Computer-assisted bounds 0.556921 <= p_c <= 0.6463 for site percolation on the square lattice
Source: hexagonmath.org/abs/2611.00065v1
Let p_c be the critical probability of Bernoulli site percolation on Z^2. We prove 0.556921 = 0.5567 (Ashton, 2026, extending van den Berg and Ermakov, 1996) and p_c <= 0.666894 (Wierman and Oberly, 2024). The lower bound uses the growth-process comparison of van den Berg and Ermakov with the exact posterior on the exploration tree. We reduce this posterior to a scalar interval recursion and explore the tree generation by generation, with a prescribed order among siblings. This passes the value 0.556788... that the same scalar computation gives when the order is not used, and the value…
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