Multi-time distribution in discrete polynuclear growth

Johansson, Kurt; Rahman, Mustazee

doi:10.48550/arxiv.1906.01053

Cited by 13 publications

(28 citation statements)

References 23 publications

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“…On a related, but different, line of recent works, efforts have also been directed to obtain exact formulae for the one-point or multi-point joint distribution of the profile at two (on-scale) separated time points. This originated with Johansson's work in Brownian LPP [34] and has been continued for geometric LPP and discrete polynuclear growth [34,35,36] (see also [24] for a replica calculation). In parallel, the works [2,39] have obtained exact asymptotic formulae for the two time distribution for the height function of TASEP (related to Exponential LPP by the standard coupling) with different initial conditions.…”

Section: Background and Related Resultsmentioning

confidence: 99%

“…Although the joint distribution of the height function at different spatial locations at a given time have been more classically studied, significant recent interest has been devoted to understanding the joint distribution of the profile at two (on-scale separated) time points starting from a general initial condition. There has been a number of recent works obtaining exact formulae for the two time joint distribution for a number of models in the KPZ universality class [34,35,36,2,39]. Typically the formulae are quite involved and it does not appear to be straightforward to extract the asymptotics of some interesting statistics for the time evolution of the interface.…”

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confidence: 99%

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Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison

Basu

Ganguly

Zhang

2019

Preprint

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We consider directed last passage percolation on Z 2 with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture in [26], we show that for the passage times from the line x + y = 0 to the points (r, r) and (n, n), denoted Xr and Xn respectively, as n → ∞ and r n is small but bounded away from zero, the covariance satisfies Cov(Xr, Xn) = Θ ( r n ) 4/3+o(1) n 2/3 , thereby establishing 4 3 as the temporal covariance exponent. This differs from the corresponding exponent for the droplet initial condition recently rigorously established in [25,3] and requires novel arguments. Key ingredients include the understanding of geodesic geometry and recent advances in quantitative comparison of geodesic weight profiles to Brownian motion using the Brownian Gibbs property. The proof methods are expected to be applicable for a wider class of initial data.

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Section: Background and Related Resultsmentioning

confidence: 99%

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confidence: 99%

Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison

Basu

Ganguly

Zhang

2019

Preprint

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“…For the models in the infinite domain, Johansson computed the two-time distribution for a directed last passage percolation model in [Joh17] and for the discrete-time TASEP in [Joh19]. Most recently, in the span of about one month, the multitime distribution was evaluated in [JR19] for a directed last passage percolation model and, independently. in [Liu19] for the (continuous-time) TASEP.…”

Section: )mentioning

confidence: 99%

“…in [Liu19] for the (continuous-time) TASEP. The work [JR19] was for the step initial condition while the work [Liu19] was for both the step or flat initial conditions. We mention that the formulas of these two papers are different, and it still remains to show that they are equal.…”

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confidence: 99%

Periodic TASEP with general initial conditions

Baik

Liu

2019

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We consider the one-dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain, and obtain explicit formulas for the multi-point distributions in the space-time plane. The formulas are given in terms of an integral involving a Fredholm determinant. We then evaluate the large-time, large-period limit in the relaxation time scale, which is the scale such that the system size starts to affect the height fluctuations. The limit is obtained assuming certain conditions on the initial condition, which we show that the step, flat, and step-flat initial conditions satisfy. Hence, we obtain the limit theorem for these three initial conditions in the periodic model, extending the previous work on the step initial condition. We also consider uniform random and uniform-step random initial conditions.

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“…Newer integrable ideas have yielded a richer set of formulas, e.g. see Johansson and Rahman[JR19]; Liu[Liu19]; and Matetski, Quastel, and Remenik[MQR16].The works discussed above provide a strong integrable framework for understanding the directed landscape. More recently, probabilistic and geometric methods have been used in conjunction with a few key integrable inputs to prove regularity results, convergence statements, and exponent estimates in such models.…”

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confidence: 99%

Disjoint optimizers and the directed landscape

Dauvergne

Zhang

2021

Preprint

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We study maximal length collections of disjoint paths, or 'disjoint optimizers', in the directed landscape. We show that disjoint optimizers always exist, and that their lengths can be used to construct an extended directed landscape. The extended directed landscape can be built from an independent collection of extended Airy sheets, which we define via last passage percolation across the Airy line ensemble. We show that the extended directed landscape and disjoint optimizers are scaling limits of the corresponding objects in Brownian last passage percolation. As two consequences of this work, we show that one direction of the Robinson-Schensted-Knuth bijection passes to the KPZ limit, and we find a criterion for geodesic disjointness in the directed landscape that uses only a single Airy line ensemble.

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Multi-time distribution in discrete polynuclear growth

Cited by 13 publications

References 23 publications

Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison

Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison

Periodic TASEP with general initial conditions

Disjoint optimizers and the directed landscape

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