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

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 one to achieve the random matrix solution with no need of this analytical continuation.