Principal eigenvalue for random walk among random traps on Z^d

Let $({\tau }_x{)}_{x\in {\mathbb{Z}}^d}$ 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 ${\omega }_{x\to y}={{\tau }_x}^{-(1-a)}{{\tau }_y}^a$ 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.