The cylinder amplitude in the Hard Dimer model on 2D Causal Dynamical Triangulations

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.

Boundary Conditions and the q-state Potts model on Random Planar Maps

We extend a recent analysis of the $q$-states Potts model on an ensemble of random planar graphs with $pleqslant q$ allowed, equally weighted, spins on a connected boundary. In this paper we explore the $(q<4,pleqslant q)$ parameter space of finite-sheeted resolvents and derive the associated critical exponents. By definition a value of $q$ is allowed if there is a $p=1$ solution, and we reproduce the long-known result that $q= 2(1+cos{frac{m}{n} pi})$ with $m,n$ coprime. In addition we find that there are two distinct sequences of solutions, one of which contains $p=2$ and $p=q/2$ while the other does not. The boundary condition $p=3$ appears only for $q=3$ which also has a $p=3/2$ boundary condition; we conjecture that this new solution corresponds in the scaling limit to the New boundary condition, discovered on the flat lattice by Affleck et al. We also explore Kramers-Wannier duality for $q=3$ in this context and explicitly construct the known boundary conditions; we show that the mixed boundary condition is dual to a boundary condition on dual graphs that corresponds to Affleck et als identification of the New boundary condition on fixed lattices. On the other hand we find that the mixed boundary condition of the dual, and the corresponding New boundary condition of the original theory are not described by conventional resolvents.

Sums of Random Matrices and the Potts Model on Random Planar Maps

We compute the partition function of the $q$-states Potts model on a random planar lattice with $pleq q$ allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix model representation in the planar limit, generalising a result by Voiculescu for the addition of random matrices to a situation beyond free probability theory. We show that the partition functions with $p$ and $q-p$ colours on the boundary are related algebraically. Finally, we investigate the phase diagram of the model when $0leq qleq 4$ and comment on the conformal field theory description of the critical points.

Aspects of dynamical dimensional reduction in multigraph ensembles of CDT

We study the continuum limit of a radially reduced approximation of Causal Dynamical Triangulations (CDT), so-called multigraph ensembles, and explain why they serve as realistic toy models to study the dimensional reduction observed in numerical simulations of four-dimensional CDT. We present properties of this approximation in two, three and four dimensions comparing them with the numerical simulations and pointing out some common features with 2+1 dimensional Horava-Lifshitz gravity.

Spectral dimension flow on continuum random multigraph

We review a recently introduced effective graph approximation of causal dynamical triangulations (CDT), the multigraph ensemble. We argue that it is well suited for analytical computations and that it captures the physical degrees of freedom which are important for the reduction of the spectral dimension as observed in numerical simulations of CDT. In addition multigraph models allow us to study the relationship between the spectral dimension and the Hausdorff dimension, thus establishing a link to other approaches to quantum gravity

Multigraph models for causal quantum gravity and scale dependent spectral dimension

We study random walks on ensembles of a specific class of random multigraphs which provide an effective graph ensemble for the causal dynamical triangulation (CDT) model of quantum gravity. In particular, we investigate the spectral dimension of the multigraph ensemble for recurrent as well as transient walks. We investigate the circumstances in which the spectral dimension and Hausdorff dimension are equal and show that this occurs when rho, the exponent for anomalous behaviour of the resistance to infinity, is zero. The concept of scale dependent spectral dimension in these models is introduced. We apply this notion to a multigraph ensemble with a measure induced by a size biased critical Galton-Watson process which has a scale dependent spectral dimension of two at large scales and one at small scales. We conclude by discussing a specific model related to four dimensional CDT which has a spectral dimension of four at large scales and two at small scales.

Dynamical dimensional reduction in toy models of 4D causal quantum gravity

In recent years several approaches to quantum gravity have found evidence for a scale dependent spectral dimension of space-time varying from four at large scales to two at small scales of order of the Planck length. The first evidence came from numerical results on four-dimensional causal dynamical triangulations (CDT) [Ambjorn et al., Phys. Rev. Lett. 95 (2005) 171]. Since then little progress has been made in analytically understanding the numerical results coming from the CDT approach and showing that they remain valid when taking the continuum limit. Here we argue that the spectral dimension can be determined from a model with fewer degrees of freedom obtained from the CDTs by radial reduction. In the resulting toy model we can take the continuum limit analytically and obtain a scale dependent spectral dimension varying from four to two with scale and having functional behaviour exactly of the form which was conjectured on the basis of the numerical results.

The Spectrum of FZZT Branes Beyond the Planar Limit

Minimal string theory has a number of FZZT brane boundary states; one for each Cardy state of the minimal model. It was conjectured by Seiberg and Shih that all branes in a minimal string theory could be expressed as a linear combination of the brane associated to the identity operator of the minimal model with complex shifts in the boundary cosmological constant. Subsequently it was found that this identification of FZZT branes does not hold exactly for some cylinder amplitudes but was spoiled by terms that are associated with vanishing worldsheet area and are therefore non-universal. In this paper we investigate this claim systematically, using both Liouville and matrix model methods, beyond the planar limit. We find that the aforementioned identification of FZZT branes is spoiled by terms that do not admit an interpretation as non-universal terms. Furthermore, the spoiling terms as computed using the matrix model are found to be in agreement with those coming from Liouville theory, which also suggests that these terms have universal meaning. Finally, we also investigate the identification of FZZT branes by replacing the boundary state with a sum of local operators. We find in this case that the brane associated with the identity operator appears to be special as it is the only one to correctly reproduce the correlation numbers for bulk operators on the torus.

On the spectral dimension of causal triangulations

We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases the Hausdorff dimension almost surely equals 2. The infinite causal triangulations are shown to be almost surely recurrent or, equivalently, their spectral dimension is almost surely less than or equal to 2. We also establish that for certain reduc

The spectral dimension of generic trees

We define generic ensembles of infinite trees. These are limits as $Ntoinfty$ of ensembles of finite trees of fixed size $N$, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension $d_h =2$. Our main result is that their spectral dimension is $d_s=4/3$, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.