Seminar by Ahmed El Alaoui

Shattering and chaos in pure spherical spin glasses

Given a high-dimensional probability measure, shattering is the property that this measure puts nearly all its mass, in roughly equal proportions, on an exponential number of small ``clusters", and that the clusters are all far apart. In the case of mean-field disordered spin systems, this property is believed to be a barrier to efficient sampling. In this talk I will discuss the case of the pure spherical p-spin model for large p, and show the existence of a large, essentially optimal, interval of temperatures where shattering occurs. Next I will discuss the consequences of shattering on the computational tractability of sampling from the corresponding Gibbs measure. I will introduce the notion of transport disorder chaos, show that it is implied by shattering, and then show that it implies failure of "stable" sampling algorithms.