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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 

References

Alimohammadi, M., Karimipour, V. and Khorrami, M., Exact solution of a one-parameter family of asymmetric exclusion processes, Phys. Rev. E 57 (1998), 63706376.Google Scholar
Amir, G., Corwin, I. and Quastel, J., Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions, Commun. Pure Appl. Math. 64 (2011), 466537.Google Scholar
Babbitt, D. and Thomas, L., Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. An explicit Plancherel formula, Comm. Math. Phys. 54 (1977), 255278.CrossRefGoogle Scholar
Babbitt, D. and Gutkin, E., The Plancherel formula for the infinite XXZ Heisenberg spin chain, Lett. Math. Phys. 20 (1990), 9199.CrossRefGoogle Scholar
Bertini, L. and Cancrini, N., The stochastic heat equation: Feynman–Kac formula and intermittence, J. Stat. Phys. 78 (1995), 13771401.Google Scholar
Bethe, H., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. (On the theory of metals. I. Eigenvalues and eigenfunctions of the linear atom chain), Z. Physik 71 (1931), 205226.CrossRefGoogle Scholar
Bogoliubov, N. M., Bullough, R. K. and Timonen, J., Critical behavior for correlated strongly coupled Boson systems in 1 + 1 dimensions, Phys. Rev. Lett. 25 (1994), 39333936.CrossRefGoogle Scholar
Bogoliubov, N. M., Izergin, A. G. and Kitanine, N. A., Correlation functions for a strongly correlated Boson system, Nuclear Phys. B 516 (1998), 501528.Google Scholar
Borodin, A., Schur dynamics of the Schur processes, Adv. Math. 228 (2011), 22682291.Google Scholar
Borodin, A. and Corwin, I., Discrete time q-TASEPs, Int. Math. Res. Not. doi:10.1093/imrn/rnt206.Google Scholar
Borodin, A. and Corwin, I., Macdonald processes, Probab. Theory Related Fields 158 (2014), 225400.Google Scholar
Borodin, A., Corwin, I. and Ferrari, P. L., Free energy fluctuations for directed polymers in random media in 1 + 1 dimension, Comm. Pure Appl. Math. 67 (2014), 11291214.Google Scholar
Borodin, A., Corwin, I., Ferrari, P. L. and Vető, B., Height fluctuations for the stationary KPZ equation, Preprint (2014), arXiv:1407.6977.Google Scholar
Borodin, A., Corwin, I., Gorin, V. and Shakirov, S., Observables of Macdonald processes, Trans. Amer. Math. Soc., to appear. Preprint (2013), arXiv:1306.0659.Google Scholar
Borodin, A., Corwin, I., Petrov, L. and Sasamoto, T., Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz, Preprint (2014), arXiv:1407.8534.Google Scholar
Borodin, A., Corwin, I. and Remenik, D., Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity, Comm. Math. Phys. 324 (2013), 215232.Google Scholar
Borodin, A., Corwin, I. and Sasamoto, T., From duality to determinants for q-TASEP and ASEP, Ann. Probab. , to appear. Preprint (2012), arXiv:1207.5035.Google Scholar
Borodin, A. and Ferrari, P. L., Anisotropic growth of random surfaces in 2 + 1 dimensions, Comm. Math. Phys. 325 (2014), 603684.Google Scholar
Borodin, A. and Gorin, V., Lectures on integrable probability, Preprint (2012), arXiv:1212.3351.Google Scholar
Borodin, A. and Petrov, L., Nearest neighbor Markov dynamics on Macdonald processes, Probab. Surveys 11 (2014), 158.Google Scholar
Calabrese, P. and Caux, J. S., Dynamics of the attractive 1D Bose gas: analytical treatment from integrability, J. Stat. Mech. 2007 (2007), doi:10.1088/1742-5468/2007/08/P08032.Google Scholar
Calabrese, P., Le Doussal, P. and Rosso, A., Free-energy distribution of the directed polymer at high temperature, Eur. Phys. Lett. 90 (2010), doi:10.1209/0295-5075/90/20002.Google Scholar
Calabrese, P. and Le Doussal, P., An exact solution for the KPZ equation with flat initial conditions, Phys. Rev. Lett. 106 (2011), doi:10.1103/PhysRevLett.106.250603.Google Scholar
Calabrese, P. and Le Doussal, P., The KPZ equation with flat initial condition and the directed polymer with one free end, J. Stat. Mech. 2012 (2012), doi:10.1088/1742-5468/2012/06/P06001.Google Scholar
Corwin, I., The Kardar–Parisi–Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012).Google Scholar
Corwin, I., O’Connell, N., Seppäläinen, T. and Zygouras, N., Tropical combinatorics and Whittaker functions, Duke. Math. J. 163 (2014), 465663.Google Scholar
Corwin, I. and Petrov, L., The $q$-PushASEP: A new integrable traffic model in $1+1$dimension, Preprint (2013), arXiv:1308.3124.Google Scholar
