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Full replica symmetry breaking in the Sherrington-Kirkpatrick model

  • Patrick Lopatto
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Abstract

We prove that, at zero external field and for every inverse temperature β>1, the Parisi measure of the Sherrington-Kirkpatrick model is supported on an interval [0,qβ], with a smooth density on (0,qβ) and an atom at qβ. The primary difficulty is to exclude gaps in the support. On such a gap, the optimality conditions for the Parisi measure would force the second derivative of the associated self-consistency function Γ to cross zero downward. We establish that it can only cross upward by showing that at a zero of Γ'', the derivative Γ''' is a positive linear combination of two nonnegative covariances, one of which is strictly positive.

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Cite

BibTeXRIS
BibTeX
@misc{lopatto2026full,
  title = {{Full replica symmetry breaking in the Sherrington-Kirkpatrick model}},
  author = {Patrick Lopatto},
  howpublished = {arXiv},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2607.11756},
  eprint = {2607.11756},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.11756},
}
RIS
TY  - UNPB
TI  - Full replica symmetry breaking in the Sherrington-Kirkpatrick model
AU  - Patrick Lopatto
PY  - 2026
DA  - 2026/07/13
PB  - arXiv
DO  - 10.48550/arxiv.2607.11756
UR  - https://arxiv.org/abs/2607.11756
AB  - We prove that, at zero external field and for every inverse temperature β>1, the Parisi measure of the Sherrington-Kirkpatrick model is supported on an interval [0,q_β], with a smooth density on (0,q_β) and an atom at q_β. The primary difficulty is to exclude gaps in the support. On such a gap, the optimality conditions for the Parisi measure would force the second derivative of the associated self-consistency function Γ to cross zero downward. We establish that it can only cross upward by showing that at a zero of Γ'', the derivative Γ''' is a positive linear combination of two nonnegative covariances, one of which is strictly positive.
SN  - 2331-8422
ER  -