Abstract
We study first-passage percolation models and their higher dimensional analogs—models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to +∞. As corollaries we show that in any dimensiond≥2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or +∞. Proofs employ simple large-deviation estimates and ergodic arguments.
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References
R. Burton and M. Keane, Density and uniqueness in percolation,Commun. Math. Phys. 121:501 (1989).
R. Durrett,Lecture Notes on Particle Systems and Percolation (Wadsworth, Pacific Grove, California, 1988).
R. Durrett,Probability. Theory and Examples, 2nd ed. (Wadsworth, Pacific Grove, California, 1996).
J. Fröhlich, Introduction to mathematical theory of disordered systems, inGauge Theories, Critical Phenomena and Disordered Systems, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986).
H. Kesten, Aspects of first passage percolation, inLecture Notes in Mathematics, Vol. 1180 (Springer, Berlin, 1986).
H. Kesten, Surfaces with minimal random weights and maximal flows. A higher dimensional version of first-passage percolation,Illinois J. Math. 31:99 (1987).
U. Krengel,Ergodic Theorems (de Gruyter, Berlin, 1985).
C. Licea and C. Newman,Ann. Prob., to appear.
C. Newman and M. Piza,Ann. Prob. 23:977 (1995).
C. Newman and L. Schulman, Infinite clusters in percolation models,J. Stat. Phys. 26:613 (1981).
R. Smythe and J. C. Wierman,First Passage Percolation on the Square Lattice (Springer, Berlin, 1978).
J. Wehr and J. Woo, Absence of geodesics in first passage percolation on a half-plane (1996), preprint.
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Communicated by J. L. Lebowitz
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Wehr, J. On the number of infinite geodesics and ground states in disordered systems. J Stat Phys 87, 439–447 (1997). https://doi.org/10.1007/BF02181495
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DOI: https://doi.org/10.1007/BF02181495