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≥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 λqp for sufficiently large even p produces a unique r-RSB-to-(r+1)-RSB transition as λ varies near a critical value. Since such covariances exist for every r≥0, this yields transitions of arbitrary depth. The main idea of the proof is that, for large p, the perturbation λqp is negligible for overlaps bounded away from $1$, while its effect is concentrated at overlaps very close to $1$.
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BibTeX
@misc{aronow2026replica,
title = {{Replica symmetry breaking at arbitrary depth}},
author = {P. M. Aronow and Patrick Lopatto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.05317},
eprint = {2609.05317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05317},
}
RIS
TY - UNPB
TI - Replica symmetry breaking at arbitrary depth
AU - P. M. Aronow
AU - Patrick Lopatto
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2609.05317
UR - https://arxiv.org/abs/2609.05317
AB - 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≥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 λq^p for sufficiently large even p produces a unique r-RSB-to-(r+1)-RSB transition as λ varies near a critical value. Since such covariances exist for every r≥0, this yields transitions of arbitrary depth. The main idea of the proof is that, for large p, the perturbation λq^p is negligible for overlaps bounded away from $1$, while its effect is concentrated at overlaps very close to $1$.
ER -