On the monotonicity of the critical time in the Constrained-degree percolation model

TL;DR

This paper investigates the monotonicity of the critical time in the Constrained-degree percolation model across different dimensions, providing numerical evidence and comparing critical exponents with Bernoulli percolation.

Contribution

It extends the study of the Constrained-degree percolation model to higher dimensions and demonstrates the monotonicity of the critical time with respect to the constrained degree.

Findings

Critical time is monotonically non-increasing with the constrained degree in 3 and 4 dimensions.

The lowest constrained degree for phase transition is 3.

Critical exponents match those of Bernoulli percolation.

Abstract

The Constrained-degree percolation model was introduced in [B.N.B. de Lima, R. Sanchis, D.C. dos Santos, V. Sidoravicius, and R. Teodoro, Stoch. Process. Appl. (2020)], where it was proven that this model has a non-trivial phase transition on a square lattice. We study the Constrained-degree percolation model on the $d$-dimensional hypercubic lattice ($\mathbb{Z}^d$) and, via numerical simulations, found evidence that the critical time $t_{c}^{d}(k)$ is monotonous not increasing in the constrained $k$ if $d=3,4$, like it is when $d=2$. We verify that the lowest constrained value $k$ such that the system exhibits a phase transition is $k=3$ and that the correlation critical exponent $\nu$ for the Constrained-degree percolation model and ordinary Bernoulli percolation are the same.