Statistical Mechanics of the Sub-Optimal Transport
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910024372060160 |
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| author | Piombo, Riccardo Buffa, Lorenzo Mazzilli, Dario Patelli, Aurelio |
| author_facet | Piombo, Riccardo Buffa, Lorenzo Mazzilli, Dario Patelli, Aurelio |
| contents | Statistical mechanics is a powerful framework for analyzing optimization yielding analytical results for matching, optimal transport, and other combinatorial problems. However, these methods typically target the zero-temperature limit, where systems collapse onto optimal configurations, a.k.a. the ground states. Real-world systems often occupy intermediate regimes where entropy and cost minimization genuinely compete, producing configurations that are structured yet sub-optimal. The Sub-Optimal Transport (SOT) model captures this competition through an ensemble of weighted bipartite graphs: a coupling parameter interpolates between entropy-dominated dense configurations and cost-dominated sparse structures. This crossover has been observed numerically but lacked analytical understanding. Here we develop a mean-field theory that characterizes this transition. We show that local fluctuations in Lagrange multipliers become sub-extensive in the thermodynamic limit, reducing the full model with strength constraints to an effective single-constraint problem admitting an exact solution in some intermediate regime. The resulting free energy is analytic in the coupling parameter, confirming a smooth crossover rather than a phase transition. We derive closed-form expressions for thermodynamic observables and weight distributions, validated against numerical simulations. These results establish the first analytical description of the SOT model, extending statistical mechanics methods beyond the zero-temperature regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04308 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Statistical Mechanics of the Sub-Optimal Transport Piombo, Riccardo Buffa, Lorenzo Mazzilli, Dario Patelli, Aurelio Statistical Mechanics Disordered Systems and Neural Networks Statistical mechanics is a powerful framework for analyzing optimization yielding analytical results for matching, optimal transport, and other combinatorial problems. However, these methods typically target the zero-temperature limit, where systems collapse onto optimal configurations, a.k.a. the ground states. Real-world systems often occupy intermediate regimes where entropy and cost minimization genuinely compete, producing configurations that are structured yet sub-optimal. The Sub-Optimal Transport (SOT) model captures this competition through an ensemble of weighted bipartite graphs: a coupling parameter interpolates between entropy-dominated dense configurations and cost-dominated sparse structures. This crossover has been observed numerically but lacked analytical understanding. Here we develop a mean-field theory that characterizes this transition. We show that local fluctuations in Lagrange multipliers become sub-extensive in the thermodynamic limit, reducing the full model with strength constraints to an effective single-constraint problem admitting an exact solution in some intermediate regime. The resulting free energy is analytic in the coupling parameter, confirming a smooth crossover rather than a phase transition. We derive closed-form expressions for thermodynamic observables and weight distributions, validated against numerical simulations. These results establish the first analytical description of the SOT model, extending statistical mechanics methods beyond the zero-temperature regime. |
| title | Statistical Mechanics of the Sub-Optimal Transport |
| topic | Statistical Mechanics Disordered Systems and Neural Networks |
| url | https://arxiv.org/abs/2602.04308 |