We present evidence for a phase transition in a theory of 2D causal set quantum gravity which contains a dimensionless non-locality parameter ǫ ∈ (0, 1]. The transition is between a continuum phase and a crystalline phase, characterised by a set of covariant observables. For a fixed size of the causal set the transition temperature β Causal set theory(CST) is a discrete approach to quantum gravity which combines local Lorentz invariance with a fundamental discreteness [1]. The spacetime continuum is replaced by a locally finite poset or causal set, with the order relation ≺ being the analog of the spacetime causal order. The continuum quantum gravity path integral is thus replaced by a discrete sum over causal setswhere Ω is a sample space of causal sets and S[C] is an appropriately chosen action. The combination of local Lorentz invariance with fundamental discreteness gives rise to a non-locality in the continuum approximation, making the extraction of local geometric data highly non-trivial. The recent construction of a discrete Einstein-Hilbert action, the Benincasa-Dowker action for causal sets [2], thus allows us for the first time to begin a serious study of the causal set partition function Z CST . Apart from a choice of action, Z CST also depends crucially on the sample space Ω. A natural starting choice for Ω is the collection of countable 1

The causal set theory (CST) approach to quantum gravity postulates that at the most fundamental level, spacetime is discrete, with the spacetime continuum replaced by locally finite posets or "causal sets". The partial order on a causal set represents a proto-causality relation while local finiteness encodes an intrinsic discreteness. In the continuum approximation the former corresponds to the spacetime causality relation and the latter to a fundamental spacetime atomicity, so that finite volume regions in the continuum contain only a finite number of causal set elements. CST is deeply rooted in the Lorentzian character of spacetime, where a primary role is played by the causal structure poset. Importantly, the assumption of a fundamental discreteness in CST does not violate local Lorentz invariance in the continuum approximation. On the other hand, the combination of discreteness and Lorentz invariance gives rise to a characteristic non-locality which distinguishes CST from most other approaches to quantum gravity.In this review we give a broad, semi-pedagogical introduction to CST, highlighting key results as well as some of the key open questions. This review is intended both for the beginner student in quantum gravity as well as more seasoned researchers in the field.

We reexamine the thermodynamics of anti-de Sitter (adS) black holes with Ricci flat horizons using the adS soliton as the thermal background. We find that there is a phase transition which is dependent not only on the temperature but also on the black hole area, which is an independent parameter. As in the spherical adS black hole, this phase transition is related via the adS/conformal-field-theory correspondence to a confinement-deconfinement transition in the large- N gauge theory on the conformal boundary at infinity.

We revisit the action principle for general relativity, motivated by the path integral approach to quantum gravity. We consider a spacetime region whose boundary has piecewise C 2 components, each of which can be spacelike, timelike or null and consider metric variations in which only the pullback of the metric to the boundary is held fixed. Allowing all such metric variations we present a unified treatment of the spacelike, timelike and null boundary components using Cartan's tetrad formalism. Apart from its computational simplicity, this formalism gives us a simple way of identifying corner terms. We also discuss "creases" which occur when the boundary is the event horizon of a black hole. Our treatment is geometric and intrinsic and we present our results both in the computationally simpler tetrad formalism as well as the more familiar metric formalism. We recover known results from a simpler and more general point of view and find some new ones. arXiv:1612.00149v2 [gr-qc] 1 Feb 2017

An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally finite partial order. A new topology on causal sets using "thickened antichains" is constructed. This topology is then used to recover the homology of a globally hyperbolic spacetime from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or "Hauptvermutung" of causal set theory.

We study the N -dependent behaviour of 2d causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter β, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of N . Focussing on the 2d causal set action S, we find that β S scales like N ν where the scaling exponent ν takes different values on either side of the phase transition. For β > β c we find that ν = 2 which is consistent with our analytic predictions for a non-continuum phase in the large β regime. For β < β c we find that ν = 0, consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in 2d causal set quantum gravity for N 65.

It is a common misconception that spacetime discreteness necessarily implies a violation of local Lorentz invariance. In fact, in the causal set approach to quantum gravity, Lorentz invariance follows from the specific implementation of the discreteness hypothesis. However, this comes at the cost of locality. In particular, it is difficult to define a "local" region in a manifoldlike causal set, i.e., one that corresponds to an approximately flat spacetime region. Following up on suggestions from previous work, we bridge this lacuna by proposing a definition of locality based on the abundance of m-element order-intervals as a function of m in a causal set. We obtain analytic expressions for the expectation value of this function for an ensemble of causal set that faithfully embeds into an Alexandrov interval in d-dimensional Minkowski spacetime and use it to define local regions in a manifoldlike causal set. We use this to argue that evidence of local regions is a necessary condition for manifoldlikeness in a causal set. This in addition provides a new continuum dimension estimator. We perform extensive simulations which support our claims.

We propose a family of boundary terms for the action of a causal set with a spacelike boundary. We show that in the continuum limit one recovers the Gibbons-Hawking-York boundary term in the mean. We also calculate the continuum limit of the mean causal set action for an Alexandrov interval in flat spacetime. We find that it is equal to the volume of the codimension-2 intersection of the two light-cone boundaries of the interval. 1 arXiv:1502.05388v2 [gr-qc]