Ultrametricity in the infinite-range Ising spin glass

Monte Carlo simulations have been used to study the development of ultrametricity in the low-temperature phase of the infinite-range Sherrington-Kirkpatrick model of the Ising spin glass. Equilibrium configuration overlap data have been obtained for sizes N between 16 and 512 spins at the transition temperature T=T_c and in the spin-glass phase at T=0.6T_c. At the lower temperature, the data are consistent with the emergence, in the thermodynamic limit, of isosceles triangles formed by the mutual overlaps of three configurations, as predicted by Mezard and co-workers (1984, 1987). Evidence for equilateral triangles is not clear and may require larger sizes. This behaviour is contrasted with the data at T=T_c, where all overlap distributions are found to scale with size in terms of a single exponent.