Abstract
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.
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References
Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett.103B, 207 (1981)
Durhuus, B.: Quantum theory of strings. In: Lecture Notes, Nordita-82/36, and references therein
Kesten, H.: On the time constant and path length of first-passage percolation. Adv. Appl. Prob.12, 848 (1980)
Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.74, 41 (1980)
Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheorie Verw. Geb.56, 229 (1981)
Harris, T.E.: A lower bound for the critical probability in certain percolation processes. P. Camb. Philos. Soc.56, 13 (1960)
Wierman, J.C.: Bond percolation on honeycomb and triangular lattices. Adv. Appl. Prob.13, 293 (1981)
Aizeman, M.: Surface phenomena in Ising systems and ℤ(2) gauge models (Geometric analysis, Part III) (in preparation)
Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89 (1971)
Kesten, H.: Analyticity properties and power law estimates of functions in percolation theory. J. Stat. Phys.25, 717 (1981)
Krinsky, S., Emery, V.: Upper bound on correlation functions of Ising ferromagnet. Phys. Lett.50A, 235 (1974)
Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys.77, 111 (1980)
Lieb, E.: A refinement of Simon's correlation inequality. Commun. Math. Phys.77, 127 (1980)
Aizenman, M., Newman, C.: Tree diagram bounds and the critical behavior in percolation models (in preparation)
Higuchi, Y.: Coexistence of infinite (*) clusters. Z. Wahrscheinlichkeitstheorie Verw. Geb.61, 75 (1982)
Kesten, H.: Percolation theory for mathematicians. Boston, Basel, Stuttgart: Birkhäuser 1982
Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheorie Verw. Geb.43, 39 (1978)
Dobrushin, R.L.: Gibbs states describing coexistence of phases for a three-dimensional Ising model. Theor. Prob. Appl.17, 582 (1972)
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Communicated by A. Jaffe
Work supported in part by NSF grants PHY-8301493 (M.A.), PHY-8117463 (J.T.C.) and PHY-8116101 A01 (L.C.)
Sloan Foundation Research Fellow
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Aizenman, M., Chayes, J.T., Chayes, L. et al. On a sharp transition from area law to perimeter law in a system of random surfaces. Commun.Math. Phys. 92, 19–69 (1983). https://doi.org/10.1007/BF01206313
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DOI: https://doi.org/10.1007/BF01206313