Sharpness of the phase transition for constrained-degree percolation
Ivailo Hartarsky, Roger W. C. Silva
TL;DR
We study constrained-degree percolation (CDP) on ${\mathbb Z}^d$ with degree constraint ${\kappa}\in\{2,\dots,2d\}$. The main result establishes a sharp phase transition: in the subcritical regime $p<p_c$, the one-arm probability decays exponentially as $${\theta_n(p) \le e^{-\alpha_p n}}$$, and $p_c$ coincides with the susceptibility threshold ${\hat p_c}$. The proof combines the Duminil-Copin–Raoufi–Tassion randomized algorithm framework with an innovative pivotality-transfer mechanism based on a switching path to overcome lack of FKG and finite-energy properties. A modified one-arm event $E_n(p)$ with head/tail structure and a Russo formula underpin the transfer from $U$-pivotal to $p$-pivotal; this yields a differential inequality that implies sharpness and, via a standard corollary, exponential decay of cluster volumes in the subcritical phase. The techniques developed may apply to other dynamical or dependent percolation models lacking standard monotonicity properties.
Abstract
We consider constrained-degree percolation on the hypercubic lattice. Initially, all edges are closed, and each edge independently attempts to open at a uniformly distributed random time; the attempt succeeds if, at that instant, both end-vertices have degrees strictly less than a prescribed parameter. The absence of the FKG inequality and the finite energy property, as well as the infinite range of dependency, make the rigorous analysis of the model particularly challenging. In this work, we show that the one-arm probability exhibits exponential decay in its entire subcritical phase. The proof relies on the Duminil-Copin--Raoufi--Tassion randomized algorithm method and resolves a problem of dos Santos and the second author. At the heart of the argument lies an intricate combinatorial transformation of pivotality in the spirit of Aizenman--Grimmett essential enhancements, but with unbounded range. This technique may be of use in other dynamical settings.