A random walk in random environment has transition probabilities that are sampled randomly and then held fixed during the walk. Conditional on the environment, it
is a Markov chain. For a nearest-neighbor random
walk on 
, an environment specifies a probability
vector with components 
at every vertex 
,
for the 
coordinate unit steps 
. Its transition probabilities satisfy
 |
whenever 
.
This conditional law is the quenched law, while averaging
it over the environment gives the annealed law. Independent and identically distributed
environments mean that the complete rows at distinct vertices
are independent and identically
distributed, not that entries within a row are independent.
Strict ellipticity requires every 
to be positive almost
surely. Uniform ellipticity strengthens this to a deterministic common positive lower bound.
For a fixed nonzero direction 
, directional
transience is the event 
.
