The directed landscape

Dauvergne, Duncan; Ortmann, Janosch; Virág, Bálint

doi:10.48550/arxiv.1812.00309

Cited by 72 publications

(226 citation statements)

References 39 publications

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“…A path on which the total weight is attained is called a geodesic from x to y. Lattice LPP belongs to a large family of LPP models which seem to share the same limiting behaviour of the fluctuations of their observables. Since the seminal work of Rost [36] where the first order of L(0, y) was established for y large in exponential LPP, much progress has been made in the study of these models [1,13,26,28,35].…”

Section: Introductionmentioning

confidence: 99%

“…The sup functional in (1.1) takes two elements, L and h ρ . In [13,14] it was shown that under the KPZ scaling, for various LPP models, L has a limiting object L called the Airy Sheet. If the process h has a diffusive scaling limit G, and if the sup functional is continuous with respect to the topologies in which the convergences take place, then one would hope that the limit of H N will be given by…”

Section: Introductionmentioning

confidence: 99%

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Diffusive scaling limit of the Busemann process in Last Passage Percolation

Busani¹

2021

Preprint

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In exponential last passage percolation, we consider the rescaled Busemann process, as a process parametrized by the scaled density ρ = 1/2+ µ 4 N −1/2 , and taking values in C(R). We show that these processes, as N → ∞, have a càdlàg scaling limit G = (G µ ) µ∈R , parametrized by µ and taking values in C(R). The limiting process G, which can be thought of as the perturbation of the stationary initial condition under the KPZ scaling, can be described as an ensemble of "sticky" lines. Our proof provides some insight into this limiting behaviour by highlighting a connection between the joint distribution of Busemann functions obtained by Fan and Seppäläinen in [16], and a sorting algorithm of random walks introduced by O'Connell and Yor in [31].

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Diffusive scaling limit of the Busemann process in Last Passage Percolation

Busani¹

2021

Preprint

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“…The terminology comes from the fact that last passage percolation can essentially be thought of as a metric on the plane. When the environment is a collection of independent two-sided Brownian motions B = {B i ∶ i ∈ Z}, the Brownian last passage percolation (x, m; y, n) ↦ B[(x, m) → (y, n)] has a four-parameter scaling limit, recently constructed in [9]. This limit is the directed landscape L. More recently, L was shown to be the scaling limit of other integrable models of last passage percolation [11].…”

Section: Introductionmentioning

confidence: 99%

Last passage isometries for the directed landscape

Dauvergne¹

2021

Preprint

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Consider the restriction of the directed landscape L(x, s; y, t) to a set of the form {x 1 , . . . , x k } × {s 0 } × R × {t 0 }. We show that on any such set, the directed landscape is given by a last passage problem across k locally Brownian functions. The k functions in this last passage isometry are built from certain marginals of the extended directed landscape. As applications of this construction, we show that the Airy difference profile is locally absolutely continuous with respect to Brownian local time, that the KPZ fixed point started from two narrow wedges has a Brownian-Bessel decomposition around its cusp point, and that the directed landscape is a function of its geodesic shapes.

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“…Our overall techniques for proving Theorem 1.4 heavily relies on the inputs from the Gibbs property of the KPZ line ensemble (via Proposition 4.6 and 4.7) and monotonocity property of the KPZ equation (via Proposition 4.8). Such properties are also present in many models in the KPZ universality class including the KPZ fixed point (recently constructed in [MQR16,DOV18]), asymmetric simple exclusion process (ASEP), last passage percolation, stochastic six vertex model etc. We hope that it could be possible to prove lower bound to the lower tail probability of those models using the techniques of the present paper.…”

Section: Introductionmentioning

confidence: 98%

Fractal Geometry of the Valleys of the Parabolic Anderson Equation

Ghosal¹,

Yi²

2021

Preprint

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We study the macroscopic fractal properties of the deep valleys of the solution of the (1 + 1)-dimensional parabolic Anderson equationwhere Ẇ is the time-space white noise and 0 < infx∈R u0(x) ≤ sup x∈R u0(x) < ∞. Unlike the macroscopic multifractality of the tall peaks, we show that valleys of the parabolic Anderson equation are macroscopically monofractal. In fact, the macroscopic Hausdorff dimension (introduced by Barlow and Taylor [BT89, BT92]) of the valleys undergoes a phase transition at a point which does not depend on the initial data. The key tool of our proof is a lower bound to the lower tail probability of the parabolic Anderson equation. Such lower bound is obtained for the first time in this paper and will be derived by utilizing the connection between the parabolic Anderson equation and the Kardar-Parisi-Zhang equation. Our techniques of proving this lower bound can be extended to other models in the KPZ universality class including the KPZ fixed point.

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The directed landscape

Cited by 72 publications

References 39 publications

Diffusive scaling limit of the Busemann process in Last Passage Percolation

Diffusive scaling limit of the Busemann process in Last Passage Percolation

Last passage isometries for the directed landscape

Fractal Geometry of the Valleys of the Parabolic Anderson Equation

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