Abstract
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.
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Mourrat, JC. Principal Eigenvalue for the Random Walk among Random Traps on \({\mathbb{Z}}^{\bf {\it d}}\) . Potential Anal 33, 227–247 (2010). https://doi.org/10.1007/s11118-009-9167-z
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DOI: https://doi.org/10.1007/s11118-009-9167-z