Shifted critical threshold in the loop o(N) model at arbitrarily small n

In the loop O(n) model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to.where λ, n ∈ [0, ∞). Let µ be the connective constant of the lattice and, for any n ∈ [0, ∞), let λ c (n) be the largest value of λsuch that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that λ c (n) = 1/µ when n = 0 (in this case the model corresponds to the self-avoiding walk) and that for any n ≥ 0, λ c (n) ≥ 1/µ. In this note we prove that,on Z d , with d ≥ 2, and on the hexagonal lattice, where c 0 > 0. This means that, when n is positive (even arbitrarily small), as a consequence of the mutual repulsion between the loops, a phase transition can only occur at a strictly larger critical threshold than in the self-avoiding walk.