Abstract
The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process that is conjectured to be at the core of the KPZ universality class. In this article we study two aspects the KPZ fixed point that share the same Brownian limiting behaviour: the local space regularity and the long time evolution. Most of the results that we will present here were obtained by either applying explicit formulas for the transition probabilities or applying the coupling method to discrete approximations. Instead we will use the variational description of the KPZ fixed point, allowing us the possibility of running the process starting from different initial data (basic coupling), to prove directly the aforementioned limiting behaviours.
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Notes
- 1.
Convergence in terms of a sequence of random elements in the space of continuous scalar fields on a fixed compact subset of \({\mathbb R}^2\), endowed with the uniform metric.
- 2.
Lemma 2 [22] shows that m(a) is differentiable at a = 0 if \({\mathfrak f}\) is a sum of a deterministic function \({\mathfrak h}\) with a Brownian motion.
- 3.
It also follows from Proposition 4 that \(\varDelta {\mathfrak h}_t^{+\mu }(x)-\varDelta {\mathfrak h}_t^{-\mu }(x)\) is a nondecreasing function of x.
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Acknowledgements
This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program—Universality in random structures: Interfaces, Matrices, Sandpiles (Code: ICTS/urs2019/01), and also by the CNPQ grants 421383/2016-0, 302830/2016-2 and 305356/2019-4, and by the FAPERJ grant E-26/203.048/2016. The author would like to thank Patrik Ferrari, Daniel Remenik for useful comments and enlightening discussions concerning this subject, and to an anonymous referee for the review of the article.
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Pimentel, L.P.R. (2021). Brownian Aspects of the KPZ Fixed Point. In: Vares, M.E., Fernández, R., Fontes, L.R., Newman, C.M. (eds) In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius. Progress in Probability, vol 77. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-60754-8_29
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