Higher Dimensional Energetic Causal Sets

The energetic causal set (ECS) program of Cort\^es and Smolin, whose distinguishing feature is the foundational time irreversibility of the evolution equations, was initiated ten years ago. The toy model used to show the emergence of the time-reversible phase from the time-irreversible foundational regime originally had only one spatial dimension (1d+1). The extension to two and more spatial dimensions has posed a substantial challenge, where the higher-dimensional set of solutions had measure zero (involving infinitely-specified initial conditions). This challenge is overcome here with the extension of the ECS to 2+1 dimensions, invoking a finite interaction cross-section to determine generation of new events and including an adjustable fundamental stochasticity. Successfully, as in the 1+1 dimensional case, we observe a phase transition into a time-symmetric phase, here through the emergence of crystal-like structures. Due to the irreversible evolution we also witness the discrete dynamical systems (DDS) behaviour of the 1d+1 case explored in later articles of the ECS program, in which the model is captured in and out of the limit cycles which here take the form of lattice crystals. In a companion paper we carry out a detailed parameter investigation and study the causal network underlying the set, explaining the emergence of the time-symmetric phase.
Comment: 15 pages, 19 figures, updated to improve figure quality and layout