Nicholas Read – Yale University
Wednesday, October 23rd, 10:45AM EDT
Structural analysis of Gibbs states and metastates in short-range classical spin glasses
The old problem of the equilibrium spin systems with quenched disorder remains unsolved and controversial. Different points of view concern whether the equilibrium (Gibbs) states in the low-temperature spin-glass phase of a short-range system decompose into many ordered or ``pure'' states (as suggested by replica symmetry breaking theory (RSB)) or whether instead there are only one or two pure states, related by symmetry (the basis for the scaling-droplet theory). The question does not have much meaning in a finite size system, and so Gibbs states in infinite size are essential. Only rigorous approaches seem likely to shed much light on this problem. To analyze infinite size, the concept of metastate (introduced by Aizenman, Wehr, Newman,and Stein) is essential. A metastate is a probability distribution on Gibbs states at given disorder, with certain properties. In this talk we aim to: (i) clarify a little what RSB suggests for finite-dimensional short-range systems; (ii) describe recent progress in rigorous results on these systems; (iii) point out that these results are in accord with, and certainly do not rule out, the predictions of RSB for short-range systems. The rigorous results include a new concept of decomposable metastates, and the result that any metastate can be decomposed into indecomposable metastates. Another older idea is stochastic stability; we prove that any metastate is stochastically stable, which means that the statistical properties are unchanged by addition of very weak independent disorder. This has a number of consequences that are new statements about the structure of the decomposition into pure states of a Gibbs state drawn from the metastate, and the form of the metastate. Particular scenarios such as ``dynamically-frozen'' states, widely thought to have already been ruled out heuristically, will be discussed if there is sufficient time. All results describe what is allowed in a metastate; existence questions of whether non-trivial Gibbs states actually occur will not be considered.