Hemmen, JLV; Palmer, RG, *The thermodynamic limit and the replica method for short-range random systems*,
Journal of Physics A: Mathematical and General, vol. 15 no. 12
(1982),
pp. 3881-3890 [doi] .
**Abstract:**

*Short-range spin systems with random interactions are considered. A simple proof is given showing that the free energy of almost every sample converges to the average free energy in the thermodynamic limit. A stronger criterion, thermodynamic convergence, is also demonstrated. This implies that the N to infinity and n to 0 limits may be interchanged in the replica method.*