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

TL;DR

This paper proves mean-field behavior for long- and finite-range Ising, percolation, and self-avoiding walk models using lace expansion, establishing dimension thresholds based on decay exponents.

Contribution

It introduces a simplified lace expansion analysis for these models, extending mean-field results to broader classes with power-law decaying interactions.

Findings

Mean-field behavior established for models above certain dimension thresholds.

Simplified lace expansion analysis based on trigonometric methods.

Dimension thresholds depend on decay exponent and model type.

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 $d>2(\alpha\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(\alpha\wedge2)$ 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)