An Identity in Distribution Between Full-Space and Half-Space Log-Gamma Polymers

Abstract

We prove an identity in distribution between two kinds of partition functions for the log-gamma directed polymer model: (1) the point-to-point partition function in a quadrant and (2) the point-to-line partition function in an octant. As an application, we prove that the point-to-line free energy of the log-gamma polymer in an octant obeys a phase transition depending on the strength of the noise along the boundary. This transition of (de)pinning by randomness was first predicted in physics by Kardar in 1985 and proved rigorously for zero temperature models by Baik and Rains in 2001. While it is expected to arise universally for models in the Kardar–Parisi–Zhang universality class, this is the first positive temperature model for which this transition can be rigorously established.

Boundary current fluctuations for the half‐space ASEP and six‐vertex model

“…Beyond just the phase transition studied in this paper, there has been tremendous progress in understanding half‐space models at positive temperature and their algebraic structure, see, for example, [9, 12, 13, 22, 32, 43, 44]. For some additional recent works on half‐space models, see [6, 16, 19, 33, 34, 47], and also relevant work in the physics literature [10, 11, 17].…”