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Published online by Cambridge University Press: 25 June 2025
We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these to the study of various folding problems arising from physics, including: the meander (or polymer folding) problem “enumeration of all topologically inequivalent closed nonintersecting plane curves intersecting a line through a given number of points” and a fluid membrane folding problem reformulated as that of “enumerating all vertex-tricolored triangulations of arbitrary genus, with given numbers of vertices of either color”.
1. Introduction
Our first aim of this article is to convince the reader that matrix integrals, exactly calculable or not, can always be interpreted in some sort of combinatorial way as generating functions for decorated graphs of given genus, with possibly specified vertex and/or face valencies. We show this by expressing pictorially the processes involved in computing Gaussian integrals over matrices, what physicists call generically Feynman rules. These matrix diagrammatic techniques have been first developed in the context of quantum chromodynamics in the limit of large number of colors (the size of the matrix) [1; 2], and more recently in the context of two-dimensional quantum gravity, namely the coupling of two-dimensional statistical models (matter theories) to the fluctuations of the two-dimensional space into surfaces of arbitrary topologies (gravity) [3]. These toy models for noncritical string theory are a nice testing ground for physical ideas, and have led to many confirmations of continuum field-theoretical results in quantum gravity.
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