Functional limit theorems for the Bouchaud trap model with slowly-varying traps

David Croydon, Stephen Muirhead

Published 2014-10-03Version 1

We consider the Bouchaud trap model on the integers in the case that the trap distribution has a slowly-varying tail at infinity. Our main result is a functional limit theorem for the model under the annealed law, analogous to the functional limit theorems previously established in the literature in the case of integrable or regularly-varying trap distribution. Reflecting the fact that the clock process is dominated in the limit by the contribution from the deepest-visited trap, the limit process for the model is a spatially-subordinated Brownian motion whose associated clock process is an extremal process. We identify this process as the $\alpha \to 0$ scaling limit of the FIN diffusion with parameter $\alpha$. Additionally, we study a natural `transparent' generalisation of the Bouchaud trap model with slowly-varying trap distribution.