  The $q$-model, a random walk model rich in behaviour and applications, is
investigated. We introduce and motivate the $q$-model via its application
proposed by Coppersmith {\em et al.} to the flow of stress through granular
matter at rest. For a special value of its parameters the $q$-model has a
critical point that we analyse. To characterise the critical point we imagine
that a uniform 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 behaviour analogous to thermodynamic critical phenomena.
The critical behaviour is remarkably tractable: the harvest of analytic results
includes scaling functions that describe the evolution of the variance of 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 results are
of intrinsic interest as a tractable example of a random critical point. Of the
many applications of the q-model, the critical behaviour is particularly
relevant to network models of river basins, as we briefly discuss. Finally we
discuss circumstances under which quantum network models that describe the
surface electronic states of a quantum Hall multilayer can be mapped onto the
classical $q$-model. For mesoscopic multilayers of finite circumference the
mapping fails; instead a mapping to a ferromagnetic supersymmetric spin chain
has proved fruitful. We discuss aspects of the superspin mapping and give a new
elementary derivation of it making use of operator rather than functional
methods.
