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One dimensional $$1/| j-i| ^ s$$ percolation models: The existence of a transition for s$$\leq 2$$. (English) Zbl 0604.60097
Consider a one-dimensional independent bond percolation model with $$p_ j$$ denoting the probability of an occupied bond between integer sites i and $$i\pm j$$, $$j\geq 1$$. If $$p_ j$$ is fixed for $$j\geq 2$$ and $$\lim_{j\to \infty}j^ 2p_ j>1$$, then (unoriented) percolation occurs for $$p_ 1$$ sufficiently close to 1.
This result, analogous to the existence of spontaneous magnetization in long range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur for $$p_ 1$$ close to 1 if $$\lim_{j\to \infty}j^ sp_ j>0$$ for some $$s<2$$. Analogous results are valid for one-dimensional site-bond percolation models.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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##### References:
 [1] Aizenman, M., Chayes, J. T., Chayes, L., Fröhlich, J., Russo, L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Commun. Math. Phys.92, 19-69 (1983) · Zbl 0529.60099 · doi:10.1007/BF01206313 [2] 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) · Zbl 1084.82514 [3] Aizenman, M., Newman, C. M.: Discontinuity of the percolation density in one-dimensional 1/|x?y|2 percolation models. (in preparation) · Zbl 0613.60097 [4] 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 one-dimensional classical statistical mechanics. Phys. Rev.B1, 4464-4473 (1970) [5] Dyson, F. J.: Existence of phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys.12, 91-107 (1969) · Zbl 1306.47082 · doi:10.1007/BF01645907 [6] Fröhlich, J., Spencer, T.: The phase transition in the one-dimensional Ising model with 1/r 2 interaction energy. Commun. Math. Phys.84, 87-101 (1982) · Zbl 1110.82302 · doi:10.1007/BF01208373 [7] Grimmett, G. R., Keane, M., Marstrand, J. M.: On the connectedness of a random graph. Math. Proc. Camb. Philos. Soc.96, 151-166 (1984) · Zbl 0543.60016 · doi:10.1017/S0305004100062034 [8] Newman, C. M., Schulman, L. S.: Infinite clusters in percolation models. J. Stat. Phys.26, 613-628 (1981) · Zbl 0509.60095 · doi:10.1007/BF01011437 [9] Schulman, L. S.: Long range percolation in one dimension. J. Phys. A. Lett.16, L639-L641 (1983) [10] Shamir, E.: Private communication
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