Higher order corrections for anisotropic bootstrap percolation

Abstract

We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with (1, 2)-neighbourhood and threshold r = 3. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: pc([L]2,N(1,2),3)=(loglogL)212logL-loglogLlogloglogL3logL+log92+1±o(1)loglogL6logL.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} p_c\\big ( [L]^2,{\\mathcal {N}}_{\\scriptscriptstyle (1,2)},3 \\big ) \\;= & {} \\; \\frac{(\\log \\log L)^2}{12\\log L} - \\frac{\\log \\log L \\, \\log \\log \\log L}{ 3\\log L} \\\\&+ \\frac{\\left( \\log \\frac{9}{2} + 1 \\pm o(1) \\right) \\log \\log L}{6\\log L}. \\end{aligned}$$\\end{document}We note that the second and third order terms are so large that the first order asymptotics fail to approximate p_c even for lattices of size well beyond 10^{10^{1000}}.