The cavity method is a mathematical method presented by
Marc Mézard,
Giorgio Parisi
Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods ...
and
Miguel Angel Virasoro in 1987
to derive and solve some
mean field-type models in
statistical physics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, specially adapted to disordered systems. The method has been used to compute properties of
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
s in many
condensed matter
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases, that arise from electromagnetic forces between atoms and electrons. More gen ...
and
optimization problem
In mathematics, engineering, computer science and economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...
s.
Initially invented to deal with the
Sherrington–Kirkpatrick model
In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of Spin (physics), spins at a temperature called the "freezing temperature," ''T''f. In Ferromagnetism, ferroma ...
of
spin glass
In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called the "freezing temperature," ''T''f. In ferromagnetic solids, component atoms' ...
es, 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
In the statistical physics of spin glasses and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula:
\ln Z=\lim_
or:
\ln Z = \lim_ \frac
where
Z is most commonly the partit ...
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 problem
In mathematics, engineering, computer science and economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...
s such as
k-satisfiability and
graph coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have th ...
. 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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, but is also closely related to methods from other areas such as
belief propagation
Belief propagation, also known as sum–product message passing, is a message-passing algorithm for performing inference on graphical models, such as Bayesian networks and Markov random fields. It calculates the marginal distribution for ea ...
.
References
Further reading
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Condensed matter physics
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