Optimal corrector estimates on percolation cluster
Paul Dario
Abstract
We prove optimal quantitative estimates on the first-order correctors on supercritical percolation clusters: we show that they are bounded in dimension larger than 3 and have logarithmic growth in dimension 2 in the sense of stretched exponential moments. The main ingredients are a renormalization scheme of the supercritical percolation cluster, following the works of Pisztora (Probab. Theory Related Fields 104 (1996) 427–466); large-scale regularity estimates developed by Armstrong and the author in (Comm. Pure Appl. Math. 71 (2018) 1717–1849); and a nonlinear concentration inequality of the Efron–Stein type which is used to transfer quantitative information from the environment to the correctors.
Topics & Concepts
MathematicsPercolation (cognitive psychology)LogarithmBounded functionDimension (graph theory)Supercritical fluidCluster (spacecraft)Exponential functionRenormalizationLogarithmic growthMonotonic functionNonlinear systemStatistical physicsApplied mathematicsMathematical analysisCombinatoricsMathematical physicsNeuroscienceBiologyPhysicsProgramming languageChemistryOrganic chemistryQuantum mechanicsComputer scienceAdvanced Mathematical Modeling in EngineeringNonlinear Partial Differential EquationsStochastic processes and statistical mechanics