Scaling limit of additive functionals for reversible non-gradient exclusion process: critical cases

For the reversible speed-change exclusion process $({\eta }_t{)}_{t\ge 0}$ in ${\mathbb{Z}}^d$, we study the scaling limit of additive functionals ${\mathrm{\Gamma }}_t(f)={\int }_0^{t}f({\eta }_s)\mathrm{d}s$. Concerning the local centered function $f$, the previous work [Commun. Math. Phys. 104, 1-19, 1986] by Kipnis and Varadhan and [Comm. Pure Appl. Math., 66: 649-677, 2013] by Gon{ç}alves and Jara respectively covered the cases $d\ge 3$ and $d=1$. The present paper completes the missing part $d=2$, and also develops the theory for functions with higher degree. The novelty is a quantitative homogenization of the resolvent, which allows to overcome the obstacle of correlation function in non-gradient models.