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

@article{Bakhtin2018ThermodynamicLF,
  title={Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation},
  author={Yuri Bakhtin and Liying Li},
  journal={Communications on Pure and Applied Mathematics},
  year={2018},
  volume={72}
}
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 deterministic free energy density in the infinite‐volume limit for every fixed asymptotic slope, concentration inequalities for free energy implying a bound on its fluctuation exponent, and straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this… 
Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation
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 and Gibbs measures in directed polymer models on $\mathbb{Z}^{2}$
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*
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
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
Non-existence of bi-infinite polymers
We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise
Non-existence of bi-infinite polymer Gibbs measures.
We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise
Busemann functions and semi-infinite O’Connell–Yor polymers
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
On end-point distribution for directed polymers and related problems for randomly forced Burgers equation
In this paper, we study several problems related to the theory of randomly forced Burgers equation. Our numerical analysis indicates that despite the localization effects the quenched variance of the
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
Geometry of geodesics through Busemann measures in directed last-passage percolation
We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and study the geometry of the full set of semi-infinite geodesics in a typical realization of
...
...

References

SHOWING 1-10 OF 106 REFERENCES
Limiting Results for the Free Energy of Directed Polymers in Random Environment with Unbounded Jumps
We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first
Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models
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
Viscosity Limit of Stationary Distributions for the Random Forced Burgers Equation
We prove convergence of stationary distributions for the randomly forced Burgers and Hamilton–Jacobi equations in the limit when viscosity tends to zero. It turns out that for all values of the
Ratios of partition functions for the log-gamma polymer
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-
Brownian Directed Polymers in Random Environment
We study the thermodynamics of a continuous model of directed polymers in random environment. The environment is given by a space-time Poisson point process, whereas the polymer is defined in terms
Directed polymers in a random environment: path localization and strong disorder
We consider directed polymers in random environment. Under some mild assumptions on the environment, we prove here: (i) equivalence between the decay rate of the partition function and some natural
Localization Transition for Polymers in Poissonian Medium
We study a model of directed polymers in random environment in dimension 1 + d, given by a Brownian motion in a Poissonian potential. We study the effect of the density and the strength of
Quenched point-to-point free energy for random walks in random potentials
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
On the partition function of a directed polymer in a Gaussian random environment
TLDR
In the low-dimensional cases, it is proved that for all, the renormalized partition function converges to 0 and the correlation of two independent configurations does not converge to 0.
The Burgers equation with a random force and a general model for directed polymers in random environments
Summary. The study of the Burgers equation with a random force leads via a Hopf-Cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential. The
...
...