On the joint distribution of the maximum and its position of the Airy^sub 2^ process minus a parabola

The maximal point of the ... process minus a parabola is believed to describe the scaling limit of the end-point of the directed polymer in a random medium. This 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. (ProQuest: ... denotes formulae/symbols omitted.)