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

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

2007-12-03

arxiv.org/abs/0712.0312v2

Mathematics

Probability

43 pages, many figures. Version v2 with various (minor) changes (in particular in Sections 1.4 and A.1), and Sect. 4 is shorte

Scientific paper

10.1007/s10955-008-9580-5

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)

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