math.PR — Probability
Computer-assisted bounds 0.556921 <= p_c <= 0.6463 for site percolation on the square lattice
Let p_c be the critical probability of Bernoulli site percolation on Z^2. We prove 0.556921 <= p_c <= 0.6463. The previous best rigorous bounds known to us are p_c >= 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 0.556788090 reported by Ashton as the limit of his relaxation. The upper bound uses the substitution method of Wierman, in two stages with 4x4 regions, from the line graph of Z^2 (site threshold 1/2) to Z^2. Both bounds are certified by short pure-Python programs in exact rational and integer arithmetic.