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Parisi measure of the Sherrington-Kirkpatrick model with external field
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.
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BibTeX
@misc{lopatto2026parisi,
title = {{Parisi measure of the Sherrington-Kirkpatrick model with external field}},
author = {Patrick Lopatto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00127},
eprint = {2610.00127},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00127},
}
RIS
TY - UNPB
TI - Parisi measure of the Sherrington-Kirkpatrick model with external field
AU - Patrick Lopatto
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00127
UR - https://arxiv.org/abs/2610.00127
AB - 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.
ER -