Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

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 $d>2(\alpha \wedge 2)$ for self-avoiding walk and the Ising model, and $d>3(\alpha \wedge 2)$ for percolation, where $d$ denotes the dimension and $\alpha$ 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. (2007)

43 pages, many figures. Version v2 with various (minor) changes (in particular in Sections 1.4 and A.1), and Sect. 4 is shortened. Journal of Statistical Physics (to appear)

References in corpus (3)

• Transience, Recurrence and Critical Behavior for Long-Range Percolation
• Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals
• Mean-field driven first-order phase transitions in systems with long-range interactions

Cited by in corpus (21)

• The Alexander-Orbach conjecture holds in high dimensions
• Random Currents and Continuity of Ising Model's Spontaneous Magnetization
• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Hydrogenation-induced ferromagnetism on graphite surfaces
• Critical exponents for long-range O(n) models below the upper critical dimension
• Critical two-point function for long-range $O(n)$ models below the upper critical dimension
• The discontinuity of the specific heat for the 5D Ising model
• Structures in supercritical scale-free percolation
• Critical two-point functions for long-range statistical-mechanical models in high dimensions
• Application of the lace expansion to the ${\phi }^4$ model
• Percolation on a product of two trees
• Applicability of n-vicinity method for calculation of free energy of Ising model
• Self-avoiding walk, spin systems, and renormalization
• Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ${\mathbb{Z}}^d$
• Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation
• Critical two-point function for long-range models with power-law couplings: The marginal case for $d\ge d_c$
• On the derivation of mean-field percolation critical exponents from the triangle condition
• Critical site percolation in high dimension
• Isoperimetric lower bounds for critical exponents for long-range percolation
• Expansion of the Critical Intensity for the Random Connection Model
• High-dimensional long-range statistical mechanical models have random walk correlation functions