An analytical analysis of CDT coupled to dimer-like matter

Abstract

We consider a model of restricted dimers coupled to two-dimensional causal dynamical triangulations (CDT), where the dimer configurations are restricted in the sense that they do not include dimers in regions of high curvature. It is shown how the model can be solved analytically using bijections with decorated trees. At a negative critical value for the dimer fugacity the model undergoes a phase transition at which the critical exponent associated to the geometry changes. This represents the first account of an analytical study of a matter model with two-dimensional interactions coupled to CDT. a

A restricted dimer model on a two-dimensional random causal triangulation

“…We introduce a restricted hard dimer model on a random causal triangulation that is exactly solvable and generalizes a model recently proposed by Atkin and Zohren [16]. We show that the latter model exhibits unusual behaviour at its multicritical point; in particular, its Hausdorff dimension equals 3 and not 3/2 as would be expected from general scaling arguments.…”

“…We consider an extension of the dimer system on random causal triangulations first introduced in [16]. A finite causal triangulation T of the planar disc D, is constructed as shown in Fig 1. T is the union of a central disc Σ 0 having central vertex v 0 and boundary circle S 1 , and a sequence of annuli (or time slices) Σ k , k > 1, such that Σ k is bounded by circles S k−1 and S k .…”

Trees and spatial topology change in causal dynamical triangulations

“…(16)). Since we have explicitly determined Z 0 (T ) and Z 1 (T ) we can verify using (36) that W λ,gs (X) is indeed given by formula (14).…”

“…where W λ,gs (X) is given by (14). While this determines W λ,gs (X) algebraically, one has to appeal to figure 8, say, to identify it as the disk function, as done previously.…”

“…Let us now turn to surfaces with one or more boundary components. We will first present a heuristic derivation of the disk function W λ,gs (X) in (14) using only the continuum tree amplitudes Z 0 (T ) and Z 1 (T ). Below we will see that W λ,gs (X) can also be obtained directly in the scaling limit of quadrangulations with a boundary, by relating it to the propagator with a final boundary of length 0.…”