nLab matrix model

Idea

Matrix models are physical models in which the dynamical quantities are square matrices (in certain class of matrices, e.g. hermitian), in other words, a Lagrangian/Hamiltonian depends on matrix quantities and is usually taken at the limit when the size of matrix tends to infinity.

Matrix models are studied mainly in the context of statistical mechanics (see random matrix theory) or in quantum field theory.

Related entries

• Fredholm determinant, random matrix theory, determinental process?, Kontsevich matrix model?, large N limit

• integrable system, topological recursion

Matrix models that have been argued to capture D-brane dynamics and nonperturbative effects in string theory include

• BFSS matrix model

• IKKT matrix model

• BMN matrix model

• membrane matrix model

• K-matrix model

• matrix model for the 3-brane in 6d

References

For references on the BFSS-, IKKT- and BMN matrix models see there.

First inkling of matrix models from the large N limit of QCD:

• Tohru Eguchi, Hikaru Kawai, Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory, Phys. Rev. Lett. 48, 1063 (1982) (spire:176459, doi:10.1103/PhysRevLett.48.1063)

• A. Gonzalez-Arroyo, M. Okawa, A twisted model for large $N$ lattice gauge theory, Physics Letters B Volume 120, Issues 1–3, 6 January 1983, Pages 174-178 (doi:10.1016/0370-2693(83)90647-0)

• A. Gonzalez-Arroyo, M. Okawa, Twisted-Eguchi-Kawai model: A reduced model for large- $N$ lattice gauge theory, Phys. Rev. D 27, 2397 (1983) (doi:10.1103/PhysRevD.27.2397)

Discussion of topological recursion for matrix models originates with

• Leonid Chekhov, Bertrand Eynard, Nicolas Orantin, Free energy topological expansion for the 2-matrix model, JHEP 0612:053,2006 (arXiv:math-ph/0603003)

See also

• Raghav G. Jha, Introduction to Monte Carlo for Matrix Models (arXiv:2111.02410)

• Chong-Sun Chu, A Matrix Model Proposal for Quantum Gravity and the Quantum Mechanics of Black Holes [arXiv:2406.01466]

category: physics