Heavy-Tailed Mixed p-Spin Spherical Model: Breakdown of Ultrametricity and Failure of the Parisi Formula
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909709820231680 |
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| author | Kim, Taegyun |
| author_facet | Kim, Taegyun |
| contents | We prove that the two cornerstones of mean-field spin glass theory -- the Parisi variational formula and the ultrametric organization of pure states -- break down under heavy-tailed disorder. For the mixed spherical $p$-spin model whose couplings have tail exponent $α<2$, we attach to each $p$ an explicit threshold $H_p^{*}$. If any coupling exceeds its threshold, a single dominant monomial governs both the limiting free energy and the entire Gibbs measure; the resulting energy landscape is intrinsically probabilistic, with a sharp failure of ultrametricity for $p\ge4$ and persistence of only a degenerate 1-RSB structure for $p\le3$. When all couplings remain below their thresholds, the free energy is $O(n^{-1})$ and the overlap is near zero, resulting in a trivial Gibbs geometry. For $α<1$ we further obtain exact fluctuations of order $n^{1-p}$. Our proof introduces Non-Intersecting Monomial Reduction (NIMR), an algebraic-combinatorial technique that blends convexity analysis, extremal combinatorics and concentration on the sphere, providing the first rigorous description of both regimes for heavy-tailed spin glasses with $p\ge3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heavy-Tailed Mixed p-Spin Spherical Model: Breakdown of Ultrametricity and Failure of the Parisi Formula Kim, Taegyun Probability Mathematical Physics We prove that the two cornerstones of mean-field spin glass theory -- the Parisi variational formula and the ultrametric organization of pure states -- break down under heavy-tailed disorder. For the mixed spherical $p$-spin model whose couplings have tail exponent $α<2$, we attach to each $p$ an explicit threshold $H_p^{*}$. If any coupling exceeds its threshold, a single dominant monomial governs both the limiting free energy and the entire Gibbs measure; the resulting energy landscape is intrinsically probabilistic, with a sharp failure of ultrametricity for $p\ge4$ and persistence of only a degenerate 1-RSB structure for $p\le3$. When all couplings remain below their thresholds, the free energy is $O(n^{-1})$ and the overlap is near zero, resulting in a trivial Gibbs geometry. For $α<1$ we further obtain exact fluctuations of order $n^{1-p}$. Our proof introduces Non-Intersecting Monomial Reduction (NIMR), an algebraic-combinatorial technique that blends convexity analysis, extremal combinatorics and concentration on the sphere, providing the first rigorous description of both regimes for heavy-tailed spin glasses with $p\ge3$. |
| title | Heavy-Tailed Mixed p-Spin Spherical Model: Breakdown of Ultrametricity and Failure of the Parisi Formula |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2506.23987 |