Ordering Field, Order Parameter, and Self-Avoiding Walks

We consider a grand canonical ensemble of self-avoiding walks and study its properties on a Bethe lattice of coordination number $q=3\mathrm{}$ and in one dimension, without using the $n\to 0$ trick. The study enables us to identify the ordering field as the activity for the walk ends. We also identify the order parameter of the problem for the first time: The order parameter is the probability of obtaining an infinite self-avoiding walk (reaching the "boundary"), having started at the origin in some direction.