Cavity method

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The Cavity method is a mathematical method presented by M. Mezard, Giorgio Parisi and Miguel Angel Virasoro in 1985 to solve some mean field type of models in statistical physics, specially adapted to disordered systems. It 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, it 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 graph with loops that are not too short. The different approximations that can be done with the cavity method are usually named after their equivalent with the different steps of the replica method which is mathematically more subtle and less intuitive than the cavity approach.

The cavity method has played and is playing a major role in the solution of optimization problems like the K-satisfiability and the graph coloring in present days. It has yielded not only ground states energy predictions in the average case, but also has inspired algorithmic methods for solving particular instances of an optimization problem.

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The cavity method has been created in the context of statistical physics, and is closely related to other methods and algorithms in different fields of mathematics such as belief propagation. It has been created initially in the context of spin glasses as an alternative of the replica trick.

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