Corwin, I. and Quastel, J., The renormalization fixed point of the Kardar–Parisi–Zhang universality class, Preprint (2011), arXiv:1103.3422.Google Scholar
Corwin, I. and Quastel, J., Crossover distributions at the edge of the rarefaction fan, Ann. Probab. 41 (2013), 12431314.Google Scholar
Diaconis, P. and Fill, J. A., Strong stationary times via a new form of duality, Ann. Probab. 18 (1990), 14831522.Google Scholar
Dotsenko, V., Bethe ansatz derivation of the Tracy–Widom distribution for one-dimensional directed polymers, Eur. Phys. Lett. 90 (2010), doi:10.1209/0295-5075/90/20003.Google Scholar
Dotsenko, V., Replica Bethe ansatz derivation of the GOE Tracy–Widom distribution in one-dimensional directed polymers with free boundary conditions, J. Stat. Mech. 2012 (2012), doi:10.1088/1742-5468/2012/11/P11014.CrossRefGoogle Scholar
Dotsenko, V., Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential, J. Stat. Mech. (2013), P02012.Google Scholar
Dotsenko, V., Two-time free energy distribution function in (1 + 1) directed polymers, J. Stat. Mech. (2013), P06017.Google Scholar
Dotsenko, V., Two-point free energy distribution function in (1 + 1) directed polymers, J. Phys. A: Math. Theor. 46 (2013), 355001.Google Scholar
Gabrielli, D., Jona-Lasinio, G. and Landim, C., Onsager symmetry from microscopic TP invariance, J. Stat. Phys. 96 (1999), 639652.Google Scholar
Gangolli, R. and Varadarajan, V. S., Harmonic analysis of spherical functions on real reductive spaces, Ergebnisse der Mathematik, vol. 101 (Springer, Berlin, 1988).Google Scholar
Gaudin, M., Boundary energy of a Bose gas in one dimension, Phys. Rev. A 4 (1971), 386394.Google Scholar
Gaudin, M., La fonction d’onde de Bethe (Masson, Paris, 1983).Google Scholar
Gerasimov, A., Lebedev, D. and Oblezin, S., On q-deformed gl+1-Whittaker function I, Comm. Math. Phys. 294 (2010), 97119.Google Scholar
Gerasimov, A., Lebedev, D. and Oblezin, S., On a classical limit of q-deformed Whittaker functions, Lett. Math. Phys. 100 (2012), 279290.Google Scholar
Gutkin, E., Heisenberg–Ising spin chain: Plancherel decomposition and Chebyshev polynomials, in Calogero–Moser–Sutherland models, CRM Series in Mathematical Physics (Springer, Berlin, 2000), 177192.Google Scholar
Heckman, G. J. and Opdam, E. M., Yang’s system of particles and Hecke algebras, Ann. of Math. (2) 145 (1997), 139173.Google Scholar
Helgason, S., An analogue of the Paley–Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297308.Google Scholar
Helgason, S., Groups and geometric analysis (Academic Press, London, 1984).Google Scholar
Imamura, T. and Sasamoto, T., Replica approach to the KPZ equation with half Brownian motion initial condition, J. Phys. A: Math. Theor. 44 (2011), doi:10.1088/1751-8113/44/38/385001.CrossRefGoogle Scholar
Imamura, T. and Sasamoto, T., Exact solution for the stationary Kardar–Parisi–Zhang equation, Phys. Rev. Lett. 108 (2012), doi:10.1103/PhysRevLett.108.190603.Google Scholar
Imamura, T. and Sasamoto, T., Stationary correlations for the 1D KPZ equation, J. Stat. Phys. 150 (2013), 908939.Google Scholar
Imamura, T., Sasamoto, T. and Spohn, H., On the equal time two-point distribution of the one-dimensional KPZ equation by replica, Preprint (2013), arXiv:1305.1217.Google Scholar
Kardar, M., Replica-Bethe Ansatz studies of two-dimensional interfaces with quenched random impurities, Nuclear Phys. B 290 (1987), 582602.Google Scholar
Korff, C., Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Comm. Math. Phys. 318 (2013), 173246.CrossRefGoogle Scholar
Korhonen, M. and Lee, E., The transition probability and the probability of the left-most particle’s position of the $q$-TAZRP, Preprint (2013), arXiv:1308.4769.Google Scholar
Kormos, M., Shashi, A., Chou, Y.-Z., Caux, J.-S. and Imambekov, A., Interaction quenches in the 1D Bose gas, Phys. Rev. B 88 (2013), 205131.Google Scholar
Lee, E., Transition probabilities of the Bethe Ansatz solvable interacting particle systems, J. Stat. Phys. 142 (2011), 643656.Google Scholar
Lieb, E. H. and Liniger, W., Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. Lett. 130 (1963), 16051616.Google Scholar
Liggett, T., Interacting particle systems (Spinger, Berlin, 2005).Google Scholar
Macdonald, I. G., Spherical functions of p-adic type (Publications of the Ramanujan Institute, 1971).Google Scholar
Macdonald, I. G., Symmetric functions and Hall polynomials, second edition (Oxford University Press, New York, 1999).Google Scholar
