Boundary Loop Models and 2D Quantum Gravity

We study the O(n) loop model on a dynamically triangulated disk, with a new type of boundary conditions, discovered recently by Jacobsen and Saleur. The partition function of the model is that of a gas of self and mutually avoiding loops covering the disk. The Jacobsen-Saleur (JS) boundary condition prescribes that the loops that do not touch the boundary have fugacity n ∈ [−2, 2], while the loops touching at least once the boundary are given different fugacity y. The class of JS boundary conditions, labeled by the real number y, contains the Neumann (y = n) and Dirichlet (y = 1) boundary conditions as particular cases. Here we consider the dense phase of the loop gas, where we compute the two-point boundary correlators of the L-leg operators with mixed Neumann-JS boundary condition. The result coincides with the boundary two-point function in Liouville theory, derived by Fateev, Zamolodchikov and Zamolodchikov. The Liouville charge of the boundary operators match, by the KPZ correspondence, with the L-leg boundary exponents conjectured by JS.