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

Janjigian, Christopher; Rassoul-Agha, Firas

doi:10.1007/s10955-020-02541-z

Cited by 12 publications

(12 citation statements)

References 26 publications

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“…Through this connection one can use these correctors to construct infinite path length limits (Gibbs measures and infinite geodesics) and study their properties. See the recent papers: [1], [6], [7], [9], [12], [13], [14], [16], [20], [21], [22]. As mentioned above, in Theorem 4.8 of [20] it was noticed that variational problems of the type we produce here can be used to resolve a key technical obstruction in the construction of the cocycles which are needed in order to build these infinite volume objects.…”

Section: Previous Workmentioning

confidence: 90%

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

Janjigian¹,

Sergazy²,

Rassoul-Agha³

2020

Preprint

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We prove a shape theorem and derive a variational formula for the limiting quenched Lyapunov exponent and the Green's function of random walk in a random potential on a square lattice of arbitrary dimension and with an arbitrary finite set of steps. The potential is a function of a stationary environment and the step of the walk. This potential is subject to a moment assumption whose strictness is tied to the mixing of the environment. Our setting includes directed and undirected polymers, random walk in static and dynamic random environment, and, when the temperature is taken to zero, our results also give a shape theorem and a variational formula for the time constant of both site and edge directed last-passage percolation and standard first-passage percolation.

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Section: Previous Workmentioning

confidence: 90%

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

Janjigian¹,

Sergazy²,

Rassoul-Agha³

2020

Preprint

Self Cite

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“…We do not need ergodicity in the present project and so do not prove it here. These questions are addressed in our companion paper [40].…”

Section: 1mentioning

confidence: 96%

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

Janjigian,

Rassoul-Agha

2018

Preprint

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We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planer 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/non-uniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove non-existence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function limits in directions where the limiting free energy has some regularity.

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“…(Ergodicity of a polymer model related to the discrete Toda lattice, cf. Subsection 6.2, is studied in [19].) Problem 8.4.…”

Section: Iterated Random Functionsmentioning

confidence: 99%

Detailed balance and invariant measures for discrete KdV- and Toda-type systems

Croydon¹,

Sasada²

2020

Preprint

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This article focusses on systems, discretely indexed in space and time, whose dynamics are deterministic and defined locally via lattice equations. A detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. A condition for establishing ergodicity of the dynamics is also given. These results are applied to various examples of discrete integrable systems, namely the ultra-discrete and discrete KdV equations, for which it is shown that the relevant invariant measures are of exponential/geometric and generalized inverse Gaussian form, respectively, as well as the ultradiscrete and discrete Toda lattice equations, for which the relevant invariant measures are found to be of exponential/geometric and gamma form. Ergodicity is demonstrated in the case of the KdV-type models. Links between the invariant measures of the different systems are presented, as are connections with stochastic integrable models and iterated random functions. Furthermore, a number of conjectures concerning the characterization of standard distributions are posed.

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

Cited by 12 publications

References 26 publications

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

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

Detailed balance and invariant measures for discrete KdV- and Toda-type systems

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