2025/06/30 by Taegyun Kim, Kim, Taegyun
Physics and Astronomy · Mathematics · #Theoretical and Computational Physics #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2506.23987
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 Hp*. 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≥4 and persistence of only a degenerate 1-RSB structure for p≤3. 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 n1-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≥3.