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

TL;DR

This paper establishes a functional limit theorem for the Bouchaud trap model with slowly varying trap distributions, showing the limiting process is a spatially subordinated Brownian motion driven by an extremal process.

Contribution

It extends existing limit theorems to the case of slowly varying trap tails, highlighting the dominance of the deepest trap in the asymptotic behavior.

Findings

Limit process is a spatially subordinated Brownian motion.

Clock process converges to an extremal process.

Results generalize previous theorems for regularly varying traps.

Abstract

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.