Potential Analysis

, Volume 33, Issue 3, pp 227–247 | Cite as

Principal Eigenvalue for the Random Walk among Random Traps on \({\mathbb{Z}}^{\bf {\it d}}\)

  • Jean-Christophe Mourrat


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


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

Mathematics Subject Classifications (2000)

60K37 82B41 47A75 


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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.CMIUniversité de ProvenceMarseilleFrance
  2. 2.Facultad de MatemáticasPUC de ChileMaculChile

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