Spectral characterisation of ageing: the REM-like trap model

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We review the ageing phenomenon in the context of simplest trap model, Bouchaud's REM-like trap model from a spectral theoretic point of view. We show that the generator of the dynamics of this model can be diagonalised exactly. Using this result, we derive closed expressions for correlation functions in terms of complex contour integrals that permit an easy investigation into their large time asymptotics in the thermodynamic limit. We also give a `grand canonical' representation of the model in terms of the Markov process on a Poisson point process. In this context we analyse the dynamics on various time scales.