[Paper Review] Principal eigenvalue for random walk among random traps on Z^d

Jean-Christophe Mourrat|ArXiv.org|May 6, 2008

Stochastic processes and statistical mechanics25 references3 citations

TL;DR

This paper studies the principal eigenvalue of a continuous-time random walk on ℤ^d with random, heavy-tailed holding times (traps) at each site. It establishes a phase transition in the asymptotic behavior of the eigenvalue depending on the tail index α and dimension d, showing that the eigenvalue decays as n^{−max(2,1+1/α)} for d=1 and n^{−max(2,d/α)} for d≥2, with sharp bounds under different regimes of α and d.

ABSTRACT

Let $(τ_x)_{x \in \Z^d}$ be i.i.d. random variables with heavy (polynomial) tails. Given $a \in [0,1]$, we consider the Markov process defined by the jump rates $ω_{x o y} = {τ_x}^{-(1-a)} {τ_y}^a$ between two neighbours $x$ and $y$ in $\Z^d$. We give the asymptotic behaviour of the principal eigenvalue of the generator of this process, with Dirichlet boundary condition. The prominent feature is a phase transition that occurs at some threshold depending on the dimension.

Motivation & Objective

• To understand the asymptotic behavior of the principal eigenvalue of the generator of a continuous-time random walk on ℤ^d with random, i.i.d. holding times (traps) at each site.
• To investigate how the tail behavior of the trap distribution (power-law with index α) and the spatial dimension d affect the spectral gap of the process.
• To identify a phase transition in the decay rate of the principal eigenvalue, depending on the interplay between α and d, particularly when α is small or large relative to d.
• To establish sharp bounds on the eigenvalue under different regimes, including cases where the eigenvalue scales as 1/n² or slower, depending on the environment's tail behavior.
• To demonstrate that the eigenvalue behavior is independent of the parameter a ∈ [0,1], which interpolates between different jump rate dynamics, despite differing heuristics.

Proposed method

• Define a continuous-time Markov process on ℤ^d with jump rates ω_{x→y} = τ_x^{-(1−a)} τ_y^a between nearest neighbors, where τ_x are i.i.d. heavy-tailed random variables.
• Study the principal eigenvalue λ_n of the generator −ℒ_n restricted to a box B_n = {−n,…,n}^d with Dirichlet boundary conditions.
• Use the generalized inverse h(x) = inf{y : 1/F(y) ≥ x} of the tail distribution F(y) = ℙ(τ_0 > y), which satisfies h ∈ RV_{1/α} under regular variation assumption.
• Apply variational principles and spectral bounds via test functions (e.g., sine functions) to derive upper bounds on λ_n, using energy and L² norms.
• Use the distinguished path method with weighted paths to derive lower bounds on λ_n, showing limitations in high dimensions when α < d.
• Analyze the asymptotic behavior of maxima and sums of τ_x using extreme value theory and regular variation, particularly relating h(n^d) to the typical scale of the largest trap in B_n.

Experimental results

Research questions

• RQ1 How does the principal eigenvalue λ_n of the generator of a random walk among random traps on ℤ^d decay as the box size n → ∞?
• RQ2 What is the dependence of the decay rate of λ_n on the tail index α of the i.i.d. trap distribution and the spatial dimension d?
• RQ3 Does the parameter a ∈ [0,1], which controls the jump rate dynamics, affect the asymptotic behavior of λ_n?
• RQ4 What is the critical threshold for α that triggers a phase transition in the decay rate of λ_n, and how does it depend on d?
• RQ5 Can the distinguished path method yield sharp lower bounds on λ_n, and if not, why does it fail in high dimensions when α < d?

Key findings

• For d = 1, the principal eigenvalue satisfies lim_{n→∞} −ln(λ_n)/ln(n) = max(2, 1 + 1/α) almost surely.
• For d ≥ 2, the decay rate is lim_{n→∞} −ln(λ_n)/ln(n) = max(2, d/α) almost surely.
• When d ≥ 2 and α > d/2, or d = 1 and α > 1, the eigenvalue decays as Θ(1/n²), with constants k₁, k₂ > 0 such that k₁/n² ≤ λ_n ≤ k₂/n² for large n.
• When α < 1 and d ≠ 2, the eigenvalue scales as λ_n ≍ 1/(n h(n)) for d = 1 and λ_n ≍ 1/h(n^d) for d ≥ 3, where h(x) ∼ x^{1/α} is the generalized inverse of the tail distribution.
• For d = 2 and α < 1, the eigenvalue satisfies λ_n^∘ ≍ 1/(ln(n) h(n²)) under the environment law, showing a logarithmic correction.
• The eigenvalue behavior is independent of a ∈ [0,1], indicating that the phase transition depends only on d and α, not on the jump rate parameter a.