Macdonald, I. G., Orthogonal polynomials associated with root systems, Sem. Lothar. Combin. 45 (2000), B45a.Google Scholar
McGuire, J. B., Study of exactly soluble one-dimensional N-body problems, J. Math. Phys. 5 (1964), 622.Google Scholar
O’Connell, N., Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), 437458.Google Scholar
O’Connell, N. and Pei, Y., A q-weighted version of the Robinson–Schensted algorithm, Electron. J. Probab. 18 (2013), 125.Google Scholar
O’Connell, N. and Yor, M., Brownian analogues of Burke’s theorem, Stoch. Proc. Appl. 96 (2001), 285304.Google Scholar
Okounkov, A., Infinite wedge and random partitions, Selecta Math. 7 (2001), 5781.Google Scholar
Okounkov, A. and Reshetikhin, N., Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), 581603.Google Scholar
Oxford, S., The Hamiltonian of the quantized nonlinear Schrödinger equation, PhD thesis, UCLA (1979).Google Scholar
Povolotsky, A. M., Bethe ansatz solution of zero-range process with non-uniform stationary state, Phys. Rev. E 69 (2004), doi:10.1103/PhysRevE.69.061109.Google Scholar
Povolotsky, A. M., On integrability of zero-range chipping models with factorized steady state, J. Phys. A: Math. Theor. 46 (2013), 465205.Google Scholar
Povolotsky, A. M., Priezzhev, V. B. and Hu, C.-K., The asymmetric avalanche process, J. Stat. Phys. 111 (2003), 11491182.Google Scholar
Prolhac, S. and Spohn, H., Two-point generating function of the free energy for a directed polymer in a random medium, J. Stat. Mech. (2011), doi:10.1088/1742-5468/2011/01/P01031.Google Scholar
Prolhac, S. and Spohn, H., The one-dimensional KPZ equation and the Airy process, J. Stat. Mech. (2011), doi:10.1088/1742-5468/2011/03/P03020.Google Scholar
Prolhac, S. and Spohn, H., The propagator of the attractive delta-Bose gas in one dimension, J. Math. Phys. 52 (2011), doi:10.1063/1.3663431.CrossRefGoogle Scholar
Sasamoto, T. and Wadati, M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A 31 (1998), 60576071.Google Scholar
Schütz, G. M., Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys. 88 (1997), 427445.Google Scholar
Tabatabaei, F. and Schütz, G. M., Shocks in the asymmetric exclusion process with internal degree of freedom, Phys. Rev. E 74 (2006), doi:10.1103/PhysRevE.74.051108.Google Scholar
Takeyama, Y., A discrete analogue of period delta Bose gas and affine Hecke algebra, Preprint (2012), arXiv:1209.2758.Google Scholar
Tracy, C. and Widom, H., Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (2008), 815844; Erratum: Comm. Math. Phys. 304 (2011), 875–878.Google Scholar
Tracy, C. and Widom, H., Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129154.Google Scholar
Tracy, C. and Widom, H., On ASEP with step Bernoulli initial condition, J. Stat. Phys. 137 (2009), 825838.Google Scholar
Tracy, C. and Widom, H., The Bose gas and asymmetric simple exclusion process on the half-line, J. Stat. Phys. 150 (2013), 112.Google Scholar
Tsilevich, N. V., Quantum inverse scattering method for the q-Boson model and symmetric functions, Funct. Anal. Appl. 40 (2006), 207217.Google Scholar
van Den Ban, E. P. and Schlichtkrull, H., The most continuous part of the Plancherel decomposition for a reductive symmetric space, Ann. of Math. (2) 145 (1997), 267364.Google Scholar
van Diejen, J. F., On the Plancherel formula for the (discrete) Laplacian in a Weyl chamber with repulsive boundary conditions at the walls, Ann. Inst. H. Poincaré 5 (2004), 135168.Google Scholar
van Diejen, J. F., Diagonalization of an integrable discretization of the repulsive delta Bose gas on the circle, Comm. Math. Phys. 267 (2006), 451476.Google Scholar
van Diejen, J. F. and Emsiz, E., Diagonalization of the infinity q-Boson system, J. Funct. Anal. 266 (2014), 58015817.Google Scholar
van Diejen, J. F. and Emsiz, E., The semi-infinity q-Boson system with boundary interaction, Lett. Math. Phys. 104 (2014), 103113.Google Scholar
Yang, C. N., Some exact results for the many body problem in one dimension with repulsive delta function interaction, Phys. Rev. Lett. 19 (1967), 13121314.Google Scholar
Yang, C. N., S matrix for the one dimensional N-body problem with repulsive or attractive delta-function interaction, Phys. Rev. 168 (1968), 19201923.CrossRefGoogle Scholar
Yudson, V. I., Dynamics of integrable quantum systems, Zh. Èksper. Teoret. Fiz. 88 (1985), 17571770.Google Scholar