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Local criteria for global connectivity comparisons: beyond stochastic domination

Johannes Bäumler, Benedikt Jahnel, Jonas Köppl, Bas Lodewijks, Lily Reeves, András Tóbiás

TL;DR

The paper develops a robust local-to-global criterion for comparing percolation models that goes beyond stochastic domination, proving that a site-wise advantage in connecting to neighbor subsets propagates to global connectivity events such as $o$ connected to $\partial B_n$. Central to the approach is a discrete interpolation inequality coupled with a breadth-first exploration of the out-component, enabling a transfer from local inequalities $\mathbf{P}[N(o) \cap A \neq \emptyset] \le \mathbf{Q}[N(o) \cap A \neq \emptyset]$ to global bounds $\mathbb{P}[o \rightsquigarrow \partial B_n]$. The authors then apply this framework to degree-constrained percolation models, including directed and bidirectional $2dp$-nearest-neighbor graphs, deriving asymptotically optimal bounds on critical parameters and resolving open questions in high dimensions. This provides a versatile tool for analyzing percolation in networks with local constraints and demonstrates new thresholds for complex neighbor-structure models with potential applications in statistical physics and network theory.

Abstract

We introduce a site-wise domination criterion for local percolation models, which enables the comparison of one-arm probabilities even in the absence of stochastic domination. The method relies on a local-to-global principle: if, at each site, one model is more likely than the other to connect to a subset of its neighbors, for all nontrivial such subsets, then this advantage propagates to connectivity events at all scales. In this way, we obtain a robust alternative to stochastic domination, applicable in all cases where the latter works and in many where it does not. As a main application, we compare classical Bernoulli bond percolation with degree-constrained models, showing that degree constraints enhance percolation, and obtain asymptotically optimal bounds on critical parameters for degree-constrained models.

Local criteria for global connectivity comparisons: beyond stochastic domination

TL;DR

The paper develops a robust local-to-global criterion for comparing percolation models that goes beyond stochastic domination, proving that a site-wise advantage in connecting to neighbor subsets propagates to global connectivity events such as connected to . Central to the approach is a discrete interpolation inequality coupled with a breadth-first exploration of the out-component, enabling a transfer from local inequalities to global bounds . The authors then apply this framework to degree-constrained percolation models, including directed and bidirectional -nearest-neighbor graphs, deriving asymptotically optimal bounds on critical parameters and resolving open questions in high dimensions. This provides a versatile tool for analyzing percolation in networks with local constraints and demonstrates new thresholds for complex neighbor-structure models with potential applications in statistical physics and network theory.

Abstract

We introduce a site-wise domination criterion for local percolation models, which enables the comparison of one-arm probabilities even in the absence of stochastic domination. The method relies on a local-to-global principle: if, at each site, one model is more likely than the other to connect to a subset of its neighbors, for all nontrivial such subsets, then this advantage propagates to connectivity events at all scales. In this way, we obtain a robust alternative to stochastic domination, applicable in all cases where the latter works and in many where it does not. As a main application, we compare classical Bernoulli bond percolation with degree-constrained models, showing that degree constraints enhance percolation, and obtain asymptotically optimal bounds on critical parameters for degree-constrained models.

Paper Structure

This paper contains 9 sections, 12 theorems, 46 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. The exploration process
  4. Discrete interpolation inequality
  5. Strict comparisons and exponential decay
  6. Applications
  7. Directed nearest-neighbor models
  8. Bidirectional nearest-neighbor models
  9. Directed $k$-nearest-neighbor models with additional geometry constraints

Key Result

Theorem 2.1

If $\mathbf{P}$, $\mathbf{Q}$ are such that then,

Figures (1)

  • Figure 1: Conditioning on all vertices in $B_n \setminus \{a\}$ reduces the question of the origin $o$ being connected to the boundary $\partial B_n$ to a local question: If $a$ is pivotal and $A$ is the set of its neighbors from which one can reach the boundary, then the event $\{o \rightsquigarrow \partial B_n\}$ occurs if and only if $\{N(a)\cap A\neq\emptyset\}$ occurs.

Theorems & Definitions (23)

  • Theorem 2.1: Local-to-global
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm:local-to-global']}
  • proof : Proof of Proposition \ref{['prop:discrete-interpolation']}
  • Theorem 2.3: Strict comparison
  • Proposition 2.4: grimmett_dependent_1998
  • proof : Proof of Theorem \ref{['thm:strict-comparison']}
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 13 more