On perturbations that preserve the connectivity properties in tree percolations

We consider a general bond percolation on an infinite locally finite tree, where the edge retention probabilities $p_e$ are replaced by $\min \{1,q_{\mathrm{\mid }e\mathrm{\mid }}{p}_e\}$, where $\{q_n{\}}_{n\ge 1}$ is a sequence of positive perturbation factors and $\mathrm{\mid }e\mathrm{\mid }$ denotes the distance between the edge $e$ and the root. If the original percolation model admits infinite clusters, it is of interest to investigate under which perturbations $0<q_n\le 1$ this connectivity property is preserved. Conversely, if the original percolation does not admit infinite clusters, we are led to study the stability of such a property under perturbations satisfying $q_n\ge 1$. In both cases, under minimal assumptions on the original model, we show that the percolative behaviour is stable against certain quantitative non-trivial perturbations. We also discuss an application of our results to the Erdős similarity conjecture for Cantor sets.