by $v$, denoted by $a^-_{v} (w)$.

The neutral atmosphere (an elementary move which does not change the length of the walk) of a walk $w$ is similarly given by $a^0_{v} (w)$, if it changes the energy of a walk by $v$. If the endpoint elementary move in figure \[figure1\] is used, then $a^0_{v} (w)=0$ by default (since there are no neutral elementary moves implemented), but in general assume that a more general set of elementary moves (for example BFACF elementary moves) is used to sample walks, and in that case the neutral atmosphere may have positive size.

Implementation of GAS-sampling
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Suppose that an elementary move is implementated on the state space of self-avoiding walks from the origin, and assume the implementation is irreducible (that is, the elementary move gives a connected graph on the state space of walks).

Suppose that the sequence $$\phi_N = \LA w_0,w_1,w_2,\ldots,w_n,\ldots,w_N\RA
\label{eqn1}     $$ is realised after $N$ steps and that the atmospheres of the states $w_n$ have sizes (or *statistics*) $a_{v}^+(w_n)$, $a_{v}^0(w_n)$ and $a_{v}^-(w_n)$. These elementary moves may be classified as follows, depending on whether they increase or decrease the lengths of the walks (or are neutral), or whether they increase or decrease, or 