Lyapounov norms for random walks in low disorder and dimension greater than three

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.