  Markov chain Monte Carlo (MCMC) algorithms have played a significant role in
statistics, physics, machine learning and others, and they are the only known
general and efficient approach for some high-dimensional problems. The random
walk Metropolis (RWM) algorithm as the most classical MCMC algorithm, has had a
great influence on the development and practice of science and engineering. The
behavior of the RWM algorithm in high-dimensional problems is typically
investigated through a weak convergence result of diffusion processes. In this
paper, we utilize the Mosco convergence of Dirichlet forms in analyzing the RWM
algorithm on large graphs, whose target distribution is the Gibbs measure that
includes any probability measure satisfying a Markov property. The abstract and
powerful theory of Dirichlet forms allows us to work directly and naturally on
the infinite-dimensional space, and our notion of Mosco convergence allows
Dirichlet forms associated with the RWM chains to lie on changing Hilbert
spaces. Through the optimal scaling problem, we demonstrate the impressive
strengths of the Dirichlet form approach over the standard diffusion approach.
