[Paper Review] Percolation, boundary, noise: an experiment

Boris Tsirelson|ArXiv.org|Jun 14, 2005

Theoretical and Computational Physics5 references3 citations

TL;DR

This experimental study investigates whether the scaling limit of two-dimensional critical site percolation on the triangular lattice is stable under strong, localized perturbations along a line—specifically, by removing and re-sampling a narrow strip of hexagons near the equator. Using numerical simulations of exploration paths in finite domains, the results suggest that the distance between paths remains small as the strip width decreases, indicating strip stability and supporting the conjecture that the scaling limit may be a black noise.

ABSTRACT

The scaling limit of the critical percolation, is it a black noise? The answer depends on stability to perturbations concentrated along a line. This text, containing no proofs, reports experimental results that suggest the affirmative answer.

Motivation & Objective

• To investigate whether the scaling limit of critical percolation is stable under strong, localized perturbations concentrated along a line.
• To determine if such stability implies that the scaling limit forms a noise, specifically a 'black noise' as defined in stochastic analysis.
• To resolve a foundational question about whether the configuration in a domain is uniquely determined by its restrictions to subdomains separated by a line.
• To test the hypothesis that percolation's scaling limit is not disrupted by infinitesimal strip perturbations, which would imply it is not a black noise.

Proposed method

• Simulate critical site percolation on a triangular lattice within a rhombus-shaped domain, with deterministic boundary coloring (left white, right black).
• Construct exploration paths via the exploration process, which trace the interface between black and white clusters.
• Remove and re-generate percolation data in k consecutive rows near the equator (a narrow strip), then recompute the exploration path.
• Measure the distance between the original and perturbed exploration paths using a discrete approximation to the metric defined in [3, eq. (2)], minimizing over monotonic reparameterizations.
• Use dynamic programming to efficiently compute the path distance, with a pre-processing step to downsample paths while preserving accuracy within ±0.03.
• Run 250 independent trials per parameter set (n, k), where n controls domain size and k controls strip width, and report median path distances.

Experimental results

Research questions

• RQ1 Is the scaling limit of 2D critical percolation stable under strong perturbations confined to an infinitesimal strip along a line?
• RQ2 Does the failure of the σ-field decomposition condition F_{r,t} = F_{r,s} ∨ F_{s,t} imply that percolation does not yield a noise?
• RQ3 Can the scaling limit be classified as a black noise if it remains stable under such concentrated perturbations?
• RQ4 Does the observed decay of path distance with decreasing strip width ε = k/n support the existence of a non-trivial noise structure?
• RQ5 Can microscopic perturbations be absorbed by the strong perturbations in the limit, preserving macroscopic stability?

Key findings

• The median distance between exploration paths decreases as the strip width ε = k/n decreases, indicating increasing stability under perturbations.
• For k=1 and n=16, the median distance is 0.22; for n=1024, it drops to 0.08, suggesting a power-law decay with exponent near 1/3.
• The distance remains bounded and systematically decreases with increasing n, even for fixed k, indicating that the effect of the perturbation diminishes at large scales.
• The results are robust across multiple trials and parameter sets, with sampling error around 7% of the median value.
• The systematic error in distance computation is bounded by ±0.03 due to path approximation, supporting the reliability of the observed trend.
• The observed stability under strong, localized perturbations supports the conjecture that the scaling limit of percolation may be a black noise.