Boundary loop models and 2D quantum gravity

Abstract

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.

Boundary transitions of the model on a dynamical lattice

“…In the dense phase, where the Kac labels are defined by (2.7), one obtains, taking into account that the identity boundary operator for the ordinary boundary condition has 'wrong' dressing, α = r − s(1 − θ) → h = h r,s (dense phase). (7.11) From (7.10) and (7.11) we determine the scaling dimensions of the L-leg boundary operators (3.9): These conformal weights are in accord with the results of [7], [13], [14], [3]. We remind that the scaling dimensions are determined up to a symmetry of the Kac parametrization:…”

“…The UV and the IR limits are explored by taking respectively large and small values of the bulk and boundary cosmological constants. Our method of solution is based on the mapping to the O(n) matrix model [11,12] and on the techniques developed in [13,14]. Using the Ward identities of the matrix model, we were able to evaluate the two-point functions of the boundary changing operators for finite bulk and boundary deformations away from the anisotropic special transitions.…”

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

“…Beyond this threshold, the standard construction yields vanishing random measures [29,45]. The issue of mathematically constructing singular Liouville measures beyond the phase transition (i.e., for γ > 2) and deriving the corresponding (nonstandard dual) KPZ formula has been investigated in [9,28,29], giving the first mathematical understanding of the so-called duality in Liouville quantum gravity; see [4,5,21,27,32,44,[48][49][50]54] for an account of physical motivations. However, the rigorous construction of random measures at criticality, that is, for γ = 2, does not seem to ever have been carried out.…”

Boundary changing operators in theO(n) matrix model

“…where we used the Kac notation 2. 18. When L open lines are inserted, loops touching both boundaries are forbidden and the scaling dimension of the JS-JS boundary operators depends only on k I , k J and L,…”