The Annals of Probability

Boundedness of level lines for two-dimensional random fields

Kenneth S. Alexander

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Abstract

Every two-dimensional incompressible flow follows the level lines of some scalar function $\psi$ on $\mathbb{R}^2$; transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when $\psi$ is a stationary random field. We show that under mild hypotheses there is only one possible alternative to bounded level lines: the "treelike" random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line "winding through every region of the plane." If the random field has the FKG property, then only bounded level lines are possible. For stationary $C^2$ Gaussian random fields with covariance function decaying to 0 at $\infty$, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at $\infty$ "slower than exponentially," and only bounded level lines are possible if the covariance function is nonnegative.

Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 1653-1674.

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1041903201

Digital Object Identifier
doi:10.1214/aop/1041903201

Mathematical Reviews number (MathSciNet)
MR1415224

Zentralblatt MATH identifier
0866.60084

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C70: Transport processes 82B43: Percolation [See also 60K35]

Keywords
Random field Lagrangian trajectory incompressible flow percolation statistical topography minimal spanning tree level line

Citation

Alexander, Kenneth S. Boundedness of level lines for two-dimensional random fields. Ann. Probab. 24 (1996), no. 4, 1653--1674. doi:10.1214/aop/1041903201. https://projecteuclid.org/euclid.aop/1041903201


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References

  • 1 ADLER, R. A. 1981. The Geometry of Random Fields. Wiley, New York.
  • 2 ALDOUS, D. and STEELE, J. M. 1992. Asy mptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247 258.
  • 3 ALEXANDER, K. S. 1995. Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87 104.
  • 4 ALEXANDER, K. S. 1995. Simultaneous uniqueness of infinite clusters in stationary random labeled graphs. Comm. Math. Phy s. 168 39 55.
  • 5 ALEXANDER, K. S. and MOLCHANOV, S. A. 1994. Percolation of level sets for two-dimensional random fields with lattice sy mmetry. J. Statist. Phy s. 77 627 643.
  • 6 AVELLANEDA, M., ELLIOT, F., JR. and APELIAN, C. 1993. Trapping, percolation and anomalous diffusion of particles in a two-dimensional flow. J. Statist. Phy s. 72 1227 1304.
  • 7 BELy AEV, YU. K. 1972. Point processes and first passage problems. In Proc. Sixth Berkeley Sy mp. Math. Statist. Probab. 3 1 17. Univ. California Press, Berkeley.
  • 8 BURTON, R. and KEANE, M. 1989. Density and uniqueness in percolation. Comm. Math. Phy s. 121 501 505.
  • 9 CRAMER, H. and LEADBETTER, M. R. 1967. Stationary and Related Stochastic Processes. ´Wiley, New York.
  • 10 Dy M, H. and MCKEAN, H. P. 1976. Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic Press, New York.
  • 11 GANDOLFI, A., KEANE, M. and NEWMAN, C. M. 1992. Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 511 527.
  • 12 GANDOLFI, A., KEANE, M. and RUSSO, L. 1988. On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab. 16 1147 1157.
  • 13 ISICHENKO, M. B. 1992. Percolation, statistical topography, and transport in random media. Rev. Modern Phy s. 64 961 1043.
  • 14 JAIN, N. C. and MARCUS, M. B. 1978. Continuity of subgaussian processes. In Probability Z. on Banach Spaces. Advances in Probability and Related Topics J. Kuelbs, ed. 4 81 196. Dekker, New York.
  • 15 KRENGEL, U. 1985. Ergodic Theorems. de Gruy ter, Berlin.
  • 16 LEVINSON, N. 1940. Gap and Density Theorems. Amer. Math. Soc., Providence, RI.
  • 17 MARCUS, M. B. and SHEPP, L. A. 1970. Continuity of Gaussian processes. Trans. Amer. Math. Soc. 151 377 392.
  • 18 MOLCHANOV, S. A. and STEPANOV, A. K. 1983. Percolation in random fields. I, II, III. Teoret. Mat. Fiz. 55 246 256, 419 430; 67 177 185 Theoret. and Math. Phy s. 55 478 484, Z. 592 599; 67 434 439 1983.
  • 19 NEWMAN, C. M. and SCHULMAN, L. S. 1981. Infinite clusters in percolation models. J. Statist. Phy s. 26 613 628.
  • 20 PITT, L. D. 1982. Positively correlated normal variables are associated. Ann. Probab. 10 496 499.
  • 21 PREPARATA, F. P. and SHAMOS, M. I. 1985. Computational Geometry: An Introduction. Springer, New York.
  • 22 WHy BURN, G. T. 1942. Analy tic Topology. Coll. Publ. 28. Amer. Math. Soc., Providence, RI.
  • 23 YADRENKO, M. I. 1983. Spectral Theory of Random Fields. Optimization Software Inc., New York.
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