A random walk that cannot cross its own path behaves fundamentally differently from one that can
Take a grid random walk and add one rule: never revisit a cell. That single constraint changes everything.
A self-avoiding walk (SAW) models a polymer chain — each step is a monomer, the walk is the backbone, and physical polymers cannot overlap themselves. SAWs expand faster than ordinary random walks; how much faster is one of the deepest open problems in mathematical physics.
The left canvas runs ordinary random walks; the right runs SAWs. Both start from center and overlay many independent walks. The SAW ensemble is visibly more extended.
Figure 1. Side-by-side comparison of ensemble shapes. The ordinary random walk (left) is compact and revisits old territory. The self-avoiding walk (right) spreads further from the origin because it cannot fold back on itself.
A random walk of $N$ steps has end-to-end distance $\sim N^{0.5}$. In 2D, the SAW exponent jumps to exactly $\nu = 3/4$, one of the few exact results in the theory (via conformal field theory and SLE). In 3D, $\nu \approx 0.588$ — the Flory value, estimated in 1949.
On a log-log plot the slope is the exponent. The kinetic-growth method used here is biased and gives an effective exponent of 0.59-0.65 rather than 0.75; unbiased sampling needs algorithms like the pivot method. Still, it is clearly above 0.5.
Figure 2. Log-log plot of end-to-end distance vs. chain length. Blue circles: self-avoiding walks. Gray circles: ordinary random walks. Dashed lines: best-fit power laws. The steeper slope for SAWs indicates a more extended structure.
A SAW can paint itself into a corner — all four neighbors visited, no move possible. On a 2D square lattice most SAWs eventually trap, but they survive surprisingly long. Short walks almost never trap; trapping probability rises slowly with length.
Figure 3. Left: a single SAW advancing on the lattice. The red cell marks a dead end. Right: histogram of walk lengths at trapping. Most walks survive far longer than a naive counting argument would suggest.
The 3D exponent $\nu \approx 0.588$ predicts how polymer radius scales with chain length, confirmed by neutron scattering. Despite 60+ years of work, no exact formula for the number of SAWs of length $N$ on a square lattice is known.