Open Access
2023 On the Besicovitch-stability of noisy random tilings
Gayral Léo, Sablik Mathieu
Author Affiliations +
Electron. J. Probab. 28: 1-38 (2023). DOI: 10.1214/23-EJP917


In this paper, we introduce a framework for studying a subshift of finite type (SFT) with noise, allowing some amount of forbidden patterns to appear. Using the Besicovitch distance, which permits a global comparison of configurations, we then study the closeness of measures on noisy configurations to the non-noisy case as the amount of noise goes to 0. Our first main result is the full classification of the (in)stability in the one-dimensional case. Our second main result is a stability property under Bernoulli noise for higher-dimensional periodic SFTs, which we finally extend to an aperiodic example through a variant of the Robinson tiling.

Funding Statement

This work was supported by the ANR “Difference” project (ANR-20-CE40-0002) and the CIMI LabEx “Computability of asymptotic properties of dynamical systems” project (ANR-11-LABX-0040). The second author wishes to thank Siamak Taati for many fruitful references on the stability of tilings.


Download Citation

Gayral Léo. Sablik Mathieu. "On the Besicovitch-stability of noisy random tilings." Electron. J. Probab. 28 1 - 38, 2023.


Received: 10 September 2021; Accepted: 7 February 2023; Published: 2023
First available in Project Euclid: 9 February 2023

Digital Object Identifier: 10.1214/23-EJP917

Primary: 37A50 , 37B51
Secondary: 60K35 , 82B43

Keywords: Besicovitch distance , percolation , Robinson tiling , stability , Subshift of finite type

Vol.28 • 2023
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