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.
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.
"On the Besicovitch-stability of noisy random tilings." Electron. J. Probab. 28 1 - 38, 2023. https://doi.org/10.1214/23-EJP917