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

@inproceedings{Janjigian2019BUSEMANNFA, title={BUSEMANN FUNCTIONS AND GIBBS MEASURES IN DIRECTED POLYMER MODELS ON Z2 BY CHRISTOPHER JANJIGIAN*}, author={Christopher Janjigian and Firas Rassoul-Agha}, year={2019} }

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 vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/nonuniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin–Lanford–Ruelle equations). We also prove nonexistence of covariant or deterministically directed…

## Figures from this paper

## 6 Citations

Busemann functions and semi-infinite O’Connell–Yor polymers

- MathematicsBernoulli
- 2020

We prove that given any fixed asymptotic velocity, the finite length O'Connell-Yor polymer has an infinite length limit satisfying the law of large numbers with this velocity. By a Markovian property…

Dynamic polymers: invariant measures and ordering by noise

- MathematicsProbability Theory and Related Fields
- 2021

We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow on polymers pinned at…

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

- Mathematics
- 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…

Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape

- Mathematics
- 2019

Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work arXiv:1812.00309 of…

Negative correlation of adjacent Busemann increments

- Mathematics
- 2021

We consider i.i.d. last-passage percolation on Z2 with weights having distribution F and time-constant gF . We provide an explicit condition on the large deviation rate function for independent sums…

Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation

- Mathematics, Computer Science
- 2022

New probabilistic arguments are introduced to derive optimal-order central moment bounds in planar directed last-passage percolation in i.i.d. exponential weights for both zero and near-stationary boundary conditions.

## References

SHOWING 1-10 OF 69 REFERENCES

Localization and Perron--Frobenius theory for directed polymers

- Mathematics
- 2009

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time.
We study two main objects based…

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…

Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

- Mathematics
- 2013

We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting…

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-…

A shape theorem and semi-infinite geodesics for the Hammersley model with random weights

- Mathematics
- 2010

In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also…

Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation

- MathematicsCommunications on Pure and Applied Mathematics
- 2018

The first goal of this paper is to prove multiple asymptotic results for a time‐discrete and space‐continuous polymer model of a random walk in a random potential. These results include: existence of…

Busemann Functions and Infinite Geodesics in Two-Dimensional First-Passage Percolation

- Mathematics
- 2012

We study first-passage percolation on $${\mathbb{Z}^2}$$Z2, where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of…

Geodesics and the competition interface for the corner growth model

- Mathematics
- 2015

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

Busemann functions and equilibrium measures in last passage percolation models

- Mathematics
- 2009

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a…

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

- Mathematics
- 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…