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In the first part of the article our subject of interest is a simplesymmetric random walk on the integers which faces a random risk to be killed.This risk is described by random potentials, which in turn are defined by asequence of independent and identically distributed non-negative randomvariables. To determine the risk of taking a walk in these potentials weconsider the decay of the Green function. There are two possible tools todescribe this decay: The quenched Lyapunov exponents and the annealed Lyapunovexponents. It turns out that on the integers we can state a precise relationbetween these two. The main tool for proving this is due to a special pathproperty on the integers: if the random walk travels from one point to anotherone in finite time all the points between have to be passed in finite time aswell. This allows using Ergodic Theorems. In the second part we show that therelation is also true in the case of Lyapunov exponents for simple symmetricrandom walks on d-regular trees and for random walks with drift.

The relation between quenched and annealed Lyapunov exponents in random potential on trees