Thermal flow in the gravitational O(n) model

We study the massless flow from the critical point (dilute loops) to the low-temperature phase (dense loops) of the O(n) loop gas model when the model is coupled to 2D gravity. The flow is generated by the gravitationally dressed thermal operator \Phi_{1,3} coupled to the renormalized loop tension \lambda ~ T-T_c. We find that the susceptibility as a function of the thermal coupling \lambda and the cosmological constant \mu satisfies a simple transcendental equation.