Suppose that the integers are assigned the random variables {ω x , µ x } (taking values in the unit interval times the space of probability measures on R + ), which serve as an environment. This environment defines a random walk {X t } (called a RWREH) which, when at x, waits a random time distributed according to µ x and then, after one unit of time, moves one step to the right with probability ω x , and one step to the left with probability 1 − ω x . We prove large deviation principles for X

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... /t, both quenched (i.e., conditional upon the environment), with deterministic rate function, and annealed (i.e., averaged over the environment). As an application, we show that for random walks on Galton-Watson trees, quenched and annealed rate functions along a ray differ.