Phase Transition in Growing Causal Networks and Possible Dynamical Preference of Spatial Dimension

We study phase transitions in growing causal networks in which spatial connectivityemerges from an underlying temporal order. Introducing a “zero-connectivity” regime, wedemonstrate the existence of a percolation transition with analytically estimated criticalpointαc =1− 12k .Numerical simulations confirm the transition and yield critical exponents consistent withthree-dimensional percolation behavior.Wefurther consider a scalar ϕ4 field theory defined on the critical network. Renormalization group arguments indicate that within the interval 2 < ds < 4, where non-trivial infraredfixed points may exist, the value ds ≈ 3 occupies a dynamically favorable regime associatedwith non-mean-field behavior. While this does not constitute a proof of dimensional selection, it suggests a possible mechanism by which three-dimensional space could emerge as acritical phenomenon from purely causal constraints, with time remaining fundamental in allphases.