 Solution

By Tonelli theorem,If $p<p_c$, part (a) bounds the summand by $e^{-c\Vert x{\Vert }_{\infty }}$. There are only polynomially many vertices at each radius, so the series converges and $\chi (p)<\infty$.

Conversely, suppose $\chi (p)<\infty$. Then $p<1$. Choose $q>p$ so close to $p$ thatUse the standard sprinkling coupling for Bernoulli percolation: first expose the $p$-open clusters, then independently open each remaining edge with probability $\alpha =(q-p)/(1-p)$. Explore the $q$-cluster of the origin cluster by following sprinkled edges. Each discovered $p$-cluster has at most $2d$ times its number of vertices as many incident edges, so the exploration is dominated by a Galton-Watson process of mean at most $2d\alpha \chi (p)<1$. This process dies out almost surely, and hence there is no infinite $q$-open cluster. Thus $q\le p_c$, and $p<q$ implies $p<p_c$. Therefore