  We study partition functions of random Bergman metrics, with the actions
defined by a class of geometric functionals known as `stability functions'. We
introduce a new stability invariant - the critical value of the coupling
constant - defined as the minimal coupling constant for which the partition
function converges. It measures the minimal degree of stability of geodesic
rays in the space the Bergman metrics, with respect to the action. We calculate
this critical value when the action is the $\nu$-balancing energy, and show
that $\gamma_k^{\rm crit} = k - h$ on a Riemann surface of genus $h$.
