Replica symmetry breaking at arbitrary depth

Abstract

We study finite-step replica symmetry breaking (RSB) in mixed $p$-spin Ising spin glasses at positive temperature and zero external field. For every integer $k\ge 1$, we construct polynomial mixtures whose Parisi measures are exactly $k$-RSB, with each such phase persisting on an open set of coefficients. More generally, for a broad class of polynomial covariances with an $r$-RSB Parisi measure, adding the interaction term $\lambda q^p$ for sufficiently large even $p$ produces a unique $r$-RSB-to-$(r+1)$-RSB transition as $\lambda$ varies near a critical value. Since such covariances exist for every $r\ge 0$, this yields transitions of arbitrary depth. The main idea of the proof is that, for large $p$, the perturbation $\lambda q^p$ is negligible for overlaps bounded away from $1$, while its effect is concentrated at overlaps very close to $1$.

arXiv comment41 pages; simplified arguments