Nonergodicity of local, length-conserving Monte Carlo algorithms for the self-avoiding walk

It is proved that every dynamic Monte Carlo algorithm for the self-avoiding walk based on a finite repertoire of local, N-conserving elementary moves is nonergodic (here N is the number of bonds in the walk). Indeed, for large N, each ergodic class forms an exponentially small fraction of the whole space. This invalidates (at least in principle) the use of the Verdier-Stockmayer algorithm and its generalizations for high-precision Monte Carlo studies of the self-avoiding walk.