open access

https://hdl.handle.net/2115/79837

Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for d >= d(c)

英語

2019-12

Springer

Communications in mathematical physics

372(2)

543-572

Consider the long-range models on Z(d) of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as |x|(-d-alpha) for some alpha > 0. In the previous work (Chen and Sakai in Ann Probab 43:639-681, 2015), we have shown in a unified fashion for all alpha not equal 2 that, assuming a bound on the "derivative" of the n-step distribution (the compoundzeta distribution satisfies this assumed bound), the critical two-point function G(pc) (x) decays as |x|(alpha boolean AND 2-d) above the upper-critical dimension d(c) = (alpha boolean AND 2)m, where m = 2 for self-avoiding walk and the Ising model and m = 3 for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the n-step distribution, that G(pc) (x) for the marginal case alpha = 2 decays as |x|(2-d)/ log |x| whenever d >= d(c) (with a large spread-out parameter L). This solves the conjecture in Chen and Sakai (2015), extended all the way down to d = d(c), and confirms a part of predictions in physics (Brezin et al. in J Stat Phys 157:855-868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.

学術雑誌論文

Accepted Manuscript

This is a post-peer-review, pre-copyedit version of an article published in Communications in mathematical physics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00220-019-03385-9

0010-3616