Lyapounov norms for random walks in low disorder and dimension greater than three. (English) Zbl 1163.60050

Summary: We consider a simple random walk on \(\mathbb Z^{d}\), \(d > 3\). We also consider a collection of i.i.d. positive and bounded random variables \((V_\omega(x))_{x\in\mathbb Z^d}\), which will serve as a random potential. We study the annealed and quenched cost to perform long crossing in the random potential \(-(\lambda+\beta V_\omega(x))\), where \(\lambda \) is positive constant and \(\beta > 0\) is small enough. These costs are measured by the Lyapounov norms. We prove the equality of the annealed and the quenched norm.


60K37 Processes in random environments
60G50 Sums of independent random variables; random walks
60J45 Probabilistic potential theory
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