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

Zygouras, N.

Probab. Theory Relat. Fields 143, No. 3-4, 615-642 (2009).

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.

MSC:

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

Keywords:

random walk; random potential; Lyapounov norms

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