Recursive Packing Bounds for Supercritical Disconnection in Bernoulli Site Percolation

This paper establishes a quantitative upper bound on the supercritical disconnection probability for Bernoulli site percolation on infinite, connected, locally finite graphs by introducing a recursive packing number that counts essentially independent local witnesses for the disconnection event.

The Gist

Imagine you are standing in a vast, infinite city made of blocks (the graph). Each block has a light switch. You flip a switch for every block in the city: with probability $p$, the light turns ON (open); otherwise, it stays OFF (closed).

In this city, if a block is ON, it can "talk" to its neighbors if they are also ON. If a chain of ON blocks stretches out forever, that block is part of an infinite highway to the horizon.

The Big Question

The paper asks a specific question: If you pick a group of blocks (a set $S$), what are the odds that none of them can reach the infinite highway?

If the probability of a light turning ON ($p$) is low, the city is mostly dark, and it's easy for everyone to be cut off. But what if $p$ is high (the "supercritical" regime)? The city is mostly lit, and infinite highways are everywhere. Intuitively, it seems impossible for a whole group of blocks to be cut off.

However, the author, Zhongyang Li, wants to know: Exactly how small is the probability of this "disconnection" event? Can we put a precise number on it?

The Problem with "One Size Fits All"

In the past, mathematicians could only give answers for very specific, perfectly symmetrical cities (like a grid or a perfect tree). If the city had weird shapes or irregular streets, the math broke down.

Li's paper introduces a new tool that works for any city, no matter how weird or irregular, as long as every block has a finite number of neighbors.

The Solution: The "Recursive Packing" Strategy

To solve this, Li invents a concept called the Recursive Packing Number. Let's break this down with a metaphor.

The Metaphor: The "Witness" Game

Imagine you are trying to prove that a specific group of blocks ($S$) might get cut off from the infinite highway. To do this, you need to find "witnesses"—special blocks within your group that act as spies.

1. The First Spy: You pick a block from your group. You draw a circle (a "witness ball") around it.

   • If this block can't reach the infinite highway, it's almost certainly because it can't even reach the edge of its own circle.
   • If it can reach the edge of the circle, it has a good chance of reaching the infinite highway.
   • You check: Does this block have a decent chance (say, at least $c$) of reaching the infinite highway? If yes, it's a valid spy.

2. The Second Spy: Now, you remove the first spy's circle from the city. You can't use that area again. You look at the remaining blocks in your group and pick a second spy.

   • You draw a new circle around this second block.
   • You check: Even with the first circle gone, does this second block still have a decent chance of reaching the infinite highway?
   • Crucially, because the circles are far apart, the "luck" of the second block is mostly independent of the first.

3. The Packing Number: You keep doing this, picking spies one by one, removing their circles, and checking if they are still "supercritical" (likely to connect to infinity).

   • The Packing Number is simply the maximum number of spies you can fit into your group this way.

The Magic Formula

The paper proves a beautiful, simple rule:

The probability that your whole group gets cut off is roughly equal to:
(The chance a single spy fails) raised to the power of (The number of spies you packed).

Think of it like a safety net. If you have one spy, there's a small chance they fail. If you have 10 independent spies, the chance that all 10 fail at the same time is astronomically small.

The formula says:
$$\text{Disconnection Probability} \approx (1 - c)^{\text{Number of Spies}}$$

Where:

• $c$ is the minimum chance a single spy has of connecting to infinity.
• The "Number of Spies" is your Recursive Packing Number.

Why This is a Big Deal

1. It Works Everywhere: Whether your city is a perfect grid, a random mess, or a tree with weird branches, this method works. You just need to count how many "independent spies" you can pack into your group.
2. It's Explicit: The paper shows how to calculate this number for specific types of trees.
   • Example: On a regular tree (like a perfect family tree), if you pick blocks that are far enough apart, the Packing Number is exactly the number of blocks you picked.
   • Example: Even on a "decorated" tree (where the main path is regular, but the side branches are weird), the math still holds up.

The Takeaway

This paper gives us a universal "ruler" to measure how hard it is to disconnect a group of points in a random network.

Instead of getting lost in the complex geometry of the whole graph, you just need to ask: "How many independent, high-probability 'witnesses' can I pack into this group?"

The more witnesses you can pack, the exponentially smaller the chance that the whole group gets cut off from the infinite world. It turns a complex, global problem into a simple, local counting game.

Technical Summary

1. Problem Statement

The paper addresses the problem of obtaining quantitative upper bounds on the probability of supercritical disconnection in Bernoulli site percolation on infinite, connected, locally finite graphs $G = (V, E)$.

• Context: In the supercritical regime ($p > p_c^{\text{site}}(G)$), vertices belong to infinite open clusters with positive probability.
• The Event: For a set of vertices $S \subset V$, the paper studies the probability $P_p(S \nleftrightarrow \infty)$, which is the event that no vertex in $S$ connects to infinity (i.e., the entire set $S$ is disconnected from the infinite cluster).
• The Challenge: While qualitative results (existence of infinite clusters) are well-known, obtaining explicit, quantitative bounds for the disconnection of arbitrary sets $S$ (finite or infinite) on general graphs (without assuming transitivity or specific geometric symmetries) has been difficult. Previous results often relied on specific geometric assumptions (e.g., Euclidean lattices) or provided non-explicit bounds.

