The half-space Airy stat process

Betea, Dan; Ferrari, Patrik L.; Occelli, Alessandra

doi:10.48550/arxiv.2012.10337

Cited by 4 publications

(7 citation statements)

References 101 publications

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“…We also provide in Appendix A an alternative formula for F Brownian a , based on an extension of our earlier result for a = 0 in [6] (The formula for F Brownian a (s) appears in (131)). It remains to be shown that the exact formulas from [32] and our formulas in [6] and Section A are equivalent. Note that in [18,32], formulas were also obtained for the height distribution and multipoint correlations at points away from the boundary.…”

Section: Brownian Initial Datamentioning

confidence: 95%

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Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case

Barraquand,

Krajenbrink,

Doussal

2022

Preprint

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We study the Kardar-Parisi-Zhang equation on the half-line x 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter u of the dynamics, and the drift −v of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time t, it is natural to rescale parameters as (u, v) = t −1/3 (a, b) to study the critical region. In the special case a + b = 0, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase (a < 0) but not in the unbound phase. For instance, starting from the flat or droplet initial data, the height field near the boundary converges to the stationary process with a > 0 and b = 0, which is not Brownian. For a + b 0, we determine exactly the large time distribution F stat a,b of the height function h(0, t). As an application, we obtain the exact covariance of the height field in a half-line at two times 1 ≪ t 1 ≪ t 2 starting from stationary initial data, as well as estimates, when starting from droplet initial data, in the limit t 1 /t 2 → 1.

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Section: Brownian Initial Datamentioning

confidence: 95%

“…It remains to be shown that the exact formulas from [32] and our formulas in [6] and Section A are equivalent. Note that in [18,32], formulas were also obtained for the height distribution and multipoint correlations at points away from the boundary.…”

Section: Brownian Initial Datamentioning

confidence: 95%

Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case

Barraquand,

Krajenbrink,

Doussal

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…We also provide in appendix A an alternative formula for F Brownian a , based on an extension of our earlier result for a = 0 in [6] (the formula for F Brownian a (s) appears in (133)). It remains to be shown that the exact formulas from [32] and our formulas in [6] and appendix A are equivalent. Note that in [18,32], formulas were also obtained for the height distribution and multipoint correlations at points away from the boundary.…”

Section: Brownian Initial Conditionmentioning

confidence: 96%

Half-space stationary Kardar–Parisi–Zhang equation beyond the Brownian case

Barraquand¹,

Krajenbrink²,

Doussal³

2022

J. Phys. A: Math. Theor.

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We study the Kardar-Parisi-Zhang equation on the half-line $x\geqslant 0$ with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter $u$ of the dynamics, and the drift $−v$ of the initial condition at inﬁnity. We consider the ﬂuctuations of the height ﬁeld when the initial condition is given by one of these stationary processes. At large time $t$, it is natural to rescale parameters as $(u, v) = t^{−1/3}(a, b)$ to study the critical region. In the special case $a + b = 0$, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ($a < 0$) but not in the unbound phase. For instance, starting from the ﬂat or droplet initial data, the height ﬁeld near the boundary converges to the stationary process with $a>0$ and $b=0$, which is not Brownian. For $a + b > 0$, we determine exactly the large time distribution $F^{\rm stat}_{a,b} of the height function $h(0, t)$. As an application, we obtain the exact covariance of the height ﬁeld in a half-line at two times $1\ll t_1 \ll t_2$ starting from stationary initial data, as well as estimates, when starting from droplet initial data, in the limit $t_1/t_2 \to 1$.

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“…In particular, we conjecture that the large-scale limit of half-line ASEP stationary measures [35] does converge to the same limit at large scale. Note that this half-space KPZ fixed point has not been defined rigorously (unlike the full-space situation), but its multipoint distributions for some initial conditions are known [36,37]. The largescale limit of half-line KPZ equation stationary measures is described in the phase diagram of fig.…”

Section: Limits and Consequences -mentioning

confidence: 99%

Steady state of the KPZ equation on an interval and Liouville quantum mechanics

Barraquand¹,

Doussal²

2022

EPL

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We obtain a simple formula for the stationary measure of the height field evolving according to the Kardar-Parisi-Zhang equation on the interval with general Neumann-type boundary conditions and any interval size. This is achieved using the recent results of Corwin and Knizel (arXiv:2103.12253) together with Liouville quantum mechanics. Our formula allows to easily determine the stationary measure in various limits: KPZ fixed point on an interval, half-line KPZ equation, KPZ fixed point on a half-line, as well as the Edwards-Wilkinson equation on an interval.

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The half-space Airy stat process

Cited by 4 publications

References 101 publications

Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case

Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case

Half-space stationary Kardar–Parisi–Zhang equation beyond the Brownian case

Steady state of the KPZ equation on an interval and Liouville quantum mechanics

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