# Cavity method > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Cavity_method > Markdown URL: https://mediated.wiki/source/Cavity_method.md > Source: https://en.wikipedia.org/wiki/Cavity_method > Source revision: 1348316734 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Mathematical method in statistical physics}} The '''cavity method''' is a mathematical method presented by [Marc Mézard](/source/Marc_M%C3%A9zard), [Giorgio Parisi](/source/Giorgio_Parisi) and [Miguel Angel Virasoro](/source/Miguel_%C3%81ngel_Virasoro_(physicist)) in 1987{{cite book|last1=Mézard|first1=M.|last2=Parisi|first2=G.|last3=Virasoro|first3=M.|title=Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications|publisher=World Scientific Publishing Company|volume=9|year=1987|bibcode=1987sgtb.book.....M }} to derive and solve some [mean field](/source/mean_field)-type models in [statistical physics](/source/statistical_physics), specially adapted to disordered systems. The method has been used to compute properties of [ground state](/source/ground_state)s in many [condensed matter](/source/condensed_matter) and [optimization problem](/source/optimization_problem)s. Initially invented to deal with the [Sherrington–Kirkpatrick model](/source/Sherrington%E2%80%93Kirkpatrick_model) of [spin glass](/source/spin_glass)es, the cavity method has shown wider applicability. It can be regarded as a generalization of the [Bethe](/source/Hans_Bethe)–[Peierls](/source/Rudolf_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](/source/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](/source/Perturbation_theory). The application of the resulting approximation, along with an assumption that certain observables are [self-averaging](/source/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 problem](/source/optimization_problem)s such as [k-satisfiability](/source/Boolean_satisfiability_problem) and [graph coloring](/source/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](/source/statistical_physics), but is also closely related to methods from other areas such as [belief propagation](/source/belief_propagation). ==References== {{reflist}} ==Further reading== * {{cite journal|last1=Braunstein|first1=A.|last2=Mézard|first2=M.|last3=Zecchina|first3=R.|title=Survey propagation: An algorithm for satisfiability|journal=Random Structures and Algorithms|volume=27|issue=2|year=2005|pages=201–226|issn=1042-9832|doi=10.1002/rsa.20057|arxiv=cs.CC/0212002|s2cid=6601396}} * {{Cite book |last=Potters |first=Marc |title=A first course in random matrix theory: for physicists, engineers and data scientists |last2=Bouchaud |first2=Jean-Philippe |date=2020 |publisher=Cambridge University Press |isbn=978-1-108-48808-2 |location=Cambridge; New York, NY}} * {{cite journal|last1=Mézard|first1=M.|last2=Parisi|first2=G.|author-link2=Giorgio Parisi|title=The Bethe lattice spin glass revisited|journal=The European Physical Journal B|volume=20|issue=2|year=2001|pages=217–233|issn=1434-6028|doi=10.1007/PL00011099|arxiv=cond-mat/0009418|bibcode=2001EPJB...20..217M |s2cid=59494448}} * {{cite journal|last1=Mézard|first1=Marc|last2=Parisi|first2=Giorgio|author-link2=Giorgio Parisi|journal=Journal of Statistical Physics|title=The Cavity Method at Zero Temperature|volume=111|issue=1/2|year=2003|pages=1–34|issn=0022-4715|doi=10.1023/A:1022221005097|arxiv=cond-mat/0207121|bibcode=2003JSP...111....1M |s2cid=116942750}} * {{cite journal|last1=Krz̧akała|first1=Florent|last2=Montanari|first2=Andrea|last3=Ricci-Tersenghi|first3=Federico|last4=Semerjian|first4=Guilhem|last5=Zdeborová|first5=Lenka|author-link5=Lenka Zdeborova|journal=Proceedings of the National Academy of Sciences of the United States of America|title=Gibbs states and the set of solutions of random constraint satisfaction problems|volume= 104|issue=2|year=2007|pages=10318–10323|issn=0027-8424|doi=10.1073/pnas.0703685104|pmid=17567754|pmc=1965511|arxiv=cond-mat/0612365|bibcode=2007PNAS..10410318K |s2cid=10018706|doi-access=free}} * {{cite journal|last1=Advani|first1=Madhu|last2=Bunin|first2=Guy|last3=Mehta|first3=Pankaj|journal=Journal of Statistical Physics|title=Statistical physics of community ecology: a cavity solution to MacArthur's consumer resource model|volume=2018|year=2018|issue=3 |pages=033406|doi=10.1088/1742-5468/aab04e|pmid= 30636966|pmc=6329381|bibcode=2018JSMTE..03.3406A }} Category:Condensed matter physics {{CMP-stub}} --- Adapted from the Wikipedia article [Cavity method](https://en.wikipedia.org/wiki/Cavity_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cavity_method?action=history)). 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