2. Methodology

The author introduces a novel recursive packing/amplification principle that transforms local connectivity information into global disconnection estimates. The methodology consists of three main components:

A. Local Functional Characterization ($\phi_v^p$)

The paper utilizes a local functional $\phi_v^p(S)$ introduced in previous work [4]. This functional characterizes the critical threshold $p_c^{\text{site}}(G)$ using finite-volume connectivity data.

• It provides a quantitative lower bound on the probability that a vertex $v$ connects to infinity in the supercritical regime.
• Specifically, if $p > p_c$, there exists a vertex $w$ and a finite set $S_w$ such that the functional is close to 1, implying a high probability of connection to infinity.

B. Recursive Packing Number ($PK_{p, \epsilon, c}(S)$)

The core innovation is the definition of the $(p, \epsilon, c)$-packing number, denoted $PK_{p, \epsilon, c}(S)$. This is a combinatorial quantity defined recursively:

1. Selection: Select a vertex $w_1 \in S$.
2. Witness Ball: Associate a "witness ball" $B(w_1, D_1)$ around $w_1$.
3. Condition: The vertex $w_1$ must connect to infinity with probability at least $c$ in the original graph, and the event of not connecting to infinity must be captured (up to a factor $1+\epsilon$) by the event of not reaching the inner boundary of the witness ball.
4. Recursion: Remove the ball $B(w_1, D_1)$ from the graph to form a punctured graph $G_1$. Select the next vertex $w_2$ from $S \setminus B(w_1, D_1)$ such that it satisfies the same conditions within $G_1$.
5. Count: $PK_{p, \epsilon, c}(S)$ is the maximal number of vertices that can be selected in this manner.

This number essentially counts the maximum size of an essentially independent family of local witnesses for the global disconnection event.

C. Amplification Step

The proof combines the local lower bound on connection probabilities with the independence of the "witness ball" events. By selecting $k$ such vertices, the probability that all of them fail to connect to infinity decays roughly as $(1-c)^k$. The recursive structure allows the author to handle the dependencies introduced by removing the balls, bounding the error terms using the parameter $\epsilon$.

3. Key Contributions and Results

The Main Theorem (Packing Bound)

The paper establishes the following structural estimate (Theorem 1.1):
$$P_p(S \nleftrightarrow \infty) \leq \frac{\epsilon(1-c)}{c} + (1-c)^{PK_{p, \epsilon, c}(S)}$$

• Interpretation: The disconnection probability is bounded by a term depending on the packing number. Each additional admissible packing center contributes a multiplicative factor of $(1-c)$, leading to exponential decay if the packing number is large.
• Generality: This bound holds for any infinite, connected, locally finite graph.

Explicit Supercritical Estimate

By choosing the parameter $c$ based on the local functional characterization of $p_c$ (specifically using a parameter $p_1 \in (p_c, p)$), the author derives an explicit bound (Corollary 2.7):
$$P_p(S \nleftrightarrow \infty) \leq \inf_{p_1, \epsilon, \delta} \left[ \frac{\delta (\frac{1-p}{1-p_1})^{1-\epsilon}}{1 - (\frac{1-p}{1-p_1})^{1-\epsilon}} + \left(\frac{1-p}{1-p_1}\right)^{(1-\epsilon) PK_{p, \delta, c}(S)} \right]$$
This provides a concrete, computable upper bound for any graph where the packing number can be estimated.

Concrete Examples on Ray-Homogeneous Trees

To demonstrate that the packing number is not merely a formal construct but computable, the paper analyzes ray-homogeneous trees:

• Definition: Trees where the geometry along a specific geodesic ray $\gamma$ is homogeneous (the structure "behind" a vertex on the ray is isomorphic to a fixed rooted tree $F_1$, and the structure at the start is $F_0$).
• Result (Proposition 3.3): For sparse finite subsets $A$ of the ray $\gamma$ (where points are separated by a distance $R_\epsilon$), the packing number equals the cardinality of the set:
  $$PK_{p, \epsilon, c}(A) = |A|$$
• Applications:
  1. Regular Trees ($T_d$): The packing number equals the set size for sufficiently separated points on a ray.
  2. Non-Regular Decorated Spine: A tree constructed by attaching ternary trees to a bi-infinite path. Despite lacking transitivity, the packing number is exactly the cardinality for sparse sets on the spine.

4. Significance

1. Universality: Unlike previous works that required transitive or quasi-transitive graphs (like $\mathbb{Z}^d$), this result applies to arbitrary infinite, connected, locally finite graphs. It packages all geometric complexity into the graph-dependent packing number.
2. Quantitative Precision: It moves beyond qualitative statements (e.g., "disconnection probability decays exponentially") to provide explicit, quantitative bounds involving the parameters $p$, $c$, and the packing number.
3. New Mechanism: The "recursive packing/amplification" principle offers a new toolkit for percolation theory. It effectively bridges the gap between local connectivity properties (one-point estimates) and global disconnection events (many-point estimates) without relying on planar duality or specific lattice symmetries.
4. Computability: By showing that the packing number is exactly the cardinality for sparse sets on ray-homogeneous trees (including non-regular ones), the paper proves that these bounds are explicit and computable for concrete graph families, not just theoretical constructs.

In summary, Zhongyang Li provides a robust framework for estimating supercritical disconnection probabilities on general graphs, introducing a recursive packing number that serves as a precise geometric measure of how "independent" local disconnection events can be within a set.