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 $\text{Zd}$, oriented percolation in $\text{Zd}\times \text{Zp}$, and the contact process in $\text{Zd}$, with $pD(\cdot )$ being the coupling function whose range is denoted by $L<\mathrm{\infty }$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $pD(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point $\text{pc}$. We investigate the value of $\text{pc}$ when $d>6$ for percolation and $d>4$ for the other models, and $L\gg 1$. We prove in a unified way that $\text{pc}=1+C(D)+O(L^{-2d})$, 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 function $D$. Our proof is based on the lace expansion for each of these models.