Skip to main content
×
Home

Spectral theory for the $q$-Boson particle system

  • Alexei Borodin (a1) (a2), Ivan Corwin (a3) (a4) (a5), Leonid Petrov (a6) (a7) and Tomohiro Sasamoto (a8) (a9)
Abstract
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.

Copyright
References
Hide All
[AKK98]Alimohammadi M., Karimipour V. and Khorrami M., Exact solution of a one-parameter family of asymmetric exclusion processes, Phys. Rev. E 57 (1998), 63706376.
[ACQ11]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.
[BT77]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.
[BG90]Babbitt D. and Gutkin E., The Plancherel formula for the infinite XXZ Heisenberg spin chain, Lett. Math. Phys. 20 (1990), 9199.
[BC95]Bertini L. and Cancrini N., The stochastic heat equation: Feynman–Kac formula and intermittence, J. Stat. Phys. 78 (1995), 13771401.
[Bet31]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.
[BBT94]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.
[BIK98]Bogoliubov N. M., Izergin A. G. and Kitanine N. A., Correlation functions for a strongly correlated Boson system, Nuclear Phys. B 516 (1998), 501528.
[Bor11]Borodin A., Schur dynamics of the Schur processes, Adv. Math. 228 (2011), 22682291.
[BC13]Borodin A. and Corwin I., Discrete time q-TASEPs, Int. Math. Res. Not. doi:10.1093/imrn/rnt206.
[BC14]Borodin A. and Corwin I., Macdonald processes, Probab. Theory Related Fields 158 (2014), 225400.
[BCF14]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.
[BCFV14]Borodin A., Corwin I., Ferrari P. L. and Vető B., Height fluctuations for the stationary KPZ equation, Preprint (2014), arXiv:1407.6977.
[BCGS13]Borodin A., Corwin I., Gorin V. and Shakirov S., Observables of Macdonald processes, Trans. Amer. Math. Soc., to appear. Preprint (2013), arXiv:1306.0659.
[BCPS14]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.
[BCR13]Borodin A., Corwin I. and Remenik D., Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity, Comm. Math. Phys. 324 (2013), 215232.
[BCS12]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.
[BF14]Borodin A. and Ferrari P. L., Anisotropic growth of random surfaces in 2 + 1 dimensions, Comm. Math. Phys. 325 (2014), 603684.
[BG12]Borodin A. and Gorin V., Lectures on integrable probability, Preprint (2012), arXiv:1212.3351.
[BP14]Borodin A. and Petrov L., Nearest neighbor Markov dynamics on Macdonald processes, Probab. Surveys 11 (2014), 158.
[CC07]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.
[CLR10]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.
[CL11]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.
[CL12]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.
[Cor12]Corwin I., The Kardar–Parisi–Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012).
[COSZ14]Corwin I., O’Connell N., Seppäläinen T. and Zygouras N., Tropical combinatorics and Whittaker functions, Duke. Math. J. 163 (2014), 465663.
[CP13]Corwin I. and Petrov L., The inline-graphic$q$-PushASEP: A new integrable traffic model in inline-graphic$1+1$dimension, Preprint (2013), arXiv:1308.3124.
[CQ11]Corwin I. and Quastel J., The renormalization fixed point of the Kardar–Parisi–Zhang universality class, Preprint (2011), arXiv:1103.3422.
[CQ13]Corwin I. and Quastel J., Crossover distributions at the edge of the rarefaction fan, Ann. Probab. 41 (2013), 12431314.
[DF90]Diaconis P. and Fill J. A., Strong stationary times via a new form of duality, Ann. Probab. 18 (1990), 14831522.
[Dot10]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.
[Dot12]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.
[Dot13a]Dotsenko V., Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential, J. Stat. Mech. (2013), P02012.
[Dot13b]Dotsenko V., Two-time free energy distribution function in (1 + 1) directed polymers, J. Stat. Mech. (2013), P06017.
[Dot13c]Dotsenko V., Two-point free energy distribution function in (1 + 1) directed polymers, J. Phys. A: Math. Theor. 46 (2013), 355001.
[GJL99]Gabrielli D., Jona-Lasinio G. and Landim C., Onsager symmetry from microscopic TP invariance, J. Stat. Phys. 96 (1999), 639652.
[GV88]Gangolli R. and Varadarajan V. S., Harmonic analysis of spherical functions on real reductive spaces, Ergebnisse der Mathematik, vol. 101 (Springer, Berlin, 1988).
[Gau71]Gaudin M., Boundary energy of a Bose gas in one dimension, Phys. Rev. A 4 (1971), 386394.
[Gau83]Gaudin M., La fonction d’onde de Bethe (Masson, Paris, 1983).
[GLO10]Gerasimov A., Lebedev D. and Oblezin S., On q-deformed gl+1-Whittaker function I, Comm. Math. Phys. 294 (2010), 97119.
[GLO12]Gerasimov A., Lebedev D. and Oblezin S., On a classical limit of q-deformed Whittaker functions, Lett. Math. Phys. 100 (2012), 279290.
[Gut00]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.
[HO97]Heckman G. J. and Opdam E. M., Yang’s system of particles and Hecke algebras, Ann. of Math. (2) 145 (1997), 139173.
