Replica Symmetry Breaking, Shattering, and Metastability
The statics and dynamics of mean-field models of spin glasses have been studied in-depth by the physics community since the '70s. At the heart of this is the trade-off between the notions of replica symmetry breaking, shattering, and metastability. I will survey the current mathematical understanding of these ideas in the “simple” case of the p-spin model. I will start by providing a working definition of replica symmetry breaking and survey our current understanding of the Gibbs measure in this phase. I'll then turn to our recent joint works with Ben Arous and Gamarnik-Kizildag on the “replica symmetric” phase. Here we prove the existence of a "shattering phase". This phase was first conjectured by Kirkpatrick-Thirumalai in the 80s and does not exist in classical (non-disordered) spin systems as it involves in interplay between two distinct sources of entropy—classical “microcanonical entropy” as well as a “configurational entropy” (now called the landscape complexity). I first discuss this picture from the perspective of spherical p-spin glasses where there is a concrete geometric picture. We then show that in these models, metastable states exist up to an even higher temperature as first predicted by Barrat–Burioni–Mezard. I will then review recent progress in the hypercube setting with Gamarnik and Kizildag which establishes the existence of a strong shattering in Ising p-spin models, verifying the original prediction of Kirkpatrick-Thirumalai from the 80s. I will end by presenting a series of open questions and conjectures surrounding sharp phase boundaries for shattering and metastability.
This talk will touch on joint work with: A. Auffinger (Northwestern), G. Ben Arous (Courant), D. Gamarnik (MIT), and E. Kizildag (Univeristy of Illinois, Urbana-Chamagne), R. Gheissari (Northwestern), and I. Tobasco (Rutgers). I will not assume prior knowledge in statistical physics.