Weakly constrained-degree percolation on the hypercubic lattice

TL;DR

This paper studies a constrained percolation model on high-dimensional lattices, providing bounds on the critical time for phase transition and revealing asymptotic behavior for large constraints and dimensions.

Contribution

It offers the first quantitative bounds on the critical time for the constrained-degree percolation model in dimensions three and higher, establishing nontrivial phase transitions.

Findings

Critical time bounds are established for all $d extgreater 3$ and most $ppa$.

Asymptotic critical time is $1/(2d)$ for large constraints and dimensions.

Improved bounds for Bernoulli mixed site-bond percolation critical curve.

Abstract

We consider the Constrained-degree percolation model on the hypercubic lattice, $\mathbb L^d=(\mathbb Z^d,\mathbb E^d)$ for $d\geq 3$. It is a continuous time percolation model defined by a sequence, $(U_e)_{e\in\mathbb E^d}$, of i.i.d. uniform random variables in $[0,1]$ and a positive integer (constraint) $\kappa$. Each bond $e\in\mathbb E^d$ tries to open at time $U_e$; it succeeds if and only if both its end-vertices belong to at most $\kappa -1$ open bonds at that time. Our main results are quantitative upper bounds on the critical time, characterising a phase transition for all $d\geq 3$ and most nontrivial values of $\kappa$. As a byproduct, we obtain that for large constraints and dimensions the critical time is asymptotically $1/(2d)$. For most cases considered it was previously not even established that the phase transition is nontrivial. One of the ingredients of our proof is an improved upper bound for the critical curve, $s_{\mathrm{c}}(b)$, of the Bernoulli mixed site-bond percolation in two dimensions, which may be of independent interest.