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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.

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M. Alimohammadi , V. Karimipour and M. Khorrami , Exact solution of a one-parameter family of asymmetric exclusion processes, Phys. Rev. E 57 (1998), 63706376.

G. Amir , I. Corwin and J. Quastel , Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions, Commun. Pure Appl. Math. 64 (2011), 466537.

D. Babbitt and L. Thomas , Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. An explicit Plancherel formula, Comm. Math. Phys. 54 (1977), 255278.

D. Babbitt and E. Gutkin , The Plancherel formula for the infinite XXZ Heisenberg spin chain, Lett. Math. Phys. 20 (1990), 9199.

L. Bertini and N. Cancrini , The stochastic heat equation: Feynman–Kac formula and intermittence, J. Stat. Phys. 78 (1995), 13771401.

H. Bethe , 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.

N. M. Bogoliubov , A. G. Izergin and N. A. Kitanine , Correlation functions for a strongly correlated Boson system, Nuclear Phys. B 516 (1998), 501528.

A. Borodin , Schur dynamics of the Schur processes, Adv. Math. 228 (2011), 22682291.

A. Borodin and I. Corwin , Macdonald processes, Probab. Theory Related Fields 158 (2014), 225400.

A. Borodin , I. Corwin and P. L. Ferrari , Free energy fluctuations for directed polymers in random media in 1 + 1 dimension, Comm. Pure Appl. Math. 67 (2014), 11291214.

A. Borodin , I. Corwin and D. Remenik , Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity, Comm. Math. Phys. 324 (2013), 215232.

A. Borodin and P. L. Ferrari , Anisotropic growth of random surfaces in 2 + 1 dimensions, Comm. Math. Phys. 325 (2014), 603684.

A. Borodin and L. Petrov , Nearest neighbor Markov dynamics on Macdonald processes, Probab. Surveys 11 (2014), 158.

P. Calabrese and J. S. Caux , Dynamics of the attractive 1D Bose gas: analytical treatment from integrability, J. Stat. Mech. 2007 (2007), doi:10.1088/1742-5468/2007/08/P08032.

P. Calabrese , P. Le Doussal and A. Rosso , Free-energy distribution of the directed polymer at high temperature, Eur. Phys. Lett. 90 (2010), doi:10.1209/0295-5075/90/20002.

P. Calabrese and P. Le Doussal , An exact solution for the KPZ equation with flat initial conditions, Phys. Rev. Lett. 106 (2011), doi:10.1103/PhysRevLett.106.250603.

I. Corwin , The Kardar–Parisi–Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012).

I. Corwin , N. O’Connell , T. Seppäläinen and N. Zygouras , Tropical combinatorics and Whittaker functions, Duke. Math. J. 163 (2014), 465663.

I. Corwin and J. Quastel , Crossover distributions at the edge of the rarefaction fan, Ann. Probab. 41 (2013), 12431314.

P. Diaconis and J. A. Fill , Strong stationary times via a new form of duality, Ann. Probab. 18 (1990), 14831522.

V. Dotsenko , 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.

V. Dotsenko , 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.

D. Gabrielli , G. Jona-Lasinio and C. Landim , Onsager symmetry from microscopic TP invariance, J. Stat. Phys. 96 (1999), 639652.

M. Gaudin , Boundary energy of a Bose gas in one dimension, Phys. Rev. A 4 (1971), 386394.

A. Gerasimov , D. Lebedev and S. Oblezin , On q-deformed gl+1-Whittaker function I, Comm. Math. Phys. 294 (2010), 97119.

A. Gerasimov , D. Lebedev and S. Oblezin , On a classical limit of q-deformed Whittaker functions, Lett. Math. Phys. 100 (2012), 279290.

E. Gutkin , Heisenberg–Ising spin chain: Plancherel decomposition and Chebyshev polynomials, in Calogero–Moser–Sutherland models, CRM Series in Mathematical Physics (Springer, Berlin, 2000), 177192.

G. J. Heckman and E. M. Opdam , Yang’s system of particles and Hecke algebras, Ann. of Math. (2) 145 (1997), 139173.

S. Helgason , An analogue of the Paley–Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297308.

T. Imamura and T. Sasamoto , 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.

T. Imamura and T. Sasamoto , Exact solution for the stationary Kardar–Parisi–Zhang equation, Phys. Rev. Lett. 108 (2012), doi:10.1103/PhysRevLett.108.190603.

