# Localization and Perron--Frobenius theory for directed polymers

@article{Bakhtin2009LocalizationAP, title={Localization and Perron--Frobenius theory for directed polymers}, author={Yuri Bakhtin and Konstantin Khanin}, journal={arXiv: Probability}, year={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 on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman--Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback…

## 12 Citations

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

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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 $\mathbb{Z}^{2}$

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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…

Dynamic polymers: invariant measures and ordering by noise

- MathematicsProbability Theory and Related Fields
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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…

PR ] 9 O ct 2 01 9 BUSEMANN FUNCTIONS AND GIBBS MEASURES IN DIRECTED POLYMER MODELS ON Z

- Mathematics
- 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…

Localization of directed polymers in continuous space

- MathematicsElectronic Journal of Probability
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The first main goal of this article is to give a new metrization of the Mukherjee--Varadhan topology, recently introduced as a translation-invariant compactification of the space of probability…

Random polymers on the complete graph

- MathematicsBernoulli
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Consider directed polymers in a random environment on the complete graph of size $N$. This model can be formulated as a product of i.i.d. $N\times N$ random matrices and its large time asymptotics is…

Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation

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We consider a space-continuous and time-discrete polymer model for positive temperature and the associated zero temperature model of last passage percolation type. In our previous work, we…

Busemann functions, geodesics, and the
competition interface for directed last-passage
percolation

- MathematicsProceedings of Symposia in Applied Mathematics
- 2018

In this survey article we consider 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…

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…

Geodesics and the competition interface for the corner growth model

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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…

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USA E-mail address: bakhtin@math.gatech

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