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

Janjigian, Christopher; Rassoul-Agha, Firas

doi:10.1214/19-aop1375

Cited by 23 publications

(72 citation statements)

References 53 publications

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“…This includes various last-passage models in both discrete and continuous space, such as those studied in [2,19,24,30,33,34], and the four currently known solvable polymer models [10]. In positivetemperature polymer models, the analogous question concerns the existence of bi-infinite Gibbs measures, as discussed in [21]. These matters are left for future work.…”

Section: Related Workmentioning

confidence: 99%

Non-existence of bi-infinite geodesics in the exponential corner growth model

Balázs

Busani

Seppäläinen³

2020

Forum of Mathematics, Sigma

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Section: Related Workmentioning

confidence: 99%

Non-existence of bi-infinite geodesics in the exponential corner growth model

Balázs

Busani

Seppäläinen³

2020

Forum of Mathematics, Sigma

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“…Interestingly, a measure similar to µ was studied in [43] in the context of planar LPP and positive-temperature directed polymers, which are fully discrete models. That measure is also related via its support to exceptional disjointness of semi-infinite geodesics [44], and Theorem 1.10(a) bears a striking resemblance to [44,Thm.…”

Section: Main Results: Hausdorff Dimensions and Measure Description Of Exceptional Setsmentioning

confidence: 99%

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

Bates

Ganguly

Hammond

2022

Electron. J. Probab.

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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 [27] of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points (x, s) and (y, t) with s < t. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations x1 and x2, and consider geodesics traveling (x1, 0) → (y, 1) and (x2, 0) → (y, 1). We prove that the set of y ∈ R for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints (x, 0) and (y, 1) between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such (x, y) ∈ R 2 also has Hausdorff dimension one-half.The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in [10]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [40].

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“…The existence results of Theorem 2.5 can also be cast in a statistical mechanics framework, where the semi-infinite quenched path measures we construct via Busemann functions correspond to semi-infinite Gibbs measures that are consistent with the finite path point-to-point polymer measures Q B (m,s),(n,t) . See [18] for details in the polymer model on Z 2 .…”

Section: )mentioning

confidence: 99%

“…Under the polymer measure sets with longer length have larger probability, while the last passage model picks the longest path. With this definition we get the following result for limits of differences of passage times: Ideas from [6,7,18] can be used to show that weak convergence holds at the level of Busemann functions, namely that y, B).…”

Section: Limits Of Brownian Lppmentioning

confidence: 99%

“…The article [15] used this method for the directed log-gamma polymer [25], where the stationary model is also explicit. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Cited by 23 publications

References 53 publications

Non-existence of bi-infinite geodesics in the exponential corner growth model

Non-existence of bi-infinite geodesics in the exponential corner growth model

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

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

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