# Matrix model A matrix model is a model in which a matrix or multiple matrices are the dynamical objects. ## Refs - 't Hooft expansion - [[1974#'t Hooft]] - topological recursion - [[1983#Migdal]]: so-called loop equations determine all order results in the large-$N$ expansion of the correlators, where $N$ is the size of the matrix - [[2004#Eynard]] and [[2007#Eynard, Orantin]]: topological recursion relations - topological matrix models (no double scaling) - [[1992#Kontsevich]] - doubled scaled matrix integral - [[1990#Brezin, Kazakov]] - [[1990#Douglas, Shenker]] - [[1990#Gross, Migdal]] - $c=1$ string - [[2003#McGreevy, Verlinde]]: matrix model for $c=1$ strings as the open string field theory for an infinite number of D0-branes - [[2003#Klebanov, Maldacena, Seiberg]]: the double-scaled matrix model can be understood as arising from [[0652 ZZ brane|ZZ branes]] - Hilbert space in ETH matrix model - [[2025#Miyaji, Mori, Okuyama]] - misc topics - [[Raman2021]][](https://arxiv.org/pdf/2110.06643.pdf): replica - [[KarLamprouMarteauRosso2022]][](https://arxiv.org/pdf/2208.05974.pdf): flat space dual ## The Vandermonde determinant A matrix integral for a Hermitian matrix $H$ given by$\mathcal{M}_N=\int_{\mathbb{R}^{N^2}}[\mathrm{d} H]\, \mathrm{e}^{-N \operatorname{tr} V(H)}$can be recast by diagonalising the matrix into$\mathcal{M}_N=\int_{\mathbb{R}^N} \prod_{i=1}^N \mathrm{~d} \lambda_i \mathrm{e}^{-N^2 S[\lambda]},$where$S[\lambda]=\frac{1}{N} \sum_{i=1}^N V\left(\lambda_i\right)-\frac{1}{N^2} \sum_{i \neq j} \log \left|\lambda_i-\lambda_j\right|.$The second term is known as the Vandermonde, which comes from the Jacobian of the diagonalisation. ## Double scaling limit The double scaling limit involves taking the size of the matrix to infinity. At the same time, one focused on the edge of the spectrum. ## Related topics - [[0471 String-matrix duality]] - [[0579 Random matrix theory]]