Percolation Critical Exponents in Dimensions Three and Higher

For bond percolation on ℤ^d (d ≥ 3), define θ(p) = P(0 ↔ ∞). Question: prove that near \(p_{c}\) the observables satisfy power laws with well-defined exponents — θ(p) ∼ (p−\(p_{c}\))^β, ξ(p) ∼ |p−\(p_{c}\)|^{-ν}, etc. — and determine those exponents. In d = 2: β = 5/36, ν = 4/3 (proven via SLE). In d ≥ 3: none proven.

Harris–Kesten pinned \(p_{c}\) for d = 2 (see our historic shelf); Aizenman–Barsky and Menshikov proved general sharpness (mean-field exponents hold for p far from \(p_{c}\)); Smirnov's conformal invariance proof unlocked d = 2 via SLE. In d ≥ 3 the lace expansion gives mean-field exponents above the upper critical dimension (d ≥ 6, Hara–Slade), leaving 3 ≤ d < 6 as the genuinely open regime where lattice effects dominate.