  Using Monte Carlo simulations, we study the character of the spin-glass (SG)
state of a site-diluted dipolar Ising model. We consider systems of dipoles
randomly placed on a fraction x of all L^3 sites of a simple cubic lattice that
point up or down along a given crystalline axis. For x < 0.65 these systems are
known to exhibit an equilibrium spin-glass phase below a temperature T_sg
proportional to x. At high dilution and very low temperatures, well deep in the
SG phase, we find spiky distributions of the overlap parameter q that are
strongly sample-dependent. We focus on spikes associated with large
excitations. From cumulative distributions of q and a pair correlation function
averaged over several thousands of samples we find that, for the system sizes
studied, the average width of spikes, and the fraction of samples with spikes
higher than a certain threshold does not vary appreciably with L. This is
compared with the behaviour found for the Sherrington-Kirkpatrick model.
