On the joint distribution of the maximum and its position of the Airy2 process minus a parabola

Abstract

The maximal point of the Airy2 process minus a parabola is believed to describe the scaling limit of the end-point of the directed polymer in a random medium, which was proved to be true for a few specific cases. Recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained, one by Moreno Flores, Quastel and Remenik, and the other by Schehr. The first formula is given in terms of the Airy function and an associated operator, and the second formula is expressed in terms of the Lax pair equations of the Painleve II equation. We give a direct proof that these two formulas are the same.

15 pages, no figure, minor revision, to appear in J.Math.Phys

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Cited by in corpus (28)

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• Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
• Airy processes and variational problems
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
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• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"
• Nonintersecting Brownian bridges between reflecting or absorbing walls