The scaling limit of the longest increasing subsequence

Dauvergne, Duncan; Virág, Bálint

doi:10.48550/arxiv.2104.08210

Cited by 23 publications

(43 citation statements)

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“…These paths, looking like the stripes of a watermelon, are where the name melon comes from. This picture is expected to hold more generally, as was shown by Biane, Bougerol and O'Connell in [9,30] where the connection between non-intersecting RWs and the RSK algorithm was studied, and in fact recently used by Dauvergne and Virag in [14] to prove the convergence of various models to the Directed Landscape.…”

Section: Two Types Of Melonizationmentioning

confidence: 73%

“…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

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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: Two Types Of Melonizationmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Diffusive scaling limit of the Busemann process in Last Passage Percolation

Busani¹

2021

Preprint

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“…It is the scaling limit of random growth models in the KPZ universality class. By results of [10,21], it is related to the directed landscape L by the following formula. Letting h 0 ∶ R → R ∪ {−∞} denote the initial condition of the KPZ fixed point, we can write…”

Section: For Any Compact Set [A B] the Law Ofmentioning

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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“…is expected to converge to. The convergence to the same for ℒ and their counterparts in other integrable models have been proved across [DOV] and subsequent work by Dauvergne and Virág [DV21], making it conjecturally the universal scaling limit of such models of random geometry in two dimensions. Further, importantly, it was also shown that the geodesics in the pre-limiting models converge to their limiting counterparts in the Directed Landscape.…”

Section: The Directed Landscapementioning

confidence: 83%

“…Note that owing to length maximization, the LPP weights (•, •) don't form a metric but rather an anti-metric, i.e., it satisfies the reverse triangle inequality, ( , ) + ( , ) ( , ). The reader is encouraged to review the concept of directed metric introduced by Dauvergne and Virág [DV21] which unifies the notions of metric and antimetric.…”

Section: Kardar-parisi-zhang Universalitymentioning

confidence: 99%

Random metric geometries on the plane and Kardar-Parisi-Zhang universality

Ganguly¹

2021

Preprint

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The scaling limit of the longest increasing subsequence

Cited by 23 publications

References 0 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

Random metric geometries on the plane and Kardar-Parisi-Zhang universality

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