On perturbations that preserve the connectivity properties in tree percolations

Technical Summary: On Perturbations That Preserve the Connectivity Properties in Tree Percolations

Problem Statement
This paper investigates the stability of connectivity properties in general bond percolation models defined on infinite, locally finite trees. Specifically, the authors examine how the existence (or absence) of infinite clusters is affected when the edge retention probabilities $p_e$ are subjected to level-dependent perturbations. The perturbation replaces the original probability $p_e$ with $\min\{1, q_{|e|}p_e\}$, where $\{q_n\}_{n \ge 1}$ is a sequence of positive factors and $|e|$ denotes the graph distance of the edge from the root.

The study addresses two distinct regimes:

1. Stability of Absence: If the original model admits no infinite clusters ($P_p(I)=0$), under what conditions on perturbations $q_n \ge 1$ (which enhance connectivity) does the system remain devoid of infinite clusters?
2. Stability of Existence: If the original model admits infinite clusters (specifically, uncountable boundary clusters with positive probability), under what conditions on perturbations $0 < q_n \le 1$ (which thin connectivity) does the existence of infinite clusters persist?

The authors note that while trivial perturbations (where the product $\prod q_n$ converges to a finite non-zero limit) merely rescale probabilities, the "genuinely interesting" cases involve divergent perturbations where $\prod q_n = 0$ (cumulative thinning) or $\prod q_n = \infty$ (cumulative enhancement). Simple counterexamples demonstrate that stability is not guaranteed in general without specific structural conditions on the tree or the percolation model.

Methodology
The proofs rely heavily on Lyons' capacity characterisation of percolation on trees, a result from discrete potential theory. The core methodology involves:

• Capacity and Energy: Utilizing the relationship between the probability of percolation to a set $E \subseteq \partial T$ and the capacity of that set, $capp(E)$. This capacity is defined via an energy functional $E_p(\mu)$ associated with a gauge function $\Psi_p(x) = 1/P_p[o \to x]$.
• Lyons' Theorem: Applying the inequality $capp(E) \le P_p[o \to E] \le 2capp(E)$, which links probabilistic connectivity to potential-theoretic capacity.
• Perturbation Analysis: Analyzing how the energy functional changes under the perturbed probabilities $q p$. The authors construct specific perturbation sequences $\{q_n\}$ and demonstrate that the finiteness of the energy functional (and thus the positivity of capacity) is preserved under certain divergent conditions.
• Auxiliary Lemmas: The proof employs elementary lemmas regarding summable sequences to construct strictly increasing sequences $\{Q_n\}$ (related to the cumulative product of $q_n$) that ensure the convergence of the perturbed energy functional.

Key Contributions and Results
The paper presents Theorem A, which establishes the robustness of percolative regimes under minimal assumptions:

1. Stability in the Absence of Infinite Clusters (Part i):
   If the original model has no infinite clusters almost surely, and the perturbation sequence $q_n \ge 1$ satisfies the condition:
   $$\liminf_{n \to \infty} \left( \prod_{i=1}^n q_i \right) \cdot P_p[o \to V_n] = 0$$
   (where $V_n$ is the set of vertices at distance $n$), then the perturbed model also has no infinite clusters almost surely. The authors show that such a perturbation sequence always exists even if $\prod q_n = \infty$.

2. Stability of Uncountable Clusters (Part ii):
   If the original model has an uncountable boundary cluster with positive probability ($P_p[o \xrightarrow{u} \partial T] > 0$), there exists a perturbation sequence $q_n \in (0, 1)$ such that $\lim_{n \to \infty} q_n = 1$ and the almost-sure existence of an uncountable cluster is preserved.

   • Necessity of Conditions: The paper notes that the condition of having an uncountable cluster is necessary; if the cluster is countable (e.g., a single ray), divergent thinning perturbations can destroy percolation (as shown in Example 1.1).
   • Homogeneous Case: For homogeneous percolation ($p_e = p$), if the perturbed model percolates, the authors derive a necessary condition involving the tree's branching number and the perturbation factors:
     $$\sum_{n=1}^\infty \frac{1 - q_n p}{p^n (\# E_n) \prod_{i=1}^n q_i} < \infty$$
     where $\# E_n$ is the number of vertices at level $n$.

Significance and Applications
The authors claim that their results demonstrate that the percolative regime is "rather robust" against non-trivial, divergent perturbations, provided the underlying system is not "fragile" (i.e., relying on a single ray).

A primary application discussed is to the Erdős similarity conjecture for Cantor sets. By encoding a Cantor set $C \subset [0, 1]$ as a subtree $T_C$ of the dyadic tree, the authors use Theorem A (ii) to construct a sequence of level-dependent weights $\{q_n\}$ (with $\prod q_n = 0$) such that the perturbed percolation on the tree coding any affine copy $\lambda C + t$ retains a positive probability of connecting to the boundary.

• This leads to Corollary 4.2, which states that for any Cantor set $C$, there exists a set $B$ of zero Lebesgue measure such that $B \cap (\lambda C + t) \neq \emptyset$ for almost all $\lambda, t$.
• The paper positions this as a potential avenue for studying the Erdős similarity problem, noting that while the "largeness" of the intersection is established, the continuity of this property with respect to $(\lambda, t)$ remains an open question for a full resolution of the conjecture.

The work is distinguished by its use of potential-theoretic characterizations to provide concise proofs with great generality, avoiding more elementary but cumbersome combinatorial methods.