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date: 18 September 2019

Abstract and Keywords

This article discusses the connection between large N matrix models and critical phenomena on lattices with fluctuating geometry, with particular emphasis on the solvable models of 2D lattice quantum gravity and how they are related to matrix models. It first provides an overview of the continuum world sheet theory and the Liouville gravity before deriving the Knizhnik-Polyakov-Zamolodchikov scaling relation. It then describes the simplest model of 2D gravity and the corresponding matrix model, along with the vertex/height integrable models on planar graphs and their mapping to matrix models. It also considers the discretization of the path integral over metrics, the solution of pure lattice gravity using the one-matrix model, the construction of the Ising model coupled to 2D gravity discretized on planar graphs, the O(n) loop model, the six-vertex model, the q-state Potts model, and solid-on-solid and ADE matrix models.

Keywords: matrix model, quantum gravity, continuum world sheet theory, Liouville gravity, scaling relation, planar graph, lattice gravity, Ising model, six-vertex model, Potts model

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