Geodesic
One-sided reflected Brownian motions and the KPZ fixed point
Forum of Mathematics, Sigma, Tome 8 (2020)

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We consider the system of one-sided reflected Brownian motions that is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling-invariant Markov process defined in [MQR17] and believed to govern the long-time, large-scale fluctuations for all models in the KPZ universality class. Brownian last-passage percolation was shown recently in [DOV18] to converge to the Airy sheet (or directed landscape), defined there as a strong limit of a functional of the Airy line ensemble. This establishes the variational formula for the KPZ fixed point in terms of the Airy sheet. 
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     author = {Mihai Nica and Jeremy Quastel and Daniel Remenik},
     title = {One-sided reflected {Brownian} motions and the {KPZ} fixed point},
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Mihai Nica; Jeremy Quastel; Daniel Remenik. One-sided reflected Brownian motions and the KPZ fixed point. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.56

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