Rapid mixing in unimodal landscapes and efficient simulatedannealing for multimodal distributions

We consider nearest neighbor weighted random walks on the $d$-dimensional box $[n]^d$ that are governed by some function $g:[0,1] \ra [0,\iy)$, by which we mean that standing at $x$, a neighbor $y$ of $x$ is picked at random and the walk then moves there with probability $(1/2)g(n^{-1}y)/(g(n^{-1}y)