The K-process on a tree as a scaling limit of the GREM-like trap model

TL;DR

This paper introduces a new class of trap models on finite trees, establishes their infinite volume limit called the K-process, and shows it as the scaling limit of the GREM-like trap model under certain conditions.

Contribution

The paper defines the K-process as a scaling limit of GREM-like trap models on trees, providing a rigorous connection between finite models and their infinite volume limits.

Findings

The K-process is the infinite volume limit of the finite-volume trap models.

Under specific parameter conditions, the K-process describes the scaling limit of the GREM-like trap model.

The results apply to extreme time scales with fine tuning of volumes.

Abstract

We introduce trap models on a finite volume $k$-level tree as a class of Markov jump processes with state space the leaves of that tree. They serve to describe the GREM-like trap model of Sasaki and Nemoto. Under suitable conditions on the parameters of the trap model, we establish its infinite volume limit, given by what we call a $K$-process in an infinite $k$-level tree. From this we deduce that the $K$-process also is the scaling limit of the GREM-like trap model on extreme time scales under a fine tuning assumption on the volumes.