Statistical Mechanics of the Sub-Optimal Transport

TL;DR

This paper develops a mean-field theory to analytically describe the transition in sub-optimal transport systems, capturing the competition between entropy and cost in intermediate regimes, extending statistical mechanics methods.

Contribution

It provides the first analytical framework for the Sub-Optimal Transport model, revealing a smooth crossover between dense and sparse configurations.

Findings

Derived exact solutions for the free energy and weight distributions.

Validated theoretical predictions against numerical simulations.

Established the first analytical description of the SOT model.

Abstract

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.