Self-Avoiding Walk (SAW) & Flory Theory
A random walk on a 2D lattice that cannot revisit any site.
Flory (1949) minimized free energy F = R²/Nb² + N²/R^d to get:
ν = 3/(d+2)
• d=2: ν = 3/4 = 0.75 (exact, Nienhuis 1982)
• d=3: ν ≈ 0.588 (exact RG)
• d=4: ν = 1/2 (upper critical dim)
This is polymer science's most important scaling law. In good solvent, excluded volume interactions cause swelling.
The walk displayed uses a pivot algorithm (MCMC) for efficient sampling.
The log-log plot shows R vs N — the slope gives ν. Collect data at multiple N to see the power law emerge.