Exact Solution of the O(n) Model on a Random Lattice
B. Eynard, C. Kristjansen
TL;DR
The authors provide an exact solution for the O(n) model on a random lattice across |n|≤2, unveiling universality with respect to the potential and a pronounced n-independence for -2<n<2. They develop a moment-based, genus-by-genus framework, employing an auxiliary function G(p) to express the genus-zero one-loop correlator and then recursively construct higher-genus correlators and the free energy via a systematic basis of χ-functions. The work delivers explicit genus-zero and genus-one results, shows how to obtain multi-loop and higher-genus quantities, and develops a scaling (double-scaling) analysis that connects to continuum 2D gravity with rational matter, Ising on random lattices, and potential links to kdV hierarchies. Special treatment of n=±2 reinforces the robustness of the method. Overall, the paper establishes a powerful, universal methodology for solving O(n) models on random lattices and explores their critical behavior and continuum limits.
Abstract
We present an exact solution of the $O(n)$ model on a random lattice. The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed for the one-matrix model is found. In addition we find a large degree of universality with respect to $n$; namely for $n\in ]-2,2[$ the solution can be presented in a form which is valid not only for any potential, but also for any $n$ (not necessarily rational). The cases $n=\pm 2$ are treated separately. We give explicit expressions for the genus zero contribution to the one- and two-loop correlators as well as for the genus one contribution to the one-loop correlator and the free energy. It is shown how one can obtain from these results any multi-loop correlator and the free energy to any genus and the structure of the higher genera contributions is described. Furthermore we describe how the calculation of the higher genera contributions can be pursued in the scaling limit.