Emergence of Space-Time on the Planck Scale within the Scheme of Dynamical Cellular Networks and Random Graphs

This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation---as shown by computing the perihelion shift of a Mercury-...

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...

We review and extend the recently proposed model of combinatorial quantum gravity. Contrary to previous discrete approaches, this model is defined on (regular) random graphs and is driven by a purely combinatorial version of Ricci curvature, the Ollivier curvature, defined on generic metric spaces equipped with a Markov chain. It dispenses thus of notions such as simplicial complexes and Regge calculus and is ideally suited to extend quantum gravity to combinatorial structure...

Cellular automata represent physical systems where both space and time are discrete, and the associated physical quantities assume a limited set of values. While previous research has applied cellular automata in modeling chemical, biological, and physical systems, its potential for modeling topological systems, specifically network topologies, remains underexplored. This paper investigates the use of cellular automata to model decentralized network topologies, which could en...

I propose a random network model governed by a Gaussian weight corresponding to Ising link antiferromagnetism as a model for emergent quantum space-time. In this model, discrete space is fundamental, not a regularization, its spectral dimension $d_s$ is not a model input but is, rather, completely determined by the antiferromagnetic coupling constant. Perturbative terms suppressing triangles and favouring squares lead to locally Euclidean ground states that are Ricci flat "la...

We analyze the structure of two dimensional disordered cellular systems generated by extensive computer simulations. These cellular structures are studied as topological trees rooted on a central cell or as closed shells arranged concentrically around a germ cell. We single out the most significant parameters that characterize statistically the organization of these patterns. Universality and specificity in disordered cellular structures are discussed.

There are several indications (from different approaches) that Spacetime at the Plank Scale could be discrete. One approach to Quantum Gravity that takes this most seriously is the Causal Sets Approach. In this approach spacetime is fundamentally a discrete, random, partially ordered set (where the partial order is the causal relation). In this contribution, we examine how timelike and spacelike distances arise from a causal set (in the case that the causal set is approximate...

Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex degree and form very large numbers of simple polygons (triangles or squares). The models avoid the collapse phenomena that plague naive graph Hamiltonians based on triangle or square counts. More than that, statistically significant numbers of ...

Crystals arise as the result of the breaking of a spatial translation symmetry. Similarly, translation symmetries can also be broken in time so that discrete time crystals appear. Here, we introduce a method to describe, characterize, and explore the physical phenomena related to this phase of matter using tools from graph theory. The analysis of the graphs allows to visualizing time-crystalline order and to analyze features of the quantum system. For example, we explore in d...

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.