Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equations
Résumé
We study Jánossy densities of a randomly thinned Airy kernel determinantal point process. We prove that they can be expressed in terms of solutions to the Stark and cylindrical Korteweg de Vries equations; these solutions are Darboux tranformations of the simpler ones related to the gap probability of the same thinned Airy point process. Moreover, we prove that the associated wave functions satisfy a variation of AmirCorwinQuastel's integro-dierential Painlevé II equation. Finally, we derive tail asymptotics for the relevant solutions to the cylindrical Korteweg de Vries equation and show that they decompose asymptotically into a superposition of simpler solutions.
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