Busemann functions and equilibrium measures in last passage percolation models

Cator, Eric; Pimentel, Leandro P. R.

doi:10.1007/s00440-011-0363-6

Cited by 32 publications

(60 citation statements)

References 16 publications

(37 reference statements)

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“…Tightly connected to the thermodynamic limit results in [GRASY15] are results on the limits of ratios of partition functions. Logarithms of these limiting ratios are polymer counterparts of Busemann functions that compare actions of infinite geodesics to each other in zero temperature models such as first passage percolation (FPP), last passage percolation, or zero-viscosity Burgers equation, see [HN01], [CP12], [BCK14], [Bak16], [GRAS16], [GRS15], [DH14], [DH17] and [AHD15], which is a recent survey on FPP. In [GRAS16] and [RSY16] a variational approach to ratios of partition functions is described.…”

Section: Directed Polymersmentioning

confidence: 99%

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

Bakhtin

2018

Comm Pure Appl Math

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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 program is almost sure existence and uniqueness of polymer measures on one‐sided infinite paths with given endpoint and slope, and interpretation of these infinite‐volume Gibbs measures as thermodynamic limits. Moreover, we prove that marginals of polymer measures with the same slope and different endpoints are asymptotic to each other. The second goal of the paper is to develop ergodic theory of the Burgers equation with positive viscosity and random kick forcing on the real line without any compactness assumptions. Namely, we prove a one force–one solution principle, using the infinite‐volume polymer measures to construct a family of stationary global solutions for this system, and proving that each of those solutions is a one‐point pullback attractor on the initial conditions with the same spatial average. This provides a natural extension of the same program realized for the inviscid Burgers equation with the help of action minimizers that can be viewed as zero temperature limits of polymer measures. © 2018 Wiley Periodicals, Inc.

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Section: Directed Polymersmentioning

confidence: 99%

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

Bakhtin

2018

Comm Pure Appl Math

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“…See also [10,11,13,14,18], which use this method in discrete-time discrete-space models outside the exactly solvable class. The works [3,4,[7][8][9] also prove existence of Busemann functions in various related models, but using a different method based on path-straightness estimates, pioneered by Newman and coauthors [17,20]. In Section 4 we show how the results of Section 3 can be extended to show the existence of the infinite length Brownian last passage percolation model.…”

Section: Introductionmentioning

confidence: 99%

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

Alberts¹,

Rassoul-Agha²,

Simper³

2020

Bernoulli

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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 of the quenched polymer this reduces to showing the existence of Busemann functions: almost sure limits of ratios of random point-to-point partition functions. The key ingredients are the Burke property of the O'Connell-Yor polymer and a comparison lemma for the ratios of partition functions. We also show the existence of infinite length limits in the Brownian last passage percolation model.

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“…The problem of the KPZ phenomenon and universality has been one of the most active directions in statistical physics in the last decade, see [1], [2], [4], [11], [15], [16], [17], [18], [19], [20], [22], [23], [25], [26], [29], [30], [31], [36], [37], [38], [44], [46], [51], [52], [53], [54], [55], [57], and multiple other contributions. A fascinating feature of the problem is a combination of two factors: exact solvability and universality.…”

Section: Introductionmentioning

confidence: 99%

On global solutions of the random Hamilton–Jacobi equations and the KPZ problem

Bakhtin

Khanin

2018

Nonlinearity

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In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton-Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension 1+1. In the case of general viscous Hamilton-Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.

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Busemann functions and equilibrium measures in last passage percolation models

Cited by 32 publications

References 16 publications

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

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

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

On global solutions of the random Hamilton–Jacobi equations and the KPZ problem

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