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Spectral theory for the $q$-Boson particle system

Published online by Cambridge University Press:  17 September 2014

Alexei Borodin
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow 127994, Russia email borodin@math.mit.edu
Ivan Corwin
Affiliation:
Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, Providence, RI 02903, USA Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA email ivan.corwin@gmail.com
Leonid Petrov
Affiliation:
Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow 127994, Russia email lenia.petrov@gmail.com
Tomohiro Sasamoto
Affiliation:
Chiba University, Department of Mathematics, 1-33 Yayoi-cho, Inage, Chiba, 263-8522, Japan Zentrum Mathematik, Technische Universität Mun̈chen, Boltzmannstrasse 3, 85748 Garching, Germany email sasamoto@math.s.chiba-u.ac.jp
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Abstract

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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.

Type
Research Article
Copyright
© The Author(s) 2014 
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