The question of how the competition between the bulk disorder and a localized microscopic defect affects the macroscopic behavior of a system as reflected in pinning phenomena, arising for example in the context of high-temperature superconductors, is an interesting and important problem in statistical physics. In this work, we study this problem in the context of directed polymers on disordered trees.

Directed polymers in a random environment are typical examples of models used to study the behavior of a one-dimensional object interacting with a disordered environment. In the mathematical formulation of these models, paths of a directed walk on a regular lattice or tree represent the directed polymer while an i.i.d. collection of random variables attached to each vertex of the lattice/tree correspond to the disordered environment. Each path is assigned a Gibbs weight as the sum of the random variables of the visited vertices. The polymer's interaction with the disordered environment is controlled by a parameter, $\beta$, which represents the inverse temperature.

We consider the directed polymer on a disordered tree model by adding a fixed potential $u$ to each vertex on the left-most branch of the tree which represents the localized microscopic defect. The polymer must choose between roughly following the localized microscopic defect, or finding the best path(s) through the bulk disorder. As $u$ is increased, one expects a pinning transition at some critical $u_c$ where the polymer begins to follow the localized microscopic defect.

We compute the free energy of the model and determine the critical point $u_c$.

This is a joint work with Neal Madras.