Non-rigidity Properties of the Coulomb Gas

We prove existence of infinite volume $d$-dimensional Coulomb gases which are not number rigid for $d\ge 3$. This makes the Coulomb gas the Gibbs point process with the longest range pairwise interaction (i.e.\ with the smallest $s$ in the interaction kernel $g(x)=\mathrm{\mid }x{\mathrm{\mid }}^{-s}$) for which number non-rigidity has been proved in $d\ge 3$. We rule out properties stronger than number rigidity for the two-dimensional Coulomb gas.