O(n) vector model at n=-1, -2 on random planar lattices: a direct combinatorial derivation

TL;DR

This paper provides a combinatorial derivation of the O(n) vector model at n=-1 and n=-2 on random planar lattices, simplifying the solution process without requiring analytical continuation.

Contribution

It introduces a combinatorial reformulation that directly solves the model at n=-1 and n=-2, bypassing the need for analytical continuation in the matrix approach.

Findings

Achieves direct combinatorial solutions for n=-1 and n=-2.

Simplifies the analysis of the O(n) model on random lattices.

Provides insights into the structure of the loop gas model at specific n values.

Abstract

The O(n) vector model with logarithmic action on a lattice of coordination 3 is related to a gas of self-avoiding loops on the lattice. This formulation allows for analytical continuation in n: critical behaviour is found in the real interval [-2,2]. The solution of the model on random planar lattices, recovered by random matrices, also involves an analytic continuation in the number n of auxiliary matrices. Here we show that, in the two cases n=-1, -2, a combinatorial reformulation of the loop gas problem allows to achieve the random matrix solution with no need of this analytical continuation.