Ground States of the Mean-Field Spin Glass with 3-Spin Couplings

• URL: http://arxiv.org/abs/2501.13205v1
• Date: Wed, 22 Jan 2025 20:34:39 GMT
• Title: Ground States of the Mean-Field Spin Glass with 3-Spin Couplings
• Authors: Stefan Boettcher, Ginger E. Lau,
• Abstract summary: We use optimization methods to determine with low systematic error ground state configurations of the mean-field $p$-spin glass model with $p=3$.<n>For the bond-diluted case, the energy density and its distribution appear to scale with $ln N/N$ and $1/N$ corrections, respectively.<n>This would suggest that all $p$-spin models with $pgeq3$ exhibit the same ground-state corrections as in REM.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We use heuristic optimization methods in extensive computations to determine with low systematic error ground state configurations of the mean-field $p$-spin glass model with $p=3$. Here, all possible triplets in a system of $N$ Ising spins are connected with a bond. This model has been of recent interest, since it exhibits the ``overlap gap condition'', which should make it prohibitive to find ground states asymptotically with local search methods when compared, for instance, with the $p=2$ case better-known as the Sherrington-Kirkpatrick model (SK). Indeed, it proves more costly to find good approximations for $p=3$ than for SK, even for our heuristic. Compared to SK, the ground-state behavior for $p=3$ is quite distinct also in other ways. For SK, finite-size corrections for large system sizes $N\to\infty$ of both, the ensemble average over ground state energy densities and the width of their distribution, vary anomalously with non-integer exponents. In the $p=3$ case here, the energy density and its distribution appear to scale with $\ln N/N$ and $1/N$ corrections, respectively. The distribution itself is consistent with a Gumbel form. Even more stark is the contrast for the bond-diluted case, where SK has shown previously a notable variation of the anomalous corrections exponent with the bond density, while for $p=3$ no such variation is found here. Hence, for the 3-spin model, all measured corrections scale the same as for the random energy model (REM), corresponding to $p=\infty$. This would suggest that all $p$-spin models with $p\geq3$ exhibit the same ground-state corrections as in REM.

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