# Uniqueness and Ergodicity of Stationary Directed Polymers on $$\mathbb {Z}^2$$

@article{Janjigian2018UniquenessAE, title={Uniqueness and Ergodicity of Stationary Directed Polymers on \$\$\mathbb \{Z\}^2\$\$}, author={Christopher Janjigian and Firas Rassoul-Agha}, journal={arXiv: Probability}, year={2018} }

We study the ergodic theory of stationary directed nearest neighbor polymer models on $\mathbb Z^2$, with i.i.d. weights. Such models are equivalent to specifying a stationary distribution on the space of weights and cocycles that satisfy certain consistency conditions. We show that for prescribed weight distribution and cocycle mean vector, there is at most one such distribution which is ergodic under the $e_1$ or $e_2$ shift. Further, if the weights have more than two moments and the cocycle…

## 4 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES

### Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^{2}$

- Mathematics
- 2018

We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the…

### BUSEMANN FUNCTIONS AND GIBBS MEASURES IN DIRECTED POLYMER MODELS ON Z2 BY CHRISTOPHER JANJIGIAN*

- Mathematics
- 2019

### Stationary cocycles and Busemann functions for the corner growth model

- Mathematics
- 2015

We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable…

### Ratios of partition functions for the log-gamma polymer

- Mathematics
- 2015

We introduce a random walk in random environment associated to an underlying directed polymer model in 1 + 1 dimensions. This walk is the positive temperature counterpart of the competition in-…

### The Strict-Weak Lattice Polymer

- Mathematics
- 2014

We introduce the strict-weak polymer model, and show the KPZ universality of the free energy fluctuation of this model for a certain range of parameters. Our proof relies on the observation that the…

### Scaling for a one-dimensional directed polymer with boundary conditions

- Mathematics
- 2009

We study a (1+1)-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights. Among directed polymers, this model is special in the same way as the…

### The Kardar-Parisi-Zhang equation and universality class

- Mathematics
- 2011

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or…

### Quenched point-to-point free energy for random walks in random potentials

- Mathematics
- 2012

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and steps of the walk. The potential is subject to a…

### Random-walk in Beta-distributed random environment

- Mathematics
- 2015

We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in $$\mathbb {Z}$$Z which performs nearest neighbour jumps with…

### Characterizing stationary 1+1 dimensional lattice polymer models

- Mathematics
- 2017

Motivated by the study of directed polymer models with random weights on the square integer lattice, we define an integrability property shared by the log-gamma, strict-weak, beta, and inverse-beta…