Condensation of the invariant measures of the supercritical zero range processes

For $\alpha\geq 1$, let $g:\mathbb N\to\mathbb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k)=(k/k-1)^\alpha$, $k\geq 2$. Consider the symmetric nearest neighbour zero range process on the discrete torus $\mathbb T_L$ in which a particle jumps from a site, occupied by $k$ particles, to one of its neighbors with rate $g(k)$. Armend\'ariz and Loulakis\cite{al09} proved a strong form of the equivalence of ensembles for the invariant measure of the supercritical zero range process with $\alpha>2$. We generalize their result to all $\alpha\geq 1$.