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Scaling limits of the Bouchaud and Dean trap model on Parisi's tree in ergodic and aging time scales

Luiz Renato Fontes, Andrea Hernández

TL;DR

This work provides a rigorous dynamical analysis of the Bouchaud and Dean trap model on Parisi's tree by introducing a continuity theorem for cascading jump evolutions and applying it to obtain scaling limits in ergodic and aging regimes. The authors identify limiting dynamics as cascading K-processes (ergodic) and cascading aging processes, built from stable subordinators across levels, and establish three scaling regimes with precise rescalings of volumes and time. Aging results are obtained through explicit limiting no-jump functions $f_j(t/t_w)$ expressed via products of Beta variables, revealing aging behavior consistent with, yet distinct from, GLTM/GREM theories. The findings provide a mathematically rigorous framework for multi-level aging in trap models and connect to existing REM/GREM-like aging literature while clarifying how cascading structure governs dynamics across time scales.

Abstract

We take scaling limits of the Bouchaud and Dean trap model on Parisi's tree in time scales where the dynamics is either ergodic (close to equilibrium) or aging (far from equilibrium). These results follow from a continuity theorem formulated for a certain kind of process on trees, which we call a cascading jump evolution, defined in terms of a collection of jump functions, with a cascading structure given by the tree.

Scaling limits of the Bouchaud and Dean trap model on Parisi's tree in ergodic and aging time scales

TL;DR

This work provides a rigorous dynamical analysis of the Bouchaud and Dean trap model on Parisi's tree by introducing a continuity theorem for cascading jump evolutions and applying it to obtain scaling limits in ergodic and aging regimes. The authors identify limiting dynamics as cascading K-processes (ergodic) and cascading aging processes, built from stable subordinators across levels, and establish three scaling regimes with precise rescalings of volumes and time. Aging results are obtained through explicit limiting no-jump functions expressed via products of Beta variables, revealing aging behavior consistent with, yet distinct from, GLTM/GREM theories. The findings provide a mathematically rigorous framework for multi-level aging in trap models and connect to existing REM/GREM-like aging literature while clarifying how cascading structure governs dynamics across time scales.

Abstract

We take scaling limits of the Bouchaud and Dean trap model on Parisi's tree in time scales where the dynamics is either ergodic (close to equilibrium) or aging (far from equilibrium). These results follow from a continuity theorem formulated for a certain kind of process on trees, which we call a cascading jump evolution, defined in terms of a collection of jump functions, with a cascading structure given by the tree.

Paper Structure

This paper contains 27 sections, 10 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous results
  3. Relation to the Sasaki and Nemoto/GREM-like Trap Model
  4. Model and results
  5. Limiting dynamics
  6. Scaling regimes and main results
  7. Aging results
  8. Continuity theorem for cascading jump evolutions
  9. Cascading families of jump functions
  10. Cascading jump evolutions
  11. Alternative representation
  12. Alternative construction of ${\mathcal{X}}_k$.
  13. Continuity theorem
  14. Gap condition.
  15. Coupling of the exponential random variables of ${\mathcal{K}}_k$ and ${\mathcal{K}}_k^{(n)}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Under the conditions stipulated above, namely tau (with the convergent $L_j$'s) and fine tuning, we have that $\tilde{{\mathcal{Z}}}^{(n)}(\cdot)$ converges in distribution as $n\to\infty$ to $\tilde{{\mathcal{Z}}}$, the cascading K-process with index $\bar{\alpha}_k$.

Figures (1)

  • Figure 1: Descriptive illustration of $Z_1(t)$ for a given $t$, based on the timeless clock process $S_1$ and the ordinary clock process $C_1$. The crosses in the $x$-axis represent the marks of the Poisson processes $N^1_{x_1}$, $1\leq x_1\leq M_1$, and the locations of the jumps of both processes; jump sizes are represented in full lines; in this picture, $s$ corresponds to the $j$-th mark of $N^1_{x_1}$, and we may say that $s$ is labeled $x_1$. Notice that the labels of the successive jumps are iid random variables, uniform in $\{1,\ldots, M_1\}$.

Theorems & Definitions (38)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • ...and 28 more