Dynamics and critical behavior of the q model - art. no. 026107

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.