Relative Critical Sharpness

• Relative Critical Sharpness is a quantitative concept that defines the abrupt change in one‐arm probabilities as the bond parameter crosses the critical threshold.
• The methodology uses a coupling-based, two-scale inequality that leverages stochastic domination and iterative chunking to derive both exponential decay in the subcritical and mean-field behavior in the supercritical regimes.
• In high-dimensional settings, this approach provides precise near-critical scaling, predicting decay rates like n⁻² modified by exponential factors dependent on the distance from criticality.

Relative critical sharpness in Bernoulli percolation refers to the precise characterization of the transition in the one-arm (finite-volume connectivity) probability, $\theta_n(p)$, as the bond parameter $p$ crosses the critical threshold $p_c$. The key insight is that not only does $\theta_n(p)$ exhibit a rapid change from exponential decay to mean-field lower bounds at $p_c$, but the nature of this transition and its scaling with the relative distance $\delta = p-p_c$ can be encoded in a single coupling-based two-scale inequality. This approach yields quantitative, relative statements connecting phase behavior directly to the distance to criticality and is especially precise in high-dimensional settings.

1. Bernoulli Percolation Model and Key Quantities

Consider an infinite, vertex-transitive, locally finite graph $G=(V,E)$, with prototypical examples including the hypercubic lattice $\mathbb{Z}^d$. In Bernoulli bond percolation, each edge $e \in E$ is independently open with probability $p \in [0,1]$. The product measure on $p$0 is denoted $p$1.

Key geometric objects are:

• Graph distance: $p$2 is the shortest-path length between vertices $p$3.
• Ball/Sphere: $p$4, $p$5 for a fixed root $p$6.

The central probability is the one-arm probability: $p$7 This measures the probability that $p$8 is connected to distance $p$9. The infinite-volume connectivity is $p_c$0. The critical probability is defined by

$p_c$1

The relative distance to criticality is $p_c$2.

The non-connection probabilities are $p_c$3 and $p_c$4.

2. The Coupling-Based Two-Scale Sharpness Inequality

A central advance is the derivation of a coupling-based, two-scale sharpness inequality, applicable for all $p_c$5 and integers $p_c$6: $p_c$7 Here, $p_c$8 is explicit and arises from the estimate $p_c$9 for appropriate $\theta_n(p)$0. The strategy involves:

• Selecting $\theta_n(p)$1 so that $\theta_n(p)$2 is small,
• Shifting $\theta_n(p)$3,
• Applying the bound at doubled scale $\theta_n(p)$4 versus $\theta_n(p)$5,
• The factor $\theta_n(p)$6 reflects an iterative halving (“chunking”) argument.

This inequality is proved by establishing a stochastic domination (coupling) between a conditioned inhomogeneous percolation measure (conditioned on absence of long arms to distance $\theta_n(p)$7) and a corresponding unconditioned but reduced-parameter measure. Utilizing this, one obtains for edge-weights $\theta_n(p)$8: $\theta_n(p)$9 Iterating in “blocks” of size $p_c$0 yields

$p_c$1

and the parameter shift is at most $p_c$2 after $p_c$3 steps. Reformulation in terms of $p_c$4 leads directly to the two-scale inequality (Vanneuville, 2022).

3. Subcritical Exponential Decay and Supercritical Mean-Field Behavior

The coupling inequality yields the classical sharpness results in both subcritical and supercritical regimes:

Subcritical Case ($p_c$5):

As $p_c$6, one selects $p_c$7 and $p_c$8 so that

$p_c$9

Applying the inequality: $\delta = p-p_c$0 with $\delta = p-p_c$1. Thus, $\delta = p-p_c$2 decays exponentially in $\delta = p-p_c$3 for $\delta = p-p_c$4.

Supercritical Case ($\delta = p-p_c$5):

Taking $\delta = p-p_c$6 in the inequality and noting $\delta = p-p_c$7: $\delta = p-p_c$8 Letting $\delta = p-p_c$9 gives

$G=(V,E)$0

A refinement (see Corollary 1.2 in (Vanneuville, 2022)) retrieves the classical mean-field lower bound $G=(V,E)$1.

4. Near-Critical Sharpness and High Dimensions

In high-dimensional lattices, particularly $G=(V,E)$2 with $G=(V,E)$3, further quantitative results are available:

• At criticality, $G=(V,E)$4 (Hara–Slade, Kozma–Nachmias, Fitzner–van der Hofstad).
• For small $G=(V,E)$5,

$G=(V,E)$6

Selecting $G=(V,E)$7 ensures $G=(V,E)$8. This choice ensures $G=(V,E)$9, yielding via the two-scale inequality

$\mathbb{Z}^d$0

This recovers the upper bound for $\mathbb{Z}^d$1 in the near-critical window. The correlation length scales as $\mathbb{Z}^d$2, and inside this window,

$\mathbb{Z}^d$3

5. Summary Table of Regimes

Regime	Behavior of $\mathbb{Z}^d$4	Relation to $\mathbb{Z}^d$5
Subcritical $\mathbb{Z}^d$6	$\mathbb{Z}^d$7	Exponential decay
Supercritical $\mathbb{Z}^d$8	$\mathbb{Z}^d$9	Mean-field lower bound
Near-critical, high $e \in E$0	$e \in E$1	$e \in E$2

These behaviors, in all cases, are consequences of the central two-scale coupling inequality, which encodes the sharp transition in one-arm probabilities as $e \in E$3 changes sign (Vanneuville, 2022).

6. Significance and Extensions

The coupling-based approach to relative critical sharpness provides a unified and quantitative framework for understanding phase transitions in percolation. It bypasses differential inequalities, instead leveraging explicit stochastic domination rooted in coupling arguments, as inspired by Russo (1982). The framework yields sharp quantitative bounds in both subcritical and supercritical regimes and remains robust in high dimensions where near-critical scaling can be precisely captured. This suggests potential for extension to other random media models that admit similar exploration and coupling constructions. A plausible implication is the possibility of deriving analogous sharpness results for models beyond Bernoulli percolation, particularly where couplings or stochastic domination enable two-scale decomposition of connectivity probabilities.