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Updated: Aug 4, 2026
10:37
Spatial Separation of Molecular Conformers and Clusters
Published on: January 9, 2014
Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses
1Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France. guilhem@1pt.ens.fr
Summary
We introduce a cluster expansion for dilute systems, simplifying entropy and free energy calculations in complex problems like K-satisfiability and the Viana-Bray model. This method confirms the replica symmetric ansatz below the percolation threshold.
Area of Science:
- Statistical Mechanics
- Computational Physics
- Theoretical Computer Science
Background:
- Dilute systems present computational challenges in statistical mechanics and optimization.
- Existing methods like the replica approach have limitations in handling these systems.
Purpose of the Study:
- To develop a systematic cluster expansion for dilute systems.
- To apply this expansion to calculate the entropy of the K-satisfiability problem and the free energy of the Viana-Bray model.
- To validate the replica symmetric (RS) ansatz in specific regimes.
Main Methods:
- A systematic cluster expansion for dilute systems in the highly dilute phase.
- Application to K-satisfiability problem (satisfiable phase) and Viana-Bray model (paramagnetic phase).
- Derivation of series expansions in the average connectivity parameter.
Main Results:
- The cluster expansion yields results identical to the replica approach with a replica symmetric (RS) ansatz for the K-satisfiability problem's entropy.
- The method simplifies the computation of finite-size corrections, crucial for optimization problems.
- Calculations confirm the exactness of the RS ansatz below the percolation threshold.
Conclusions:
- The developed cluster expansion is a powerful tool for analyzing dilute systems.
- It provides a rigorous validation of the RS ansatz in certain regimes.
- The findings suggest potential revisions for the RS ansatz in other transition regions.
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