Taiwanese Journal of Mathematics

A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice

Abstract

The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a $d$-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 62 pages.

Dates
First available in Project Euclid: 18 October 2019

https://projecteuclid.org/euclid.twjm/1571364135

Digital Object Identifier
doi:10.11650/tjm/190904

Citation

Handa, Satoshi; Kamijima, Yoshinori; Sakai, Akira. A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice. Taiwanese J. Math., advance publication, 18 October 2019. doi:10.11650/tjm/190904. https://projecteuclid.org/euclid.twjm/1571364135

References

• M. Aizenman and D. J. Barsky, Sharpness of the phase transition in percolation models, Comm. Math. Phys. 108 (1987), no. 3, 489–526.
• M. Aizenman, D. J. Barsky and R. Fernández, The phase transition in a general class of Ising-type models is sharp, J. Statist. Phys. 47 (1987), no. 3-4, 343–374.
• M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Statist. Phys. 36 (1984), no. 1-2, 107–143.
• R. Bauerschmidt, H. Duminil-Copin, J. Goodman and G. Slade, Lectures on self-avoiding walks, in: Probability and Statistical Physics in Two and More Dimensions, 395–467, Clay Math. Proc. 15, Amer. Math. Soc., Providence, RI, 2012.
• B. Bollobás and O. Riordan, Percolation, Cambridge University Press, New York, 2006.
• D. Brydges and T. Spencer, Self-avoiding walk in $5$ or more dimensions, Comm. Math. Phys. 97 (1985), no. 1-2, 125–148.
• L.-C. Chen, S. Handa, M. Heydenreich, Y. Kamijima and A. Sakai, An attempt to prove mean-field behavior for nearest-neighbor percolation in $7$ dimensions, in preparation.
• H. Duminil-Copin and V. Tassion, A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model, Comm. Math. Phys. 343 (2016), no. 2, 725–745.
• W. Feller, An Introduction to Probability Theory and its Applications I, Third edition, John Wiley & Sons, New York, 1968.
• R. Fitzner and R. van der Hofstad, Generalized approach to the non-backtracking lace expansion, Probab. Theory Related Fields 169 (2017), no. 3-4, 1041–1119.
• ––––, Mean-field behavior for nearest-neighbor percolation in $d > 10$, Electron. J. Probab. 22 (2017), no. 43, 65 pp.
• J. Fröhlich, B. Simon and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Comm. Math. Phys. 50 (1976), no. 1, 79–95.
• G. Grimmett, Percolation, Second edition, Grundlehren der Mathematischen Wissenschaften 321, Springer-Verlag, Berlin, 1999.
• T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals, J. Statist. Phys. 59 (1990), no. 5-6, 1469–1510.
• ––––, Mean-field critical behaviour for percolation in high dimensions, Comm. Math. Phys. 128 (1990), no. 2, 333–391.
• ––––, Self-avoiding walk in five or more dimensions I: The critical behaviour, Comm. Math. Phys. 147 (1992), no. 1, 101–136.
• ––––, The lace expansion for self-avoiding walk in five or more dimensions, Rev. Math. Phys. 4 (1992), no. 2, 235–327.
• N. Madras and G. Slade, The Self-avoiding Walk, Probability and its Applications, Birkhäuser Boston, Boston, MA, 1993.
• M. V. Men'shikov, Coincidence of critical points in percolation problems, Dokl. Akad. Nauk SSSR 288 (1986), no. 6, 1308–1311.
• B. G. Nguyen and W.-S. Yang, Triangle condition for oriented percolation in high dimensions, Ann. Probab. 21 (1993), no. 4, 1809–1844.
• A. Sakai, Mean-field critical behavior for the contact process, J. Statist. Phys. 104 (2001), no. 1-2, 111–143.
• ––––, Lace expansion for the Ising model, Comm. Math. Phys. 272 (2007), no. 2, 283–344.
• ––––, Application of the lace expansion to the $\varphi^4$ model, Comm. Math. Phys. 336 (2015), no. 2, 619–648.
• G. Slade, The Lace Expansion and its Applications, Lecture Notes in Mathematics 1879, Springer-Verlag, Berlin, 2006.