Statistical mechanics, structure of matter Definition

Self-avoiding walk

Definition

Let $d \ge 1$ and $n \ge 0$ be integers, and let $\abs{\cdot}$ be the Euclidean norm on $\R^d$ . An $n$ -step self-avoiding walk on $\Z^d$ starting at $0$ is a sequence $\omega = (\omega_0, \omega_1, \ldots, \omega_n)$ of points of $\Z^d$ such that:

$\omega_0 = 0$ ;
$\abs{\omega_i - \omega_{i-1}} = 1$ for $1 \le i \le n$ ;
$\omega_i \ne \omega_j$ for $0 \le i < j \le n$ .

Let $c_n$ be the number of $n$ -step self-avoiding walks. The uniform probability measure on these $c_n$ walks is also called the self-avoiding walk with $n$ steps.

Remarks

Example: in $\Z^2$ , the walk $(0,0), (1,0), (1,1), (0,1)$ satisfies all three clauses, and it is one of the $c_3 = 36$ walks with three steps. Non-example: $(0,0), (1,0), (0,0)$ satisfies clauses 1 and 2 but not clause 3, since it returns to $0$ . Cutting an $(n + m)$ -step walk after $n$ steps gives $c_{n+m} \le c_n c_m$ , so by Fekete's lemma the connective constant $\mu_d = \lim_{n \to \infty} c_n^{1/n}$ exists. Walks that only move in positive coordinate directions are self-avoiding, and walks that never reverse their last step include all self-avoiding walks, so $d \le \mu_d \le 2d - 1$ .

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Random walk concept
Self-avoiding walk definition

Tag	`RDFA0SYHEQ`
URI	`m2://RDFA0SYHEQ`
Kind	definition
Writing	W0 stub
MSC	82B41, 60K35, 05A16
Defines	Self-avoiding walk
Uses	Random walk
Changed	2 November 2026

Self-avoiding walk

Definition

Remarks

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