Abstract
We consider a simple random walk on Z d, d > 3. We also consider a collection of i.i.d. positive and bounded random variables \(\left(V_\omega(x)\right)_{x\in Z^d}\) , 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 λ is positive constant and β > 0 is small enough. These costs are measured by the Lyapounov norms. We prove the equality of the annealed and the quenched norm.
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Zygouras, N. Lyapounov norms for random walks in low disorder and dimension greater than three. Probab. Theory Relat. Fields 143, 615–642 (2009). https://doi.org/10.1007/s00440-008-0139-9
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DOI: https://doi.org/10.1007/s00440-008-0139-9