Directed polymers in high dimensions

TL;DR

This paper investigates directed polymers in high dimensions, revealing that the critical dimension is d=4, where physical quantities exhibit singular behavior, indicating a special role for this dimension in the system's phase transition.

Contribution

The study identifies d=4 as the upper critical dimension for directed polymers and the KPZ equation, providing a detailed perturbation analysis of singular behaviors near this dimension.

Findings

Finite size amplitude of free energy scales as (4-d)^(1/2)

Dimension d=4 is the upper critical dimension for the system

Singular behavior emerges as d approaches 4

Abstract

We study directed polymers subject to a quenched random potential in d transversal dimensions. This system is closely related to the Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful analysis of the perturbation theory we show that physical quantities develop singular behavior for d to 4. For example, the universal finite size amplitude of the free energy at the roughening transition is proportional to (4-d)^(1/2). This shows that the dimension d=4 plays a special role for this system and points towards d=4 as the upper critical dimension of the Kardar-Parisi-Zhang problem.