Spectral theory for the q-Boson particle system
Author(s)Borodin, Alexei; Corwin, Ivan; Petrov, Leonid; Sasamoto, Tomohiro
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Cambridge University Press
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.
Author's final manuscript