Janossy densities and Darboux transformations for the Stark and cylindrical KdV equations

Claeys, Tom;Glesner, Gabriel;Ruzza, Giulio;Tarricone, Sofia
(2024) Communications in Mathematical Physics — Vol. 405, n° 113 (2024)

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  • Claeys, Tomorcid-logoUCLouvain
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  • Glesner, GabrielUCLouvain
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  • Ruzza, GiulioUCLouvain
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  • Tarricone, SofiaUCLouvain
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Abstract
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 Amir–Corwin–Quastel’s integro-differential 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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Claeys, T., Glesner, G., Ruzza, G., & Tarricone, S. (2024). Janossy densities and Darboux transformations for the Stark and cylindrical KdV equations. Communications in Mathematical Physics, 405(113). https://doi.org/10.1007/s00220-024-04988-7 (Original work published 2024)