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8. Busemann functions and LPP

    1.     A key theme is to explore connections between multipath LPP partition functions and Busemann functions defined in several directions. Busemann functions have long been instrumental in the study of geodesics, current fluctuations, and stationary measures in exactly solvable models such as TASEP and exponential/geometric LPP. Suitable multipath analogues of Busemann functions might prove fruitful in scaling limits of LPP/polymer models.

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      Problem 8.1.

      The Airy line ensemble and its Gibbs property [MR3152753] play a central role in integrable probability. One intriguing question is what happens when random path segments are resampled using the Brownian Gibbs property. As DL is a functional of the Airy line ensemble, how the DL changes under such resamplings?


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          Problem 8.2.

          Beyond the original construction via Brownian last passage percolation, alternate descriptions of the DL have begun to emerge, leveraging tools such as Busemann functions, Airy line ensembles, and stationary variants of LPP. Each of these viewpoints can shed light on the universality and rich geometric behavior inherent in the KPZ class. In discrete integrable systems, “melon” constructions often refer to nonintersecting random walks obtained as a functional of independent random walks. These naturally have connections to the Robinson–Schensted–Knuth (RSK) correspondence and its various extensions. Investigating how “stationary melons” behave under a Busemann-oriented perspective and how they embed in the directed landscape is an interesting direction.

              Cite this as: AimPL: All roads to the KPZ universality class, available at http://aimpl.org/roadtokpz.