Using the Griffiths-Simon construction of the $varphi^4$ model and the lace expansion for the Ising model, we prove that, if the strength $lambdage0$ of nonlinearity is sufficiently small for a large class of short-range models in dimensions $d>4$, t
hen the critical $varphi^4$ two-point function $langlevarphi_ovarphi_xrangle_{mu_c}$ is asymptotically $|x|^{2-d}$ times a model-dependent constant, and the critical point is estimated as $mu_c=mathscr{hat J}-fraclambda2langlevarphi_o^2rangle_{mu_c}+O(lambda^2)$, where $mathscr{hat J}$ is the massless point for the Gaussian model.
The aim of this short article is to convey the basic idea of the original paper [3], without going into too much detail, about how to derive sharp asymptotics of the gyration radius for random walk, self-avoiding walk and oriented percolation above the model-dependent upper critical dimension.
We consider the critical spread-out contact process in Z^d with dge1, whose infection range is denoted by Lge1. In this paper, we investigate the r-point function tau_{vec t}^{(r)}(vec x) for rge3, which is the probability that, for all i=1,...,r-1,
the individual located at x_iin Z^d is infected at time t_i by the individual at the origin oin Z^d at time 0. Together with the results of the 2-point function in [van der Hofstad and Sakai, Electron. J. Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper-critical dimension 4. We also prove partial results for dle4 in a local mean-field setting.
We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index alpha>0 converges to e^{-C|k|^{alphawedge2}} for some Cin(0,infty) above the upper-critica
l dimension 2(alphawedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at alpha=2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.
We provide a complete proof of the diagrammatic bounds on the lace-expansion coefficients for oriented percolation, which are used in [arXiv:math/0703455] to investigate critical behavior for long-range oriented percolation above 2min{alpha,2} spatial dimensions.
