Friday, April 6, 2012

1204.1180 (Lung-Chi Chen et al.)

Asymptotic behavior of critical two-point functions for long-range
statistical-mechanical models in high dimensions

Lung-Chi Chen, Akira Sakai
We consider long-range self-avoiding walk, percolation and the Ising model on the d-dimensional integer lattice whose pair-potential decays in powers of distance with exponent d+a. The upper-critical dimension d_c is 2min{a,2} for self-avoiding walk and the Ising model, and 3min{a,2} for percolation. Let a be not equal to 2 and assume heat-kernel bounds on the transition probability of the underlying random walk. We prove that, for d>d_c (and the spread-out parameter sufficiently large), the critical two-point function G(x) for each model is asymptotically |x|^{min{a,2}-d} times a constant, which is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between a<2 and a>2.
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