The Self-avoiding Walk Spanning a Strip

We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as β → β_c- of the probability measure on all finite length walks ω with the probability of ω proportional to β^{{pipe}ω{pipe}} where {pipe}ω{pipe} is the number of steps in ω. (β_c is the reciprocal of the connective constant.) The self-avoiding walk in a strip {z : 0 < Im(z) < y} is defined by considering all self-avoiding walks ω in the strip which start at the origin and end somewhere on the top boundary with probability proportional to β_c^{{pipe}ω{pipe}}. We prove that this probability measure may be obtained by conditioning the SAW in the half plane to have a bridge at height y. This observation is the basis for simulations to test conjectures on the distribution of the endpoint of the SAW in a strip and the relationship between the distribution of this strip SAW and SLE_8/3.