This blog is part of class project to explore various analytic techniques used by modern intelligence analysts.
They give a detailed description there, including an exact solution for 1D, 2D and 3D cubic lattices. I've also seen the exact solution in The Critical Point: I don't think that I've seen much discussion of the Gaussian model in standard texts on statistical mechanics, possibly because it has a problem at low temperatures see below.
So, it can be misleading calling it a Hamiltonian!
This is fairly straightforward Berlin and Kac show how to do it for a cubic lattice, a square lattice, and, in particular, for a linear chain, if you assume periodic wraparound boundaries.
I don't do that. Don't ask me what happens if you try to simulate the system below this temperature!
Finally, I'll turn to your question about sampling in Monte Carlo. The usual approach is to choose one spin at random, define a trial move which consists of updating that spin to give a new state of the system, and accept or reject that move on the basis of the energy change associated with that spin change.
If you reject, you keep the spin at its original value and count this state again, in the computation of simulation averages. The range is chosen empirically, to give a reasonable acceptance rate of trial moves; it is kept fixed throughout the simulation.
The essential aspect is to satisfy detailed balance: If you do this, then the acceptance criterion should be the Metropolis formula based only on the change in the neighbour interactions, i.
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