Authors’ abstract: We consider self-avoiding walk and percolation in \(\mathbb{Z}^d\), oriented percolation in \(\mathbb{Z}^d\times\mathbb{Z}_+\), and the contact process in \(\mathbb{Z}^d\), with \(p D(\cdot)\) 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 \(p_c\). We investigate the value of \(p_c\) when \(d>6\) for percolation and \(d>4\) for the other models, and \(L\gg1\). We prove in a unified way that \(p_c=1+C(D)+O(L^{-2d})\), 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 \(p_c=1+cL^{-d}+O(L^{-d-1})\), where \(c\) is a model-dependent constant. Our proof is based on the lace expansion for each of these models.