Which measures of spin-glass overlaps are informative?

The nature of equilibrium states in disordered materials is often studied using an overlap function P(q), the probability of two configurations having similarity q. Exact sampling simulations of a two-dimensional proxy for three-dimensional spin glasses indicate that common measures of P(q) in smaller samples do not decide between theoretical pictures. Strong corrections result from P(q) being an average over many scales, as seen in a toy droplet model. However, the median of the integrals of sample-dependent P(q) curves shows promise for deciding the thermodynamic behavior.