Brownian Aspects of the KPZ Fixed Point

Pimentel, Leandro P. R.

doi:10.1007/978-3-030-60754-8_29

Cited by 7 publications

(1 citation statement)

References 20 publications

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“…Besides being a natural question from the point of view of interacting particle systems, a second motivation for this work comes from the slew of recent papers on the existence of stationary evolutions of the KPZ and related equations [2,3,5,6,14,16,17,[26][27][28]32]. Although these results are about stochastic PDEs, and Theorem 1.1 is about a discrete process, there are some similarities at a conceptual level.…”

Section: Review Of the Literaturementioning

confidence: 99%

Existence of stationary ballistic deposition on the infinite lattice

Chatterjee

2022

Random Struct Algorithms

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Ballistic deposition is one of the many models of interface growth that are believed to be in the KPZ universality class, but have so far proved to be largely intractable mathematically. In this model, blocks of size one fall independently as Poisson processes at each site on the d$$ d $$‐dimensional lattice, and either attach themselves to the column growing at that site, or to the side of an adjacent column, whichever comes first. It is not hard to see that if we subtract off the height of the column at the origin from the heights of the other columns, the resulting interface process is Markovian. The main result of this article is that this Markov process has at least one invariant probability measure. We conjecture that the invariant measure is not unique, and provide some partial evidence.

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Section: Review Of the Literaturementioning

confidence: 99%

Existence of stationary ballistic deposition on the infinite lattice

Chatterjee

2022

Random Struct Algorithms

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Local KPZ Behavior Under Arbitrary Scaling Limits

Chatterjee

2022

Commun. Math. Phys.

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Local stationarity in exponential last-passage percolation

Balázs

Busani

Seppäläinen

2021

Probab. Theory Relat. Fields

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We consider point-to-point last-passage times to every vertex in a neighbourhood of size $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 at distance N from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends only on $$\delta $$ δ . Through this result we show that (1) the $$\text {Airy}_2$$ Airy 2 process is locally close to a Brownian motion in total variation; (2) the tree of point-to-point geodesics from every vertex in a box of side length $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 going to a point at distance N agrees inside the box with the tree of semi-infinite geodesics going in the same direction; (3) two point-to-point geodesics started at distance $$N^{\nicefrac {2}{3}}$$ N 2 3 from each other, to a point at distance N, will not coalesce close to either endpoint on the scale N. Our main results rely on probabilistic methods only.

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Brownian Aspects of the KPZ Fixed Point

Cited by 7 publications

References 20 publications

Existence of stationary ballistic deposition on the infinite lattice

Existence of stationary ballistic deposition on the infinite lattice

Local KPZ Behavior Under Arbitrary Scaling Limits

Local stationarity in exponential last-passage percolation

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