## Hölder regularity and dimension bounds for random curves.(English)Zbl 0944.60022

Consider for every $$\delta\in ]0,1]$$ a probability measure $$\gamma_\delta$$ on a closed set of polygonal paths of step-size $$\delta$$, contained in some fixed compact subset of $$\mathbb{R}^d$$. The aim of this work is to yield general criteria giving information on the existence and the support of weak limits of $$\gamma_\delta$$, as $$\delta\downarrow 0$$. The first criterion requires a uniform control of multiple crossings of spherical shells by the paths, and guarantees the existence of some weak limit of $$\gamma_\delta$$, supported by curves having a common upper bound on their Hausdorff dimension. The second criterion requires a uniform control on simultaneous crossings of separated cylinders by the paths, and guarantees a uniform lower bound, strictly larger than 1, for the Hausdorff dimension of the curves supporting any weak limit of $$\gamma_\delta$$. Main examples are given by percolation models and random spanning trees.
Reviewer: J.Franchi (Paris)

### MSC:

 60D05 Geometric probability and stochastic geometry 28A75 Length, area, volume, other geometric measure theory 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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### References:

 [1] Michael Aizenman, On the number of incipient spanning clusters , Nuclear Phys. B 485 (1997), no. 3, 551-582. · Zbl 0925.82112 [2] 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, Percolation and Composites, pp. 1-24. · Zbl 0941.74013 [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. · Zbl 0939.60031 [4] Kenneth S. Alexander, Percolation and minimal spanning forests in infinite graphs , Ann. Probab. 23 (1995), no. 1, 87-104. · Zbl 0827.60079 [5] Kenneth S. Alexander, The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees , Ann. Appl. Probab. 6 (1996), no. 2, 466-494. · Zbl 0855.60009 [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] I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, Uniform spanning forests , preprint, http://php.indiana.edu/ rdlyons/#papers, 1998. · Zbl 1016.60009 [8] Itai Benjamini and Oded Schramm, Conformal invariance of Voronoi percolation , Comm. Math. Phys. 197 (1998), no. 1, 75-107. · Zbl 0921.60081 [9] Patrick Billingsley, Convergence of probability measures , John Wiley & Sons Inc., New York, 1968. · Zbl 0172.21201 [10] Christopher J. Bishop, Peter W. Jones, Robin Pemantle, and Yuval Peres, The dimension of the Brownian frontier is greater than $$1$$ , J. Funct. Anal. 143 (1997), no. 2, 309-336. · Zbl 0870.60077 [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. · Zbl 1038.82035 [12] Krzysztof Burdzy and Gregory F. Lawler, Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal , Ann. Probab. 18 (1990), no. 3, 981-1009. · Zbl 0719.60085 [13] John L. Cardy, Critical percolation in finite geometries , J. Phys. A 25 (1992), no. 4, L201-L206. · Zbl 0965.82501 [14] J. Cardy, The number of incipient spanning clusters in two-dimensional percolation , preprint, http://xxx.lanl.gov/list/cond-mat/9705137. · Zbl 0973.82021 [15] J. T. Chayes, L. Chayes, and C. M. Newman, The stochastic geometry of invasion percolation , Comm. Math. Phys. 101 (1985), no. 3, 383-407. · Zbl 0596.60096 [16] Monroe D. Donsker, An invariance principle for certain probability limit theorems , Mem. Amer. Math. Soc., 1951 (1951), no. 6, 12. · Zbl 0042.37602 [17] Steven R. Dunbar, Rod W. Douglass, and W. J. Camp, The divider dimension of the graph of a function , J. Math. Anal. Appl. 167 (1992), no. 2, 403-413. · Zbl 0756.28004 [18] P. Erdős and J. Gillis, Note on the transfinite diameter , J. London Math. Soc. 12 (1937), 185-192. · Zbl 0017.11505 [19] K. J. Falconer, The geometry of fractal sets , Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. · Zbl 0587.28004 [20] P. Flory, The configuration of real polymer chains , J. Chem. Phys. 17 (1949), 303-310. [21] Geoffrey Grimmett, Percolation , Springer-Verlag, New York, 1989. · Zbl 0691.60089 [22] Olle Häggström, Random-cluster measures and uniform spanning trees , Stochastic Process. Appl. 59 (1995), no. 2, 267-275. · Zbl 0840.60089 [23] Harry Kesten, Scaling relations for $$2$$D-percolation , Comm. Math. Phys. 109 (1987), no. 1, 109-156. · Zbl 0616.60099 [24] Harry Kesten and Yu Zhang, The tortuosity of occupied crossings of a box in critical percolation , J. Statist. Phys. 70 (1993), no. 3-4, 599-611. · Zbl 0943.82542 [25] J. Kondev and C. Henley, Geometrical exponents of contour loops on random Gaussian surfaces , Phys. Rev. Lett. 74 (1995), 4580-4583. [26] Robert Langlands, Philippe Pouliot, and Yvan Saint-Aubin, Conformal invariance in two-dimensional percolation , Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 1-61. · Zbl 0794.60109 [27] G. Lawler, The dimension of the frontier of planar Brownian motion , Electron. Comm. Probab. 1 (1996), no. 5, 29-47 (electronic). · Zbl 0857.60083 [28] Gregory F. Lawler, Hausdorff dimension of cut points for Brownian motion , Electron. J. Probab. 1 (1996), no. 2, approx. 20 pp. (electronic). · Zbl 0891.60078 [29] Benoit B. Mandelbrot, The fractal geometry of nature , W. H. Freeman and Co., San Francisco, Calif., 1982. · Zbl 0504.28001 [30] C. M. Newman and D. L. Stein, Spin-glass model with dimention-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), no. 3-4, 1113-1132. · Zbl 1042.82568 [32] Robin Pemantle, Choosing a spanning tree for the integer lattice uniformly , Ann. Probab. 19 (1991), no. 4, 1559-1574. · Zbl 0758.60010 [33] Yu. V. Prohorov, Convergence of random processes and limit theorems in probability theory , Teor. Veroyatnost. i Primenen. 1 (1956), 177-238. · Zbl 0075.29001 [34] L. Richardson, The problem of contiguity , General Systems: Yearbook 6, Society for the Advancement of General Systems Theory, Ann Arbor, Mich., 1961, pp. 139-187. [35] Lucio Russo, A note on percolation , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43 (1978), no. 1, 39-48. · Zbl 0363.60120 [36] H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions , Phys. Rev. Lett. 58 (1987), no. 22, 2325-2328. [37] P. D. Seymour and D. J. A. Welsh, Percolation probabilities on the square lattice , Ann. Discrete Math. 3 (1978), 227-245, in Advances in graph theory, North-Holland, Amsterdam. · Zbl 0405.60015 [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. · Zbl 0862.60092 [40] J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability , J. Appl. Probab. 22 (1985), no. 3, 556-569. JSTOR: · Zbl 0571.60019 [41] David Bruce Wilson, Generating random spanning trees more quickly than the cover time , Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), ACM, New York, 1996, pp. 296-303. · Zbl 0946.60070
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