[Hel66]Helgason S., An analogue of the Paley–Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297308.
[Hel84]Helgason S., Groups and geometric analysis (Academic Press, London, 1984).
[IS11]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.
[IS12]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.
[IS13]Imamura T. and Sasamoto T., Stationary correlations for the 1D KPZ equation, J. Stat. Phys. 150 (2013), 908939.
[ISS13]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.
[Kar87]Kardar M., Replica-Bethe Ansatz studies of two-dimensional interfaces with quenched random impurities, Nuclear Phys. B 290 (1987), 582602.
[Kor13]Korff C., Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Comm. Math. Phys. 318 (2013), 173246.
[KL13]Korhonen M. and Lee E., The transition probability and the probability of the left-most particle’s position of the inline-graphic$q$-TAZRP, Preprint (2013), arXiv:1308.4769.
[KSCCI13]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.
[Lee11]Lee E., Transition probabilities of the Bethe Ansatz solvable interacting particle systems, J. Stat. Phys. 142 (2011), 643656.
[LL63]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.
[Lig05]Liggett T., Interacting particle systems (Spinger, Berlin, 2005).
[Mac71]Macdonald I. G., Spherical functions of p-adic type (Publications of the Ramanujan Institute, 1971).
[Mac99]Macdonald I. G., Symmetric functions and Hall polynomials, second edition (Oxford University Press, New York, 1999).
[Mac00]Macdonald I. G., Orthogonal polynomials associated with root systems, Sem. Lothar. Combin. 45 (2000), B45a.
[McG64]McGuire J. B., Study of exactly soluble one-dimensional N-body problems, J. Math. Phys. 5 (1964), 622.
[O’Co12]O’Connell N., Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), 437458.
[OP13]O’Connell N. and Pei Y., A q-weighted version of the Robinson–Schensted algorithm, Electron. J. Probab. 18 (2013), 125.
[OY01]O’Connell N. and Yor M., Brownian analogues of Burke’s theorem, Stoch. Proc. Appl. 96 (2001), 285304.
[Oko01]Okounkov A., Infinite wedge and random partitions, Selecta Math. 7 (2001), 5781.
[OR03]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.
[Oxf79]Oxford S., The Hamiltonian of the quantized nonlinear Schrödinger equation, PhD thesis, UCLA (1979).
[Pov04]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.
[Pov13]Povolotsky A. M., On integrability of zero-range chipping models with factorized steady state, J. Phys. A: Math. Theor. 46 (2013), 465205.
[PPH03]Povolotsky A. M., Priezzhev V. B. and Hu C.-K., The asymmetric avalanche process, J. Stat. Phys. 111 (2003), 11491182.
[PS11a]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.
[PS11b]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.
[PS11c]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.
[SW98]Sasamoto T. and Wadati M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A 31 (1998), 60576071.
[Sch97]Schütz G. M., Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys. 88 (1997), 427445.
[TS06]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.
[Tak12]Takeyama Y., A discrete analogue of period delta Bose gas and affine Hecke algebra, Preprint (2012), arXiv:1209.2758.
[TW08]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.
[TW09a]Tracy C. and Widom H., Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129154.
[TW09b]Tracy C. and Widom H., On ASEP with step Bernoulli initial condition, J. Stat. Phys. 137 (2009), 825838.
[TW13]Tracy C. and Widom H., The Bose gas and asymmetric simple exclusion process on the half-line, J. Stat. Phys. 150 (2013), 112.
[Tsi06]Tsilevich N. V., Quantum inverse scattering method for the q-Boson model and symmetric functions, Funct. Anal. Appl. 40 (2006), 207217.
[vDS97]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.
[vDi04]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.
[vDi06]van Diejen J. F., Diagonalization of an integrable discretization of the repulsive delta Bose gas on the circle, Comm. Math. Phys. 267 (2006), 451476.
[vDE14a]van Diejen J. F. and Emsiz E., Diagonalization of the infinity q-Boson system, J. Funct. Anal. 266 (2014), 58015817.
[vDE14b]van Diejen J. F. and Emsiz E., The semi-infinity q-Boson system with boundary interaction, Lett. Math. Phys. 104 (2014), 103113.
[Yan67]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.
[Yan68]Yang C. N., S matrix for the one dimensional N-body problem with repulsive or attractive delta-function interaction, Phys. Rev. 168 (1968), 19201923.
[Yud85]Yudson V. I., Dynamics of integrable quantum systems, Zh. Èksper. Teoret. Fiz. 88 (1985), 17571770.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 23 *
Loading metrics...

Abstract views

Total abstract views: 180 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 11th December 2017. This data will be updated every 24 hours.