Multidimensional examples of the Metropolis algorithm

Consider the problem of approximating a given probability distribution on the cube $[0,1]^n$ via the use of a square lattice discretization with mesh-size $1/N$ and the Metropolis algorithm. Here the dimension $n$ is fixed and we focus for the most part on the case $n=2$. In order to understand the speed of convergence of such a procedure, one needs to control the spectral gap, $\lambda$, of the associated finite Markov chain, and how it depends on the parameter $N$. In this work, we study basic examples for which good upper-bounds and lower-bounds on $\lambda$ can be obtained via appropriate application of path techniques.