In constrained-degree percolation, we consider the $d$-dimensional hypercubic lattice and supply each edge with a clock. Each clock rings only once at an independent and identically distributed time. When the clock of an edge rings, it becomes open, provided that there are currently at most $k-1$ open edges incident with each of its vertices, where $2 \leq k \leq 2d$ is a parameter of the model. The study is made difficult by the lack of most key properties of more conventional percolation models such as monotonicity in the underlying graph, finite energy, FKG inequality, finite dependence range, etc.
We will start with the first non-perturbative argument for this model, which allows proving that the transition is non-trivial for most values of $k$ for $d\geq 3$, as well as to determine its high-dimension asymptotics. Then, we will show exponential decay in the subcritical regime.
The talk is based on joint works with Bernardo de Lima and with Roger Silva available at https://arxiv.org/abs/2010.08955 and https://arxiv.org/abs/2509.16162.