Related topics

Abstract

The weakly self-avoiding walk, known also as the self-repellent walk and as the Domb-Joyce model [Domb and Joyce (1972)], is a measure on ordinary random walks in which self-intersections are discouraged but not forbidden. The measure associates to an n-step simple random walk ω the weight

$$\mathop Q\nolimits_n\lambda \left( \omega \right) = {1 \over {Z_n \left( \lambda \right)}}\mathop \prod \limits_{0 \le s < \le n} \left[ {1 - \lambda u_{st} \left( \omega \right)} \right],$$

(10.1.1)

where 0 > λ ≤ 1, Z _n (λ) is a normalization constant, the product is over pairs of integers s and t, and v _st (ω) is 1 if ω(s) = ω(t) and otherwise is 0. Taking λ = 1 gives the uniform measure on n-step self-avoiding walks, while 0 > λ > 1 gives a measure in which self-intersections diminish the probability of a walk. Setting λ = 0 just gives simple random walk. An alternate parametrization of the interaction which appears frequently is to take

$$ \lambda = 1 - e^{ - \beta }. $$

(10.1.2)