Brightwell and Winkler introduced the graph parameters warmth and mobility in
the context of combinatorial statistical physics. They related both parameters
to lower bounds on chromatic number.

Although warmth is not a monotone graph property we show it is nevertheless
"statistically monotone" in the sense that it tends to increase with added
random edges, and that for sparse graphs ($p=O(n^{-\alpha})$, $\alpha > 0$) the
warmth is concentrated on at most two values, and for most $p$ it is
concentrated on one value. We also put bounds on warmth and mobility in the
dense regime, and as a corollary obtain that a conjecture of Lov\'asz holds for
almost all graphs.