Principal Eigenvalue for the Random Walk among Random Traps on ℤd

Jean Christophe Mourrat

doi:10.1007/s11118-009-9167-z

Principal Eigenvalue for the Random Walk among Random Traps on ℤ^d

Jean Christophe Mourrat

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let (τ)_{x∈ ℤ} ^d be i. i. d. random variables with heavy (polynomial) tails. Given a ∈ [0,1], we consider the Markov process defined by the jump rates ω_x→y = τ_x ^-(1-a)τ_y ^a between two neighbours x and y in ℤ. 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.

Original language	English (US)
Pages (from-to)	227-247
Number of pages	21
Journal	Potential Analysis
Volume	33
Issue number	3
DOIs	https://doi.org/10.1007/s11118-009-9167-z
State	Published - 2010

Keywords

Distinguished path method
Phase transition
Random walk in random environment
Spectrum
Trap model

ASJC Scopus subject areas

Analysis

Access to Document

10.1007/s11118-009-9167-z

Cite this

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abstract = "Let (τ)x∈ ℤ d be i. i. d. random variables with heavy (polynomial) tails. Given a ∈ [0,1], we consider the Markov process defined by the jump rates ωx→y = τx -(1-a)τy a between two neighbours x and y in ℤ. 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.",

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AB - Let (τ)x∈ ℤ d be i. i. d. random variables with heavy (polynomial) tails. Given a ∈ [0,1], we consider the Markov process defined by the jump rates ωx→y = τx -(1-a)τy a between two neighbours x and y in ℤ. 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.

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KW - Random walk in random environment

KW - Spectrum

KW - Trap model

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