selfavoidance

Self-avoidance refers to the principle that a path or configuration is not allowed to intersect itself. In mathematics and statistical physics, the most studied instance is the self-avoiding walk (SAW): a sequence of moves on a lattice that visits each lattice site at most once. SAWs model the excluded-volume effect of long polymer chains in good solvent conditions and provide insight into critical phenomena and universality.

Let a_n be the number of n-step SAWs from a fixed origin. The connective constant mu = lim_{n→∞} a_n^{1/n} exists and mu ≤ z, the lattice coordination number. For the square lattice z=4, mu ≈ 2.638; for the cubic lattice, mu ≈ 4.684. Exact enumeration yields a_n for finite n.

Scaling theory posits that the typical end-to-end distance R_n scales as n^nu, and a_n behaves like C mu^n n^{gamma-1}, with universal critical exponents. In two dimensions, numerical and theoretical work suggests nu = 3/4 and gamma = 43/32; these values are widely supported, though not all are proven. A related result fixes mu exactly for the hexagonal lattice (mu = sqrt(2+√2)).

Alongside SAW, related models include self-avoiding polygons (closed loops with no self-intersections) and lattice animals. In the continuum limit, SAW relates to the theory of polymers and, in 2D, to Schramm–Loewner Evolution SLE_8/3, which is conjectured to describe the scaling limit.

Computational methods include exact enumeration for small n and Monte Carlo algorithms such as the pivot algorithm and flatPERM that efficiently sample long SAWs. Applications span polymer science, materials research, and theoretical studies of phase transitions.