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On a sharp transition from area law to perimeter law in a system of random surfaces. (English) Zbl 0529.60099
Summary: We introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a Bernoulli system of random plaquettes. The transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface, spanning a prescribed loop. As such, this transition offers a generalization of the bond percolation phenomenon. At low plaquette densities, the probability for large loops is shown to decay exponentially with the loops’ area, whereas for high densities the decay is by a perimeter law. Furthermore, we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds. Thus, if a natural conjecture about the phase structure of the bond percolation model is true, then there is a sharp transition in the asymptotic behavior of the surface events. Our analysis incorporates block variables, in terms of which a non-critical system is transformed into one which is close to a trivial, high or low density, fixed point. Stochastic geometric effects like those discussed here play an important role in lattice gauge theories.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
82B43 Percolation
60F99 Limit theorems in probability theory
Full Text: DOI
[1] Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett.103B, 207 (1981)
[2] Durhuus, B.: Quantum theory of strings. In: Lecture Notes, Nordita-82/36, and references therein · Zbl 0795.17035
[3] Kesten, H.: On the time constant and path length of first-passage percolation. Adv. Appl. Prob.12, 848 (1980) · Zbl 0436.60077 · doi:10.2307/1426744
[4] Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.74, 41 (1980) · Zbl 0441.60010 · doi:10.1007/BF01197577
[5] Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheorie Verw. Geb.56, 229 (1981) · Zbl 0457.60084 · doi:10.1007/BF00535742
[6] Harris, T.E.: A lower bound for the critical probability in certain percolation processes. P. Camb. Philos. Soc.56, 13 (1960) · Zbl 0122.36403 · doi:10.1017/S0305004100034241
[7] Wierman, J.C.: Bond percolation on honeycomb and triangular lattices. Adv. Appl. Prob.13, 293 (1981) · Zbl 0457.60085 · doi:10.2307/1426685
[8] Aizeman, M.: Surface phenomena in Ising systems and ?(2) gauge models (Geometric analysis, Part III) (in preparation)
[9] Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89 (1971) · Zbl 0346.06011 · doi:10.1007/BF01651330
[10] Kesten, H.: Analyticity properties and power law estimates of functions in percolation theory. J. Stat. Phys.25, 717 (1981) · Zbl 0512.60095 · doi:10.1007/BF01022364
[11] Krinsky, S., Emery, V.: Upper bound on correlation functions of Ising ferromagnet. Phys. Lett.50A, 235 (1974) · doi:10.1016/0375-9601(74)90804-4
[12] Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys.77, 111 (1980) · doi:10.1007/BF01982711
[13] Lieb, E.: A refinement of Simon’s correlation inequality. Commun. Math. Phys.77, 127 (1980) · doi:10.1007/BF01982712
[14] Aizenman, M., Newman, C.: Tree diagram bounds and the critical behavior in percolation models (in preparation) · Zbl 0586.60096
[15] Higuchi, Y.: Coexistence of infinite (*) clusters. Z. Wahrscheinlichkeitstheorie Verw. Geb.61, 75 (1982) · Zbl 0478.60096 · doi:10.1007/BF00537226
[16] Kesten, H.: Percolation theory for mathematicians. Boston, Basel, Stuttgart: Birkhäuser 1982 · Zbl 0522.60097
[17] Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheorie Verw. Geb.43, 39 (1978) · Zbl 0363.60120 · doi:10.1007/BF00535274
[18] Dobrushin, R.L.: Gibbs states describing coexistence of phases for a three-dimensional Ising model. Theor. Prob. Appl.17, 582 (1972) · Zbl 0275.60119 · doi:10.1137/1117073
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