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Percolation Transition for Some Excursion Sets
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Olivier Garet, Université d'Orléans, France
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Abstract
We consider a random field (Xn)n ∈ Zd and investigate when the set
Ah={k ∈ Zd; |Xk| ≥ h}
has infinite clusters.
The main problem is to decide whether the critical level
hc=sup{h ∈ R ;P(Ah has an infinite cluster)>0}
is neither
0 nor +∞. Thus, we say that a percolation transition occurs.
In a first time, we show that weakly dependent Gaussian fields satisfy to a
well-known criterion implying the percolation transition.
Then, we introduce a concept of percolation along reasonable paths and
therefore prove a phenomenon of percolation transition for reasonable paths
even for strongly dependent Gaussian fields. This allows to obtain
some results of percolation transition for oriented percolation.
Finally, we study some Gibbs states associated to a perturbation of a ferromagnetic quadratic interaction.
At first, we show that a transition percolation occurs for superstable potentials. Next, we go to the the critical case and show that a transition percolation
occurs for directed percolation when d&ge 4. We also
note that the assumption of ferromagnetism can be relaxed when we deal
with Gaussian Gibbs measures, i.e. when there is no perturbation of
the quadratic interaction.
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Full text: PDF
Pages: 255-292
Published on: April 9, 2004
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Bibliography
| Antal, Peter; Pisztora, Agoston. On the chemical distance for supercritical Bernoulli percolation.
Ann. Probab. 24 (1996), no. 2, 1036--1048. MR1404543 (98b:60168) |
| Bricmont, Jean; Lebowitz, Joel L.; Maes, Christian. Percolation in strongly correlated systems: the massless Gaussian field.
J. Statist. Phys. 48 (1987), no. 5-6, 1249--1268. MR0914444 (89c:82040) |
| Cassandro, M.; Olivieri, E.; Pellegrinotti, A.; Presutti, E. Existence and uniqueness of DLR measures for unbounded spin systems.
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 4, 313--334. MR0471115 (57 #10854) |
| Dieudonné, Jean. Calcul infinitésimal.
(French) Hermann, Paris 1968 479 pp. (loose errata). MR0226971 (37 #2557) |
| Dobrushin, R. L. Gaussian random fields---Gibbsian point of view.
Multicomponent random systems,
pp. 119--151, Adv. Probab. Related Topics, 6, Dekker, New York, 1980. MR0599534 (82d:60095) |
| Doss, H.; Royer, G. Processus de diffusion associé aux mesures de Gibbs sur $ Rsp{ Zsp{ d}}$.
(French) Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 1, 107--124. MR0512335 (80h:60122) |
| Garet, Olivier. Gibbsian fields associated to exponentially decreasing quadratic potentials.
Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), no. 3, 387--415. MR1689890 (2000g:60156) |
| Garet, Olivier ; Marchand, Régine.
Asymptotic shape for the chemical distance and first-passage
percolation in random environment.
Preprint. 2003. Available at ArXiv |
| Georgii, Hans-Otto. Gibbs measures and phase transitions.
de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 1988. xiv+525 pp. ISBN: 0-89925-462-4 MR0956646 (89k:82010) |
| Georgii, Hans-Otto; Häggström, Olle; Maes, Christian. The random geometry of equilibrium phases.
Phase transitions and critical phenomena, Vol. 18,
1--142, Phase Transit. Crit. Phenom., 18, Academic Press, San Diego, CA, 2001. MR2014387 |
| Künsch, H. Gaussian Markov random fields.
J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), no. 1, 53--73. MR0539773 (81g:60056) |
| Künsch, H.
Reellwertige Zufallsfelder auf einem Gitter:
Interpolationsprobleme, Variationsprinzip und statistische Analyse.
PhD thesis, Zürich, 1980.
|
| Liggett, T. M.; Schonmann, R. H.; Stacey, A. M. Domination by product measures.
Ann. Probab. 25 (1997), no. 1, 71--95. MR1428500 (98f:60095) |
| Molchanov, S. A.; Stepanov, A. K. Percolation in random fields. I.
(Russian) Teoret. Mat. Fiz. 55 (1983), no. 2, 246--256. MR0734878 (85j:60091) |
| Molchanov, S. A.; Stepanov, A. K. Percolation in random fields. II.
(Russian) Teoret. Mat. Fiz. 55 (1983), no. 3, 419--430. MR0711007 (85j:60092) |
| Molchanov, S. A.; Stepanov, A. K. Percolation in random fields. III.
(Russian) Teoret. Mat. Fiz. 67 (1986), no. 2, 177--185. MR0851557 (87m:60117) |
| Newman, Charles M. A surface view of first-passage percolation.
Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994),
1017--1023, Birkhäuser, Basel, 1995. MR1404001 (97h:60127) |
|
Roberts, A.P. ; Knackstedt, M.A.
Structure-property correlations in model composite materials.
Phys. Rev. E, 54:2313--2328, september 1996.
|
|
Roberts, A.P. ; Teubner, M.
Transport properties of heterogeneous materials derived from
Gaussian random fields: Bounds and simulation.
Phys. Rev. E, 51:4141--4154, may 1995.
|
| Rudin, Walter. Function theory in the unit ball of $ Csp{n}$.
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 241. Springer-Verlag, New York-Berlin, 1980. xiii+436 pp. ISBN: 0-387-90514-6 MR0601594 (82i:32002) |
| Russo, Lucio. An approximate zero-one law.
Z. Wahrsch. Verw. Gebiete 61 (1982), no. 1, 129--139. MR0671248 (84e:60153) |
| Sinai, Ya. G. Introduction to ergodic theory.
Translated by V. Scheffer.
Mathematical Notes, 18.
Princeton University Press, Princeton, N.J., 1976. 144 pp. ISBN: 0-691-08182-4 MR0584788 (58 #28437) |
| Spitzer, Frank. Principles of random walk.
The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1964 xi+406 pp. MR0171290 (30 #1521) |
| van Beijeren, Henk; Sylvester, Garrett S. Phase transitions for continuous-spin Ising ferromagnets.
J. Functional Analysis 28 (1978), no. 2, 145--167. MR0484201 (58 #4128) |
| Watson, G. N. A treatise on the theory of Bessel functions.
Reprint of the second (1944) edition.
Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. viii+804 pp. ISBN: 0-521-48391-3 MR1349110 (96i:33010) |
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Electronic Journal of Probability. ISSN: 1083-6489 |
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