 (1993), 599-611.
25. J. Kondev and C. Henley, Geometrical exponents of contour loops on random Gaussian surfaces, Phys. Rev. Lett. 74 (1995), 4580-4583.
26. R. Langlands, P. Pouliot, and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. (N.S.) 30 (1994), 1-61.
27. G. Lawler, The dimension of the frontier of planar Brownian motion, Electron. Comm. Probab. 1 (1996), paper 5, 29-47 (electronic).
28. G. Lawler, Hausdorff dimension of cut points for Brownian motion, Electron. J. Probab. 1 (1996) paper 2, 20 pp. (electronic).
29. B. Mandelbrot, The Fractal Geometry of Nature, rev. ed., W. H. Freeman, San Francisco, 1983.
30. C. M. Newman and D. L. Stein, Spin-glass model with dimension-dependent ground state multiplicity, Phys. Rev. Lett. 72 (1994), 2286-2289.
31. C. M. Newman and D. L. Stein, Ground-state structure in a highly disordered spin-glass model, J. Statist. Phys. 82 (1996), 1113-1132.
32. R. Pemantle, Choosing a spanning tree for the integer lattice uniformly, Ann. Probab. 19 (1991), 1559-1574.
33. Y. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl. 1 (1956), 157-214.
34. L. Richardson, “The problem of contiguity” in General Systems: Yearbook 6, Society for the Advancement of General Systems Theory, Ann Arbor, Mich., 1961, 139-187.
35. L. Russo, A note on percolation, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43 (1978), 39-48.
36. H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett. 58 (1987), 2325-2328.
37. P. Seymour and D. Welsh, “Percolation probabilities on the square lattice” in Advances in Graph Theory, Ann. Discrete Math. 3, North-Holland, Amsterdam, 1978, 227-245.
38. H. E. Stanley, Cluster shapes at the percolation threshold: An effective cluster dimensionality and its connection with critical-point exponents, J. Phys. A 10 (1977), L211- L220.
39. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2d ed., Taylor & Francis, London, 1992.
40. J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability, J. Appl. Probab. 22 (1985), 556-569.
41. D. Wilson, “Generating random spanning trees more quickly than the cover time” in Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, Pa., 1996), ACM, New York, 1996, 296-303.