Spectral theory for the q-Boson particle system
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Borodin_Spectral theory.pdf
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Author(s)
Borodin, Alexei
Corwin, Ivan
Petrov, Leonid
Sasamoto, Tomohiro
Date Issued
September 2014
Journal
Compositio Mathematica
Publisher
Cambridge University Press
Citation
Borodin, Alexei, Ivan Corwin, Leonid Petrov, and Tomohiro Sasamoto. “Spectral Theory for the q-Boson Particle System.” Compositio Math. 151, no. 01 (September 17, 2014): 1–67.
Version
Author's final manuscript
Abstract
We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP (q-deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1112/s0010437x14007532
