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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: \\[\n  p_c\\big( [L]^2,\\mathcal{N}_{(1,2)},3 \\big) \\; = \\; \\frac{(\\log \\log L)^2}{12\\log L} \\, - \\, \\frac{\\log \\log L \\, \\log \\log \\log L}{ 3\\log L}\n  + \\frac{\\left(\\log \\frac{9}{2} + 1 \\pm o(1) \\right)\\log \\log L}{6\\log L}. \\] We ","authors_text":"Aernout C. 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