Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in d, oriented percolation in d×+, and the contact process in d, with p D(·) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y–x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point pc. We investigate the value of pc when d>6 for percolation and d>4 for the other models, and L1. We prove in a unified way that pc=1+C(D)+O(L–2d), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L–d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that pc=1+cL–d+O(L–d–1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.