The Critical Percolation Exponents in Dimensions Three and Higher

For bond percolation on ℤ^d near \(p_{c}\), define θ(p) = \(P_{p}\)(0 ↔ ∞). Question: prove that θ(p) ∼ C(p−\(p_{c}\))^β as p ↓ \(p_{c}\) with explicit β, and similarly for correlation length ν, cluster size γ, etc. In d = 2: β = 5/36, ν = 4/3 (Smirnov 2001, SLE). In d ≥ 6: β = 1, ν = 1/2 (Hara–Slade, lace expansion). In d = 3, 4, 5: nothing rigorous.

The existence of critical exponents was only proven in general by Aizenman–Barsky and Menshikov (1987). Kesten proved d = 2 exponents implicitly through his \(p_{c}\) theorem; Smirnov's conformal invariance breakthrough (2001, Fields Medal content) gave exact d = 2 values via SLE. The lace expansion (Hara–Slade) handles d ≥ 6. The intermediate dimensions 3–5 are the hardest case because neither integrable-systems tools nor mean-field approximations apply cleanly.