Year: 2026
Type: preprint
Abstract: We prove that the Parisi measure of the Sherrington--Kirkpatrick model has connected support at every positive temperature and every positive external field. When the model breaks replica symmetry, its support is an interval contained in $(0,1)$, its restriction to the interior has a smooth density that remains strictly positive up to both endpoints, and both endpoints carry atoms. The proof rests on a strict inequality for moments of spatial derivatives of the Parisi PDE, which both excludes gaps in the support and forces positivity of the density. The same estimates recover the zero-field support description and give a lower bound for the mass of its upper atom.
Source: arXiv (Cornell University)
DOI: 10.17615/y14r-cv11
Author: Patrick Lopatto
Institution: University of North Carolina at Chapel Hill
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213275213