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. Harris–Kesten pinned \(p_c\) for d = 2; Aizenman–Barsky and Menshikov proved general sharpness, and the lace expansion gives mean-field exponents above the upper critical dimension (d ≥ 6, Hara–Slade).