  We study properties of Markov chain Monte Carlo simulations of classical spin
models with local updates. We derive analytic expressions for the mean value of
the acceptance rate of single-spin-flip algorithms for the one-dimensional
Ising model. We find that for the Metropolis algorithm the average acceptance
rate is a linear function of energy. We further provide numerical results for
the energy dependence of the average acceptance rate for the 3- and 4-state
Potts model, and the XY model in one and two spatial dimensions. In all cases,
the acceptance rate is an almost linear function of the energy in the critical
region. The variance of the acceptance rate is studied as a function of the
specific heat. While the specific heat develops a singularity in the vicinity
of a phase transition, the variance of the acceptance rate stays finite.
