Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice

Understand

We study the recently introduced Inverse-Beta polymer, an exactly solvable, anisotropic finite temperature model of directed polymer on the square lattice, and obtain its stationary measure.

• In parallel we introduce an anisotropic zero temperature model of directed polymer on the square lattice, the Bernoulli-Geometric polymer, and obtain its stationary measure.
• This new exactly solvable model is dual to the Inverse-Beta polymer and interpolates between models of first and last passage percolation on the square lattice.
• Both stationary measures are shown to satisfy detailed balance.