Abstract
We consider the model of hard dimers coupled to two-dimensional causal dynamical triangulations (CDT) with all dimer types present and solve it exactly subject to a single restriction. Depending on the dimer weights there are, in addition to the usual gravity phase of CDT, two tri-critical and two dense dimer phases. We establish the properties of these phases, computing their cylinder and disk amplitudes, and their scaling limits.
Highlights
In this paper, we consider the hard dimer model on a fluctuating background in 1+1 dimensions
A Hamiltonian operator and cylinder amplitude emerge which reproduce results first obtained in [5] by a proper-time gauge calculation in 1+1D continuum gravity. It has been shown [6] that the continuum limit of causal dynamical triangulation model (CDT) is equivalent to two-dimensional projectable Horava-Lifshitz gravity [7, 8]; we will refer to this as the ‘Pure gravity’ (PG) phase of the extended models considered in this paper
When the parameters lie on the boundary of Γ do large graphs contribute to the sum and the continuum cylinder amplitude is constructed by generalising the scaling limit (45) for the two point function G
Summary
We return to the micro-canonical ensemble of surfaces with fixed temporal extent t, and marked initial and final boundaries. X, y lie inside the joint region of convergence, Γ, of the series (1) for G(g, x, y, ξ; t), graphs with large area (number of triangles), and long boundaries (number of edges), are exponentially suppressed. When the parameters lie on the boundary of Γ do large graphs contribute to the sum and the continuum cylinder amplitude is constructed by generalising the scaling limit (45) for the two point function G. When type 1 dimers are present this law does not have to be satisfied because the dimer configuration on the intermediate boundary at T must be summed over as well. As we will see below, in some phases the property (69) is still satisfied and In these phases the intermediate dimer configuration sum generates the c−0 1 prefactor. Which is the CDT analogue of the disk amplitude for planar random graph models
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