During this research in residence, we would like to continue and write up the work developed during a first research in residence at CIRM from February 27th to March 10th, 2023, followed by a workshop from 25 to 29 September of last year. The aim is to study the partition function of the Random Energy Model (REM), introduced by Bernard DERRIDA in 1980, in the complex temperature domain and to relate it to the theory of complex random functions and the Gaussian free field in two dimensions. It is a canonical model of statistical mechanics with quenched randomness, exactly soluble, related to the replica theory developed by Giorgio PARISI. The approach followed is both from theoretical physics and probability theory