Discontinuity of the percolation density in one dimensional 1/x−y^{2} percolation models
 M. Aizenman,
 C. M. Newman
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We consider one dimensional percolation models for which the occupation probability of a bond −K _{ x,y }, has a slow power decay as a function of the bond's length. For independent models — and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bondsK _{ x,y }≦p<1 and if for long bondsK _{ x,y }≦β/x−y^{2} with β≦1, regardless of how closep is to 1, ii) in models for which the above asymptotic bound holds with some β<∞, there is a discontinuity in the percolation densityM (≡P _{∞}) at the percolation threshold, iii) assuming also translation invariance, in the nonpercolative regime, the mean cluster size is finite and the twopoint connectivity function decays there as fast asC(β,p)/x−y^{2}. The first two statements are consequences of a criterion which states that if the percolation densityM does not vanish then βM ^{2}>=1. This dichotomy resembles one for the magnetization in 1/x−y^{2} Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.
 Thouless, D.J.: Longrange order in onedimensional Ising systems. Phys. Rev.187, 732 (1969)
 Anderson, P.W., Yuval, G., Hamann, D.R.: Exact results in the Kondo problem. II. Scaling theory, qualitatively correct solution, and some new results on onedimensional classical statistical models. Phys. Rev. B1, 4464 (1970)
 Anderson, P.W., Yuval, G.: Some numerical results on the Kondo problem and the inverse square onedimensional Ising model. J. Phys. C (Solid St. Phys.)4, 607 (1971)
 Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the order parameter in one dimensional 1/x−y^{2} Ising and Potts models (in preparation)
 Schulman, L.S.: Long range percolation in one dimension. J. Phys. Lett. A16, L 639 (1983)
 Newman, C.M., Schulman, L.S.: One dimensional 1/j−i^{ s } percolation models: The existence of a transition fors≦2. Commun. Math. Phys.104, 547 (1986)
 Fröhlich, J., Spencer, T.: The phase transition in the onedimensional Ising model with 1/r ^{2} interaction energy. Commun. Math. Phys.84, 87 (1982)
 Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.36, 107 (1984)
 Simon, B., Sokal, A.: Rigorous entropyenergy arguments. J. Stat. Phys.25, 679 (1981)
 Bhattacharjee, J., Chakravarty, S., Richardson, J.L., Scalapino, D.J.: Some properties of a onedimensional Ising chain with an inversesquare interaction. Phys. Rev. B24, 3862 (1981)
 Aizenman, M., Chayes, J.T., Chayes, L., Imbrie, J., Newman, C.M.: An intermediate phase with slow decay of correlations in one dimensional 1/x−y^{2} percolation, Ising and Potts models. In preparation
 Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Phys. (submitted)
 Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys.77, 111 (1980)
 Aizenman, M.: In: Statistical physics and dynamical systems: Rigorous results. Fritz, J., Jaffe, A., Szàsz, D. (eds.), pp. 453–481. Boston: Birkhäuser 1985
 Aizenman, M., Barsky, D.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. (submitted)
 Mandelbrot, B.B.: Renewal sets and random cutouts. Z. Wahrscheinlichkeitstheor. Verw. Geb.22, 145 (1972)
 Mandelbrot, B.B.: The fractal geometry of nature. San Francisco: Freeman 1982, Chap. 33
 Fortuin, C., Kasteleyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89 (1971)
 Arjas, E., Lehtonen, T.: Approximating many server queues by means of single server queues. Math. Oper. Res.3, 205 (1978)
 Esary, J., Proschan, F., Walkup, D.: Association of random variables with applications. Ann. Math. Stat.38, 1466 (1967)
 van den Berg, J., Burton, R.: Private communication
 Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Stat.28, 790 (1957)
 Gross, L.: Decay of correlations in classical lattice models at high temperature. Commun. Math. Phys.68, 9 (1979)
 Lieb, E.H.: A refinement of Simon's correlation inequality. Commun. Math. Phys.77, 127 (1980)
 Newman, C.M., Schulman, L.S.: Infinite clusters in percolation models. J. Stat. Phys.26, 613 (1981)
 Bricmont, J., Lebowitz, J.L.: On the continuity of the magnetization and energy in Ising ferromagnets. J. Stat. Phys.42, 861 (1986)
 Title
 Discontinuity of the percolation density in one dimensional 1/x−y^{2} percolation models
 Journal

Communications in Mathematical Physics
Volume 107, Issue 4 , pp 611647
 Cover Date
 19861201
 DOI
 10.1007/BF01205489
 Print ISSN
 00103616
 Online ISSN
 14320916
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 M. Aizenman ^{(1)}
 C. M. Newman ^{(2)}
 Author Affiliations

 1. Departments of Mathematics and Physics, Rutgers University, 08903, New Brunswick, NJ, USA
 2. Department of Mathematics, University of Arizona, 85721, Tucson, AZ, USA