Various combinations of spin-spin correlations averaged over the randomness in spin glasses have long been known to be long-ranged, that is falling off like a power of the distance. This implies that correlations must be long ranged in some sense already before averaging, that is they must be long ranged with finite probability also in individual samples. The motivation for the study of correlations in individual samples of the paradigmatic case of spin glasses comes from the consideration that long range correlations seem to be a common feature of several complex systems, and in some of the real life complex systems not only the averaged behaviour, but also the individual samples are important. Numerical simulations in the mean field model [1] have already showed that the distribution of the simplest spin-spin correlations is very broad at low temperatures. We performed similar studies on low dimensional (2d,3d) spin glass samples and found broad correlation distributions again. (In 2d there is no equilibrium spin glass phase, but at low temperatures we found extremely long-lived quasi-stable patterns that allowed us to measure the correlations.) The dependence of the correlations on distance in Euclidean lattices is random, with anomalously large absolute values showing up with finite probability even for large distances. The correlations extremely sensitively depend on the realization of the random couplings and on the boundary conditions, but any averaging (over the randomness, or over some spherical regions around a reference site) washes away all this structure. The talk will conclude with some speculations concerning the possible relevance of these findings to real life complex systems, such as traffic, electric power, or financial networks.
Reference [1] A. Billoire, I. Kondor, J. Lukic and E. Marinari: Large random correlations in individual mean field spin glass samples, J. Stat. Mech. P02009 (2011)