The q model, a random walk model rich in behavior and applications, is inve
stigated. We introduce and motivate the q model via its application propose
d by Coppersmith et al. to the flow of stress through granular matter at re
st. For a special value of its parameters the q model has a critical point
that we analyze. To characterize the critical point we imagine that a unifo
rm load has been applied to the top of the granular medium and we study the
evolution with depth of fluctuations in the distribution of load. Close to
the critical point explicit calculation reveals that the evolution of load
exhibits scaling behavior analogous to thermodynamic critical phenomena. T
he critical behavior is remarkably tractable: the harvest of analytic resul
ts includes scaling functions that describe the evolution of the variance o
f the load distribution close to the critical point and of the entire load
distribution right at the critical point, values of the associated critical
exponents, and determination of the upper critical dimension. These result
s are of intrinsic interest as a tractable example of a random critical poi
nt. Of the many applications of the q model, the critical behavior is parti
cularly relevant to network models of river basins, as we briefly discuss.
Finally we discuss circumstances under which quantum network models that de
scribe the surface electronic states of a quantum Hall multilayer can be ma
pped onto the classical q model. For mesoscopic multilayers of finite circu
mference the mapping fails; instead a mapping to a ferromagnetic supersymme
tric spin chain has proved fruitful. We discuss aspects of the superspin ma
pping and give an elementary derivation of it making use of operator rather
than functional methods.