Random-h Fractional-Dimensional Lattices Reveal Endpoint-Compressed Percolation Activation between Two and Three Dimensions

TL;DR

This paper introduces the random-h fractional dimension (RhFD) lattice framework, enabling the simulation of fractional-dimensional environments and revealing a non-uniform crossover in percolation thresholds between two and three dimensions.

Contribution

The authors develop RhFD, a stochastic lattice model for fractional dimensions, and demonstrate its ability to capture complex crossover behaviors in percolation properties.

Findings

RhFD recovers integer-dimensional endpoints in 2D and 3D.

Percolation threshold decreases from 2D to 3D regimes.

Endpoint-compressed activation observed during crossover.

Abstract

Non-integer dimensionality is central to fractal and complex systems, yet it is rarely represented as an explicit lattice on which classical statistical-mechanical models can be directly simulated. Here we introduce random-h fractional dimension (RhFD), a constructive lattice framework in which fractional-dimensional environments are generated by stochastic activation of local connectivity, h. In the 2D-to-3D interval, RhFD lattices are formed by recursively growing out-of-plane sites from a square base with probability \r{ho}h. Using quenched site-percolation simulations, we show that the construction recovers the integer-dimensional endpoints and yields a robust crossover in which the percolation threshold decreases from the 2D regime toward the 3D regime. The crossover is not a uniform interpolation: high-resolution scans reveal endpoint-compressed activation, with -dpc/d\r{ho}h increasing toward \r{ho}h = 1. Mass dimension increases with \r{ho}h, whereas the coordination descriptor first decreases as sparse protrusions form and then rises sharply when a dense 3D backbone emerges. RhFD provides an explicit lattice substrate for fractional-dimensional statistical mechanics and shows that geometric mass, local coordination, and critical connectivity can decouple during dimensional crossover.