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(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). © 2008 The Author(s).