The O(n) vectormodel at n = −1, −2 on random planar lattices: a direct combinatorial derivation

Abstract

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