Dynamics and critical behavior of the q model

The q model, a random walk model rich in behavior and applications, is investigated. We introduce and motivate the q model via its application proposed by Coppersmith 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 analyze. To characterize 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 behavior analogous to thermodynamic critical phenomena. The critical behavior 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 behavior 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 an elementary derivation of it making use of operator rather than functional methods.