Random energy model

In the statistical physics of disordered systems, the random energy model is a toy model of a system with quenched disorder, such as a spin glass, having a first-order phase transition. It concerns the statistics of a collection of ${\displaystyle N}$ spins (i.e. degrees of freedom ${\displaystyle {\boldsymbol {\sigma }}\equiv \{\sigma _{i}\}_{i=1}^{N}}$ that can take one of two possible values ${\displaystyle \sigma _{i}=\pm 1}$) so that the number of possible states for the system is ${\displaystyle 2^{N}}$. The energies of such states are independent and identically distributed Gaussian random variables ${\displaystyle E_{x}\sim {\mathcal {N}}(0,N/2)}$ with zero mean and a variance of ${\displaystyle N/2}$. Many properties of this model can be computed exactly. Its simplicity makes this model suitable for pedagogical introduction of concepts like quenched disorder and replica symmetry.