[Paper Review] Scaling limits and aging for asymmetric trap models on the complete graph and K-processes

S. C. Bezerra, Luiz Renato Fontes+3 more|arXiv (Cornell University)|Feb 18, 2012

Stochastic processes and statistical mechanics24 references6 citations

TL;DR

This paper establishes scaling limits for asymmetric trap models on the complete graph by introducing a novel representation using trap depth dynamics, leading to the definition of the asymmetric K process. It derives aging behavior through self-similar scaling limits, showing that the limiting process exhibits full aging via self-similarity and convergence of correlation functions.

ABSTRACT

We obtain scaling limit results for asymmetric trap models and their infinite volume counterparts, namely asymmetric K processes. Aging results for the latter processes are derived therefrom.

Motivation & Objective

• To develop a unified framework for scaling limits of asymmetric trap models on the complete graph using trap depth dynamics instead of site location.
• To introduce the asymmetric K process as the infinite-volume limit of the trap depth process, generalizing the symmetric K process from prior work.
• To derive aging results for the asymmetric K process by analyzing its self-similar scaling limit at vanishing times.
• To demonstrate that the limiting process captures full aging behavior not only through correlation functions but through the dynamics itself.
• To show that the trap depth process Z_n enables analysis of correlation functions requiring location information, overcoming limitations of clock process-based approaches.

Proposed method

• Represent the asymmetric trap model via the process $ Z_n(t) = \tau_{Y_n(t)}^{1-a} $, tracking the depth of the currently visited trap rather than the site location.
• Establish a continuity property (Lemma 2.1) linking the limiting behavior of subordinators to the limiting process, enabling convergence results for $ Z_n $.
• Use two related subordinators: one for the cumulative time and one for the jump times, with the second defined as the integral of i.i.d. exponential variables with respect to the first.
• Apply time-rescaling via $ \varepsilon \to 0 $ to derive the self-similar limit process $ \hat{Z} $, which captures aging at vanishing time scales.
• Analyze convergence of point processes $ \hat{\cal P}^{(\varepsilon)} $ and $ \hat{\Gamma}^{(\varepsilon)} $ to their limits $ \hat{\cal P} $ and $ \hat{\Gamma} $, ensuring distributional convergence.
• Use the convergence of $ \hat{Z}^{(\varepsilon)} $ to $ \hat{Z} $ and continuity of functionals to derive limits of correlation functions such as $ \hat{\Pi}(t,s) $, $ \hat{R}(t,s) $, and $ \hat{Q}(t,s) $.

Experimental results

Research questions

• RQ1 How can scaling limits for asymmetric trap models on the complete graph be derived in a unified way across different time regimes?
• RQ2 What is the infinite-volume limit process corresponding to the asymmetric trap model, and how does it generalize the symmetric K process?
• RQ3 Does the limiting process exhibit full aging behavior, and if so, how can this be characterized beyond two-time correlation functions?
• RQ4 Can correlation functions requiring location information—beyond jump times—be analyzed using the trap depth process $ Z_n $, and how does this improve upon clock process methods?
• RQ5 What is the role of self-similarity in characterizing the aging behavior of the limiting process $ \hat{Z} $, and how does it relate to the dynamics of the system?

Key findings

• The asymmetric K process $ Z $ is identified as the scaling limit of the trap depth process $ Z_n $ at times of the order of the deepest trap, generalizing the symmetric K process.
• The limit process $ \hat{Z} $ obtained at vanishing times is self-similar with index 1, indicating a full aging behavior that extends beyond correlation functions to the dynamics itself.
• Convergence of $ \hat{Z}^{(\varepsilon)} $ to $ \hat{Z} $ implies that aging correlation functions such as $ \hat{\Pi}(t,s) $, $ \hat{R}(t,s) $, and $ \hat{Q}(t,s) $ converge in distribution, with $ \hat{Q}(t,s) $ capturing the probability of novelty in the system.
• The convergence of point processes $ \hat{\cal P}^{(\varepsilon)} $ and $ \hat{\Gamma}^{(\varepsilon)} $ to their limits ensures that the limiting dynamics is well-defined and stable under time rescaling.
• The method enables analysis of correlation functions that depend on both jump times and locations, which the clock process alone cannot capture, thus extending the scope of applicability.
• The self-similarity of $ \hat{Z} $ implies that $ \hat{Q}(t,s) $ depends only on the ratio $ t/s $, although no explicit expression is available for this functional.