[Paper Review] Trap models with vanishing drift: Scaling limits and aging regimes

Nina Gantert, Peter Moerters+1 more|arXiv (Cornell University)|Mar 7, 2010

Stochastic processes and statistical mechanics11 references4 citations

TL;DR

This paper studies trap models on the integer lattice with asymptotically vanishing drift, establishing scaling limits and ageing behavior across three distinct regimes based on drift decay speed. It identifies a novel limit process—the Fontes-Isopi-Newman diffusion with drift—in a critical intermediate regime, and shows sublinear ageing (γ < 1) in all vanishing drift cases, contrasting with the γ = 1 case under constant drift.

ABSTRACT

We discuss the long term behaviour of trap models on the integers with asymptotically vanishing drift, providing scaling limit theorems and ageing results. Depending on the tail behaviour of the traps and the strength of the drift, we identify three different regimes, one of which features a previously unobserved limit process.

Motivation & Objective

• To understand the long-term behavior of continuous-time random walks on Z with heavy-tailed trap environments and vanishing drift.
• To classify the scaling limits of such models based on the rate at which the drift vanishes.
• To characterize ageing behavior via the exponent γ in the probability that the process remains motionless over time intervals of order t^γ.
• To identify a new limit process—the Fontes-Isopi-Newman diffusion with drift—in a critical regime not previously observed.
• To establish that ageing is sublinear (γ < 1) whenever the drift vanishes, in contrast to the constant drift case where γ = 1.

Proposed method

• The model is defined as a continuous-time nearest-neighbor random walk on Z, with holding times at site i proportional to i.i.d. heavy-tailed trap variables τ_i with tail index α ∈ (0,1).
• The drift is introduced via asymmetric jump rates p^N = 1/2(1 + μ/N^β), with μ ≥ 0 and β ≥ 0, leading to asymptotically vanishing drift as N → ∞.
• Scaling limits are derived by rescaling time via X^N_t = X_{Nt}, and analyzing the rescaled process in three regimes based on β: slow (β < 1), fast (β > 1), and critical (β = 1).
• Convergence to limit processes is established using point process convergence of the rescaled trap environment and uniform convergence of scale functions.
• The ageing exponent γ is derived from the asymptotic depth of the trap in which the particle is located at time t, using the distribution of the rescaled trap size τ_{X^N_t}/N^{1/(α+1)}.
• Lemmas on point process convergence and weak convergence of the process position are used to prove convergence of the rescaled trap size to the limit law.

Experimental results

Research questions

• RQ1 What are the scaling limits of trap models on Z with asymptotically vanishing drift, depending on the decay rate of the drift?
• RQ2 Does a new limit process emerge in the critical regime where the drift vanishes at a specific rate?
• RQ3 How does the ageing behavior—measured by the exponent γ in P(X_t = X_{t+s} for all 0 ≤ s ≤ t^γ)—differ between vanishing and constant drift?
• RQ4 Is the ageing exponent γ < 1 in all cases of vanishing drift, and does this contrast with the γ = 1 case under constant drift?
• RQ5 Can the asymptotic depth of the trap in which the particle is located be used to characterize the ageing exponent γ?

Key findings

• In the regime of slowly vanishing drift (β < 1), the rescaled trap model converges in law to the inverse of a stable subordinator, consistent with results for constant drift.
• In the regime of rapidly vanishing drift (β > 1), the rescaled process converges to the Fontes-Isopi-Newman diffusion, the same limit as in the driftless case.
• In the critical intermediate regime (β = 1), the rescaled process converges to a novel limit process: the Fontes-Isopi-Newman diffusion with drift, which has not been identified as a scaling limit before.
• The ageing exponent γ is strictly less than 1 in all cases of vanishing drift, indicating sublinear ageing, whereas γ = 1 under constant drift.
• The asymptotic distribution of the trap depth τ_{X^N_t}/N^{1/(α+1)} converges in law to the random variable ρ(Fin^θ_t), which determines the ageing exponent γ.
• The ageing probability P(X_t = X_{t+s} for all 0 ≤ s ≤ t^γ) converges to a value in (0,1) as t → ∞, confirming non-trivial ageing behavior in all vanishing drift regimes.