Cavity method

The cavity method is a mathematical method presented by Marc Mézard, Giorgio Parisi and Miguel Angel Virasoro in 1987 to derive and solve some mean field-type models in statistical physics, specially adapted to disordered systems. The method has been used to compute properties of ground states in many condensed matter and optimization problems.

Initially invented to deal with the Sherrington–Kirkpatrick model of spin glasses, the cavity method has shown wider applicability. It can be regarded as a generalization of the Bethe–Peierls iterative method in tree-like graphs, to the case of a graph with loops that are not too short. The cavity method can solve many problems also solvable using the replica trick but has the advantage of being more intuitive and less mathematically subtle than replica-based methods.

The cavity method proceeds by perturbing a large system with the addition of a non-thermodynamic number of additional constituents and approximating the response of the entire system perturbatively. The application of the resulting approximation, along with an assumption that certain observables are self-averaging, yields a self-consistency equation for the statistics of the added constituents. The added constituents are then considered to be the mean-field variables.

The cavity method has proved useful in solving optimization problems such as k-satisfiability and graph coloring. It has yielded not only ground states energy predictions in the average case but has also inspired algorithmic methods.

See also

The cavity method originated in the context of statistical physics, but is also closely related to methods from other areas such as belief propagation.

Further reading

• Braunstein, A.; Mézard, M.; Zecchina, R. (2005). "Survey propagation: An algorithm for satisfiability". Random Structures and Algorithms. 27 (2): 201–226. arXiv:cs.CC/0212002. doi:10.1002/rsa.20057. ISSN 1042-9832. S2CID 6601396.
• Potters, Marc; Bouchaud, Jean-Philippe (2020). A first course in random matrix theory: for physicists, engineers and data scientists. Cambridge; New York, NY: Cambridge University Press. ISBN 978-1-108-48808-2.
• Mézard, M.; Parisi, G. (2001). "The Bethe lattice spin glass revisited". The European Physical Journal B. 20 (2): 217–233. arXiv:cond-mat/0009418. Bibcode:2001EPJB...20..217M. doi:10.1007/PL00011099. ISSN 1434-6028. S2CID 59494448.
• Mézard, Marc; Parisi, Giorgio (2003). "The Cavity Method at Zero Temperature". Journal of Statistical Physics. 111 (1/2): 1–34. arXiv:cond-mat/0207121. Bibcode:2003JSP...111....1M. doi:10.1023/A:1022221005097. ISSN 0022-4715. S2CID 116942750.
• Krz̧akała, Florent; Montanari, Andrea; Ricci-Tersenghi, Federico; Semerjian, Guilhem; Zdeborová, Lenka (2007). "Gibbs states and the set of solutions of random constraint satisfaction problems". Proceedings of the National Academy of Sciences of the United States of America. 104 (2): 10318–10323. arXiv:cond-mat/0612365. Bibcode:2007PNAS..10410318K. doi:10.1073/pnas.0703685104. ISSN 0027-8424. PMC 1965511. PMID 17567754. S2CID 10018706.
• Advani, Madhu; Bunin, Guy; Mehta, Pankaj (2018). "Statistical physics of community ecology: a cavity solution to MacArthur's consumer resource model". Journal of Statistical Physics. 2018 (3): 033406. Bibcode:2018JSMTE..03.3406A. doi:10.1088/1742-5468/aab04e. PMC 6329381. PMID 30636966.