  We implement a scale-free version of the pivot algorithm and use it to sample
pairs of three-dimensional self-avoiding walks, for the purpose of efficiently
calculating an observable that corresponds to the probability that pairs of
self-avoiding walks remain self-avoiding when they are concatenated. We study
the properties of this Markov chain, and then use it to find the critical
exponent $\gamma$ for self-avoiding walks to unprecedented accuracy. Our final
estimate for $\gamma$ is $1.15695300(95)$.