T. Imamura and T. Sasamoto , Stationary correlations for the 1D KPZ equation, J. Stat. Phys. 150 (2013), 908939.

M. Kardar , Replica-Bethe Ansatz studies of two-dimensional interfaces with quenched random impurities, Nuclear Phys. B 290 (1987), 582602.

C. Korff , Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Comm. Math. Phys. 318 (2013), 173246.

M. Kormos , A. Shashi , Y.-Z. Chou , J.-S. Caux and A. Imambekov , Interaction quenches in the 1D Bose gas, Phys. Rev. B 88 (2013), 205131.

E. Lee , Transition probabilities of the Bethe Ansatz solvable interacting particle systems, J. Stat. Phys. 142 (2011), 643656.

J. B. McGuire , Study of exactly soluble one-dimensional N-body problems, J. Math. Phys. 5 (1964), 622.

N. O’Connell , Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), 437458.

N. O’Connell and M. Yor , Brownian analogues of Burke’s theorem, Stoch. Proc. Appl. 96 (2001), 285304.

A. Okounkov , Infinite wedge and random partitions, Selecta Math. 7 (2001), 5781.

A. Okounkov and N. Reshetikhin , Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), 581603.

A. M. Povolotsky , Bethe ansatz solution of zero-range process with non-uniform stationary state, Phys. Rev. E 69 (2004), doi:10.1103/PhysRevE.69.061109.

A. M. Povolotsky , On integrability of zero-range chipping models with factorized steady state, J. Phys. A: Math. Theor. 46 (2013), 465205.

A. M. Povolotsky , V. B. Priezzhev and C.-K. Hu , The asymmetric avalanche process, J. Stat. Phys. 111 (2003), 11491182.

S. Prolhac and H. Spohn , 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.

S. Prolhac and H. Spohn , The one-dimensional KPZ equation and the Airy process, J. Stat. Mech. (2011), doi:10.1088/1742-5468/2011/03/P03020.

S. Prolhac and H. Spohn , The propagator of the attractive delta-Bose gas in one dimension, J. Math. Phys. 52 (2011), doi:10.1063/1.3663431.

T. Sasamoto and M. Wadati , Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A 31 (1998), 60576071.

G. M. Schütz , Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys. 88 (1997), 427445.

F. Tabatabaei and G. M. Schütz , Shocks in the asymmetric exclusion process with internal degree of freedom, Phys. Rev. E 74 (2006), doi:10.1103/PhysRevE.74.051108.

C. Tracy and H. Widom , Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (2008), 815844; Erratum: Comm. Math. Phys. 304 (2011), 875–878.

C. Tracy and H. Widom , Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129154.

C. Tracy and H. Widom , On ASEP with step Bernoulli initial condition, J. Stat. Phys. 137 (2009), 825838.

C. Tracy and H. Widom , The Bose gas and asymmetric simple exclusion process on the half-line, J. Stat. Phys. 150 (2013), 112.

N. V. Tsilevich , Quantum inverse scattering method for the q-Boson model and symmetric functions, Funct. Anal. Appl. 40 (2006), 207217.

E. P. van Den Ban and H. Schlichtkrull , The most continuous part of the Plancherel decomposition for a reductive symmetric space, Ann. of Math. (2) 145 (1997), 267364.

J. F. van Diejen , 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.

J. F. van Diejen , Diagonalization of an integrable discretization of the repulsive delta Bose gas on the circle, Comm. Math. Phys. 267 (2006), 451476.

J. F. van Diejen and E. Emsiz , Diagonalization of the infinity q-Boson system, J. Funct. Anal. 266 (2014), 58015817.

J. F. van Diejen and E. Emsiz , The semi-infinity q-Boson system with boundary interaction, Lett. Math. Phys. 104 (2014), 103113.

C. N. Yang , Some exact results for the many body problem in one dimension with repulsive delta function interaction, Phys. Rev. Lett. 19 (1967), 13121314.

C. N. Yang , S matrix for the one dimensional N-body problem with repulsive or attractive delta-function interaction, Phys. Rev. 168 (1968), 19201923.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
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