[Paper Review] Strict inequality for the chemical distance exponent in two-dimensional critical percolation

Michael Damron, Jack Hanson+1 more|arXiv (Cornell University)|Aug 11, 2017

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper establishes the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation, proving that the expected length of the shortest horizontal crossing path in a box of side length $n$ is bounded by $Cn^{2- u}\pi_3(n)$ for some $\nu>0$, where $\pi_3(n)$ is the three-arm probability. The result implies that, with high probability, there exists a crossing significantly shorter than the lowest crossing, resolving a long-standing open question about the strict inequality of the chemical distance exponent.

ABSTRACT

We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by $Cn^{2-δ}π_3(n)$, for some $δ>0$, where $π_3(n)$ is the "three-arm probability to distance $n$." This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of $n$ with high probability. In the case of site percolation on the triangular lattice, we obtain a strict upper bound for the exponent of $4/3$. The proof builds on the strategy developed in our previous paper, but with a new iterative scheme, and a new large deviation inequality for events in annuli conditional on arm events, which may be of independent interest.

Motivation & Objective

• To establish a nontrivial upper bound on the chemical distance exponent in two-dimensional critical percolation, which remains poorly understood despite extensive study.
• To resolve the open question of whether the expected length of the shortest crossing path is strictly smaller than that of the lowest crossing, i.e., $\mathbb{E}[S_n \mid H_n] \ll \mathbb{E}[L_n \mid H_n]$.
• To provide a rigorous probabilistic framework for analyzing path length reduction in critical percolation clusters using conditional independence and iterative path modification.
• To extend the strategy from prior work by introducing a new iterative scheme and a novel large deviation inequality for arm events in annuli.
• To demonstrate that the chemical distance exponent is strictly less than 2 in the case of site percolation on the triangular lattice, with an upper bound of $n^{1+s}$ for $s<1/3$.

Proposed method

• The authors develop a new iterative scheme to construct shortcut paths around edges in the lowest crossing, conditional on the edge being part of the lowest path rather than after conditioning on the entire path.
• They introduce a conditional large deviation inequality for events in annuli given a three-arm event, which controls the probability of rare configurations that could prevent shortcut formation.
• The proof relies on a decomposition of the domain into dyadic annuli and uses a stopping time argument to control the number of regions where shortcuts can be constructed.
• A key technical tool is the use of conditional probabilities under the three-arm event $A_3(2^n)$, ensuring that the existence of a three-arm configuration supports the construction of shortcuts with positive probability.
• The method leverages the conditional independence of regions above the lowest crossing path, enabling localized modifications without disrupting the global crossing structure.
• The authors apply a martingale-type argument with a filtration adapted to the annular decomposition, bounding the probability that too few shortcuts are formed.

Experimental results

Research questions

• RQ1 Is the chemical distance exponent in two-dimensional critical percolation strictly less than 2, implying that the shortest crossing is significantly shorter than the lowest crossing on average?
• RQ2 Can a nontrivial upper bound be established for the expected length of the shortest horizontal crossing path in critical percolation on $\mathbb{Z}^2$?
• RQ3 Does the existence of a three-arm event to distance $n$ provide sufficient control to construct shortcuts that reduce path length by a power of $n$?
• RQ4 In site percolation on the triangular lattice, can the exponent of the chemical distance be bounded strictly below $4/3$?
• RQ5 Can the strategy from prior work be refined to yield a strict inequality in the scaling of the chemical distance, rather than just a subdiffusive bound?

Key findings

• The expected length of the shortest horizontal crossing path, $\mathbb{E}[S_n \mid H_n]$, is bounded above by $Cn^{2-\delta}\pi_3(n)$ for some $\delta>0$, where $\pi_3(n)$ is the three-arm probability to distance $n$, establishing a nontrivial upper bound on the chemical distance exponent.
• The ratio $\mathbb{E}[S_n \mid H_n]/\mathbb{E}[L_n \mid H_n]$ is bounded by $Cn^{-\delta}$ for some $\delta>0$, proving that the shortest crossing is significantly shorter than the lowest crossing with high probability.
• In the case of site percolation on the triangular lattice, the chemical distance exponent is strictly less than $4/3$, with the expected shortest crossing length bounded by $n^{1+s}$ for some $s<1/3$.
• The proof introduces a new large deviation inequality for arm events in annuli, conditional on the three-arm event, which may be of independent interest in percolation theory.
• The iterative construction of shortcuts, based on conditional modifications around edges in the lowest crossing, provides a robust framework for analyzing path length reduction in critical percolation.
• The result confirms that the chemical distance exponent is strictly less than 2, resolving a long-standing expectation in the field that such a strict inequality should hold.