  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.
