The "Discrete Gaussian Chain" is a model which extends the celebrated long-range 1D lsing model with 1/r^\alpha interactions. The latter model is known to have a rather intriguing phase-diagram. Instead of having +/- spins, the discrete Gaussian Chain is a random field with values in the integers Z.

After introducing this model and its history, I will describe its large scale fluctuations (described by fractional Brownian motion) and will compare its phase diagram with the case of long-range Ising model.