Infinite support of the Parisi measure in the Sherrington-Kirkpatrick model

Abstract

We prove that, at zero external field and every inverse temperature $β>1$, the Parisi measure of the Sherrington-Kirkpatrick model has infinite support. The proof proceeds by contradiction. Assuming that the Parisi measure is finitely supported, we use the Gaussian Cole-Hopf representation of the Parisi PDE solution to derive a differential inequality that is incompatible with the variational optimality conditions for the Parisi minimizer established by Jagannath and Tobasco (2017).