Boundary transitions of the O(n) model on a dynamical lattice

TL;DR

This paper provides an exact solution for the boundary phase diagram and critical exponents of the dilute O(n) loop model on a dynamical lattice using matrix model techniques, revealing boundary deformations and phase transitions.

Contribution

It introduces an exact matrix model approach to analyze boundary conditions and phase transitions in the dilute O(n) model on a dynamical lattice, extending previous numerical results.

Findings

Reproduces boundary phase diagram and critical exponents

Describes boundary deformations under bulk thermal flow

Identifies the cusp at the isotropic special transition point

Abstract

We study the anisotropic boundary conditions for the dilute O(n) loop model with the methods of 2D quantum gravity. We solve the problem exactly on a dynamical lattice using the correspondence with a large $N$ matrix model. We formulate the disk two-point functions with ordinary and anisotropic boundary conditions as loop correlators in the matrix model. We derive the loop equations for these correlators and find their explicit solution in the scaling limit. Our solution reproduces the boundary phase diagram and the boundary critical exponents obtained recently by Dubail, Jacobsen and Saleur, except for the cusp at the isotropic special transition point. Moreover, our solution describes the bulk and the boundary deformations away from the anisotropic special transitions. In particular it shows how the anisotropic special boundary conditions are deformed by the bulk thermal flow towards the dense phase.