Eliran Subag, Courant Institute at NYU

Abstract: One of the central ideas in the physical theory for mean-field spin glasses developed in the 80s was that the system decomposes into `pure states', organized in an ultrametric structure. In his seminal work Talagrand (2010) proved for a wide class of models the existence of such a decomposition -- a sequence of subsets on which the Gibbs measure asymptotically concentrates. Panchenko (2013) established the famous ultrametricity conjecture, implying, in particular, that those subsets are organized in a certain hierarchical structure. In the context of the spherical models, I will describe a new geometric picture for the above, in which the hierarchy is expressed through a tree of nested spherical sections. In particular, the pure states concentrate on spherical bands corresponding to the leaves of this tree.