The Self-Avoiding Walk Connective Constant

Let \(c_{n}\) count the number of n-step self-avoiding walks starting at the origin on ℤ². Question: determine the connective constant μ = \(lim_{n→∞}\) \(c_{n}^{1/n}\). For the hexagonal lattice, μ = √(2+√2) exactly (Duminil-Copin–Smirnov 2010). For the square lattice, μ ≈ 2.638 from numerical estimates but no exact value or rigorous proof.

Orrick considered self-avoiding polymer chains in the 1940s; Hammersley and Morton formalised the connective constant and proved submultiplicativity (\(c_{m+n}\) ≤ \(c_{m}\) · \(c_{n}\)), guaranteeing the limit exists. Nienhuis's 1982 Coulomb gas prediction gave μ ≈ 2.638 for the square lattice via Coulomb-Franck constants. Lawler–Schramm–Werner's SLE framework (2004 Fields work) confirmed fractal dimension predictions but not μ itself.