selfavoiding

Self-avoiding refers to a process or path that does not visit the same site or state more than once. In mathematics and statistical physics, the self-avoiding walk (SAW) is a model in which a path on a lattice starts at a point and moves to neighboring lattice sites, with the constraint that no site is repeated. This creates a self-avoiding path whose length is the number of steps.

The concept originated in the study of polymers, where the excluded-volume effect makes a polymer chain behave differently from a simple random walk. The model helps in understanding the statistics of long-chain molecules in solution and in computational simulations of phase behavior.

Let c_n denote the number of n-step SAWs on a given lattice. The growth is exponential, as c_n ~ mu^n n^{gamma-1}, defining the connective constant mu and the critical exponent gamma (which depend on the lattice and dimension). In two dimensions on the square lattice, mu is about 2.638, and gamma is predicted to be 43/32 by conformal field theory methods; mu and gamma are not known exactly. In three dimensions the exponents and mu are known only numerically (mu roughly 4.68).

Variants include self-avoiding polygons (closed SAWs forming a loop without self-intersections) and prudent walks (moves constrained away from the interior of the already visited region). Methods used to analyze SAWs include exact enumeration for small n and Monte Carlo algorithms such as the pivot algorithm and the pruned-enriched Rosenbluth method (PERM), which enable estimates of mu and gamma for higher dimensions.

Applications extend beyond polymers to network theory, combinatorics, and the study of phase transitions in statistical physics.