Directed polymers on a disordered tree with a defect subtree

TL;DR

This paper investigates how localized microscopic defects influence the macroscopic behavior of directed polymers on a disordered tree, identifying phase transitions and computing free energy in different defect configurations.

Contribution

It introduces a detailed analysis of phase behavior in directed polymers with localized defects, including free energy computation and phase diagram characterization.

Findings

Identifies three phases: fully pinned, partially pinned, and depinned.

Computes free energy and critical curves for single-branch defects.

Shows the disappearance of the partially pinned phase above a critical temperature.

Abstract

We study the question of how the competition between $\textit{bulk disorder}$ and a $\textit{localized microscopic defect}$ affects the macroscopic behavior of a system in the directed polymer context at the free energy level. We consider the directed polymer model on a disordered $d$-ary tree and represent the localized microscopic defect by modifying the disorder distribution at each vertex in a single path (branch), or in a subtree, of the tree. The polymer must choose between following the microscopic defect and finding the best branches through the bulk disorder. We describe three possible phases, called the $\textit{fully pinned, partially pinned}$ and $\textit{depinned}$ phases. When the microscopic defect is associated only with a single branch, we compute the free energy and the critical curve of the model, and show that the partially pinned phase does not occur. When the localized microscopic defect is associated with a non-disordered regular subtree of the disordered tree, the picture is more complicated. We prove that all three phases are non-empty below a critical temperature, and that the partially pinned phase disappears above the critical temperature.