×

Critical two-point functions for long-range statistical-mechanical models in high dimensions. (English) Zbl 1342.60162

This paper deals with self-avoiding walks, percolation and the \(d\)-dimensional Ising model. For \(d\) greater than the upper critical dimension, the authors find the asymptotical form of the critical two-point function. The two-point function indicates how likely the spins located at those two sites point in the same direction for the Ising model. If it decays fast enough to be summable, then there is no macroscopic order. The summability of the two-point function is lost as soon as the model parameter is above the critical point and, therefore, it is naturally hard to investigate critical behavior. The authors overcome those difficulties identifying an asymptotic expression of the critical two-point function for finite-range models, such as the nearest-neighbor model for long-range models, especially when the 1-step distribution for the underlying random walk decays in powers of the distance. As is shown in this paper, the decay rate of the subcritical two-point function is the same as the 1-step distribution of the underlying random walk.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43 Percolation
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Aizenman, M. (1982). Geometric analysis of \(\varphi^{4}\) fields and Ising models. I, II. Comm. Math. Phys. 86 1-48. · Zbl 0533.58034
[2] Aizenman, M. and Fernández, R. (1986). On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44 393-454. · Zbl 0629.60106
[3] Aizenman, M. and Newman, C. M. (1984). Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36 107-143. · Zbl 0586.60096
[4] Aizenman, M. and Newman, C. M. (1986). Discontinuity of the percolation density in one-dimensional \(1/| x-y|^{2}\) percolation models. Comm. Math. Phys. 107 611-647. · Zbl 0613.60097
[5] Barsky, D. J. and Aizenman, M. (1991). Percolation critical exponents under the triangle condition. Ann. Probab. 19 1520-1536. · Zbl 0747.60093
[6] Bhattacharya, R. N. and Rao, R. R. (2010). Normal Approximation and Asymptotic Expansions. Classics in Applied Mathematics 64 . SIAM, Philadelphia, PA. · Zbl 1222.41002
[7] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory . Academic Press, New York. · Zbl 0169.49204
[8] Bogdan, K. and Jakubowski, T. (2007). Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys. 271 179-198. · Zbl 1129.47033
[9] Chen, L.-C. and Sakai, A. (2008). Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Related Fields 142 151-188. · Zbl 1149.60065
[10] Chen, L.-C. and Sakai, A. (2009). Critical behavior and the limit distribution for long-range oriented percolation. II. Spatial correlation. Probab. Theory Related Fields 145 435-458. · Zbl 1176.60082
[11] Chen, L.-C. and Sakai, A. (2011). Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. Ann. Probab. 39 507-548. · Zbl 1228.60108
[12] Ginibre, J. (1970). General formulation of Griffiths’ inequalities. Comm. Math. Phys. 16 310-328.
[13] Griffiths, R. B., Hurst, C. A. and Sherman, S. (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11 790-795.
[14] Grimmett, G. (1999). Percolation , 2nd ed. Springer, Berlin. · Zbl 0926.60004
[15] Hara, T. (2008). Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36 530-593. · Zbl 1142.82006
[16] Hara, T., Heydenreich, M. and Sakai, A. One-arm exponent for the Ising ferromagnets in high dimensions. In preparation.
[17] Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333-391. · Zbl 0698.60100
[18] Hara, T., van der Hofstad, R. and Slade, G. (2003). Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 349-408. · Zbl 1044.82006
[19] Heydenreich, M., van der Hofstad, R. and Hulshof, T. (2011). High-dimensional incipient infinite clusters revisited. Preprint. Available at . arXiv:1108.4325 · Zbl 1296.82029
[20] Heydenreich, M., van der Hofstad, R. and Sakai, A. (2008). Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Stat. Phys. 132 1001-1049. · Zbl 1152.82007
[21] Kozma, G. and Nachmias, A. (2011). Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24 375-409. · Zbl 1219.60085
[22] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk . Birkhäuser, Boston, MA. · Zbl 0780.60103
[23] Sakai, A. (2004). Mean-field behavior for the survival probability and the percolation point-to-surface connectivity. J. Stat. Phys. 117 111-130. · Zbl 1206.60092
[24] Sakai, A. (2007). Lace expansion for the Ising model. Comm. Math. Phys. 272 283-344. · Zbl 1133.82007
[25] Slade, G. (2006). The Lace Expansion and Its Applications. Lecture Notes in Math. 1879 . Springer, Berlin. · Zbl 1113.60005
[26] van den Berg, J. and Kesten, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556-569. · Zbl 0571.60019
[27] van der Hofstad, R. and Slade, G. (2002). A generalised inductive approach to the lace expansion. Probab. Theory Related Fields 122 389-430. · Zbl 1002.60095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.