Temperature dependence of the Gibbs state in the random energy model (Q1871886)

Temperature chaos, that is the question on how dramatically the Gibbs state of a disordered system reacts to small changes of the temperature, is one of the actively and controversially discussed issues in the theory of disordered systems today. The present paper addresses this issue by providing a complete and rigorous answer in the context of a simple toy model, the random energy model. Interestingly, the author considers ``two ways how to raise the temperature'' that lead to a different correlation between the disorder variables. In the first, more standard version, the temperature parameter is changed while the realisation of the disorder remains untouched. In that case, it is shown that no temperature chaos takes place. The Gibbs state changes smoothly, as can be seen from the explicit formulas obtained for the overlap distribution between the two states. The second way consists in representing the Gaussian random variables defining the model as Brownian motions at time \(t=\sqrt {\beta}\) and considering two Gibbs states for different values of \(t\). This time, temperature chaos is manifest below the critical temperature: the two states get uncorrelated as soon as \(t'\) is different from \(t\). The reason for this should be intuitively clear: the change in \(t\) affects different variables differently, and modifies the order statistics of the random variables defining the model. In this situation, the author also looks on the fine structure of the change to the Gibbs measure by considering correlations between times \(t\) and \(t+\theta/N\). In that case the precise values of the overlap distribution between the two states are again computed explicitly.