If a random walk is forbidden from ever crossing its own path — the simplest model of a polymer that can't pass through itself — how much bigger does it get, and is the swelling a universal number?
▶ Launch the interactive simulationGrow ensembles of walks on the lattice and measure the end-to-end size ⟨R²⟩ ∝ N^(2ν). Naive step-by-step growth biases ν, so weight each walk by its ROSENBLUTH factor W=∏(m_k/z) (m_k = free choices at step k) to recover the true self-avoiding ensemble; the log–log slope of ⟨R²⟩ vs N gives 2ν. Three live walkers paint self-avoiding paths until they trap themselves. The ordinary (crossing-allowed) walk is shown as a control.
self-avoidance SWELLS the coil to a universal exponent: ν=¾ EXACTLY in 2-D (Nienhuis; Flory's 3/(d+2) is exact here) and ≈0.588 in 3-D, versus ν=½ for the ordinary random walk (⟨R²⟩∝N). The local 'never revisit' rule produces a long-range size change. Honest: the Rosenbluth correction is essential — naive kinetic growth undershoots to ν≈0.64; finite walk length (N≤60) leaves a few-% residual bias