Let $(\tau_x)_{ x \in \Z^d}$ 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 $\omega_{ x \to y} = { \tau_x}^{ -(1-a)} { \tau_y}^a$ 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.
Categories: math.PR