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تسجيل مستخدم جديدHigher order corrections for anisotropic bootstrap percolation
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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: [ p_cbig( [L]^2,mathcal{N}_{(1,2)},3 big) ; = ; frac{(log log L)^2}{12log L} , - , frac{log log L , log log log L}{ 3log L} + frac{left(log frac{9}{2} + 1 pm o(1) right)log log L}{6log L}. ] 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 $10^{10^{1000}}$.
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