  A graph homomorphism from the rooted $d$-branching tree $\phi: T^d \to H$ is
said to be cold if the values of $\phi$ for vertices arbitrarily far away from
the root can restrict the value of $\phi$ at the root. Warmth is a graph
parameter that measures the non-existence of cold maps. We study warmth of
random graphs $G(n,p)$, and for every $d \ge 1$, we exhibit a nearly-sharp
threshold for the existence of cold maps. As a corollary, for
$p=O(n^{-\alpha})$ warmth of $G(n,p)$ is concentrated on at most two values. As
another corollary, a conjecture of Lov\'asz relating mobility to chromatic
number holds for "almost all" graphs. Finally, our results suggest new
conjectures relating graph parameters from statistical physics with graph
parameters from equivariant topology.
