Abstract
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d>2(α ∧2) for self-avoiding walk and the Ising model, and d>3(α ∧2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886–1944, 2005).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aizenman, M.: Geometric analysis of φ 4 fields and Ising models. I, II. Commun. Math. Phys. 86(1), 1–48 (1982)
Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108(3), 489–526 (1987)
Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44(3–4), 393–454 (1986)
Aizenman, M., Fernández, R.: Critical exponents for long-range interactions. Lett. Math. Phys. 16(1), 39–49 (1988)
Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36(1–2), 107–143 (1984)
Aizenman, M., Newman, C.M.: Discontinuity of the percolation density in one-dimensional 1/| x−y | 2 percolation models. Commun. Math. Phys. 107(4), 611–647 (1986)
Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47(3–4), 343–374 (1987)
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. 111(4), 505–531 (1987)
Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the magnetization in one-dimensional 1/| x−y | 2 Ising and Potts models. J. Stat. Phys. 50(1–2), 1–40 (1988)
Barsky, D.J., Aizenman, M.: Percolation critical exponents under the triangle condition. Ann. Probab. 19(4), 1520–1536 (1991)
Berg, J.v.d., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 22(3), 556–569 (1985)
Berger, N.: Transience, recurrence and critical behavior for long-range percolation. Commun. Math. Phys. 226(3), 531–558 (2002)
Biskup, M., Chayes, L., Crawford, N.: Mean-field driven first-order phase transitions in systems with long-range interactions. J. Stat. Phys. 122(6), 1139–1193 (2006)
Borgs, C., Chayes, J.T., van der Hofstad, R., Slade, G., Spencer, J.: Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab. 33(5), 1886–1944 (2005)
Bovier, A., Felder, G., Fröhlich, J.: On the critical properties of the Edwards and the self-avoiding walk model of polymer chains. Nucl. Phys. B 230(1, FS10), 119–147 (1984)
Brydges, D., Spencer, T.: Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys. 97(1–2), 125–148 (1985)
Chayes, J.T., Chayes, L.: Inequality for the infinite-cluster density in Bernoulli percolation. Phys. Rev. Lett. 56(16), 1619–1622 (1986)
Chen, L.-C., Sakai, A.: Limit distribution and critical behavior for long-range oriented percolation. Probab. Theory Related Fields 142, 151–188 (2008)
Cheng, Y.: Long range self-avoiding random walks above critical dimension. PhD thesis, Temple University, August 2000
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1966)
Fernández, R., Fröhlich, J., Sokal, A.D.: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Texts and Monographs in Physics. Springer, Berlin (1992)
Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50, 79–85 (1976)
Griffiths, R.B.: Correlations in Ising ferromagnets I. J. Math. Phys. 8(3), 478–483 (1967)
Grimmett, G.: Percolation, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321. Springer, Berlin (1999)
Hara, T.: Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36(2), 530–593 (2008)
Hara, T., Slade, G.: Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128(2), 333–391 (1990)
Hara, T., Slade, G.: The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys. 4(2), 235–327 (1992)
Hara, T., Slade, G.: Self-avoiding walk in five or more dimensions. I. The critical behaviour. Commun. Math. Phys. 147(1), 101–136 (1992)
Hara, T., van der Hofstad, R., Slade, G.: Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31(1), 349–408 (2003)
Heydenreich, M., van der Hofstad, R.: Random graph asymptotics on high-dimensional tori. Commun. Math. Phys. 270(2), 335–358 (2007)
Hofstad, R.v.d.: Spread-out oriented percolation and related models above the upper critical dimension: induction and superprocesses. In: Ensaios Matemáticos [Mathematical Surveys], vol. 9, pp. 91–181. Sociedade Brasileira de Matemática, Rio de Janeiro (2005)
Hofstad, R.v.d., Slade, G.: A generalised inductive approach to the lace expansion. Probab. Theory Relat. Fields 122(3), 389–430 (2002)
Lebowitz, J.L.: GHS and other inequalities. Commun. Math. Phys. 35, 87–92 (1974)
Madras, N., Slade, G.: The Self-Avoiding Walk.Probability and Its Applications. Birkhäuser Boston, Boston (1993)
Menshikov, M.V.: Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288(6), 1308–1311 (1986)
Newman, C.M., Schulman, L.S.: One-dimensional 1/| j−i | s percolation models: the existence of a transition for s≤2. Commun. Math. Phys. 104(4), 547–571 (1986)
Sakai, A.: Applications of the lace expansion to statistical-mechanical models. Preprint (2007). To appear as a chapter in Analysis and Stochastics of Growth Processes (Oxford)
Sakai, A.: Lace expansion for the Ising model. Commun. Math. Phys. 272(2), 283–344 (2007)
Schulman, L.S.: Long range percolation in one dimension. J. Phys. A 16(17), L639–L641 (1983)
Slade, G.: The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys. 110(4), 661–683 (1987)
Slade, G.: The Lace Expansion and its Applications. Springer Lecture Notes in Mathematics, vol. 1879. Springer, Berlin (2006)
Yang, W.-S., Klein, D.: A note on the critical dimension for weakly self-avoiding walks. Probab. Theory Relat. Fields 79(1), 99–114 (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Heydenreich, M., van der Hofstad, R. & Sakai, A. Mean-Field Behavior for Long- and Finite Range Ising Model, Percolation and Self-Avoiding Walk. J Stat Phys 132, 1001–1049 (2008). https://doi.org/10.1007/s10955-008-9580-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9580-5