From Parisi to Boltzmann

par Nicola Kistler

jeudi 26 févr. 2026, 14:00 → 15:00 Europe/Paris

435 (ENS)

The Sherrington-Kirkpatrick model for spin glasses is traditionally viewed as a Gaussian field on the N-dimensional hypercube, with covariance given by the overlap, i.e. the scalar product of two configurations. 
In this talk, we shall take a rather un-orthodox point of view: instead of interaction disorder as the main source of randomness, 
we shall use thermal fluctuations to drive probabilistic phenomena.
The framework cogently links the Thouless-Anderson-Palmer equations (TAP), the Generalized Random Energy Models (GREM)
by Derrida, the limiting Derrida-Ruelle cascades, and the Parisi solution, thereby completely bypassing Guerra's interpolations
and the Ghirlanda-Guerra identities. 
TAP equations are treated as a genuine dynamical system: TAP solutions emerge algorithmically through carefully engineered 
GREM-like seeds which promote quenched magnetizations to functional degrees of freedom. 
This gives rise to a hierarchy of quenched central limit theorems for the effective fields and perceptron-like Hamiltonians 
which are then analyzed via quenched large deviations principles yielding classical Boltzmann-Gibbs variational principles. 
The approach refines and extends the renormalization group treatment of Dotsenko, implements Bolthausen's insight of 
"contraction toward ultrametricity", and provides a rigorous foundation for the dynamical theories of Sompolinsky, 
and Sommers-de Dominicis-Gabay.  It shows that the Parisi solution — and in particular: the spontaneous organisation into 
hierarchical structures — is not an ansatz or mathematical artefact, but a law of nature: the unique, self-consistent structure 
that any complex disordered system is compelled to obey. The same principles in fact apply to Hopfield and multi-species models, 
the perceptron, or, for that matter, any model where TAP-like equations steer the thermodynamics. 
Time permitting, we shall also discuss visionary yet largely overlooked conjectures by Mézard and Virasoro which identify the origin
of ultrametricity in purely random matrix terms.