Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

Abstract

We develop spectral theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result that implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system. Owing to a Markov duality with the q-Hahn TASEP (a discrete-time generalization of TASEP with particles’ jump distribution being the orthogonality weight for the classical q-Hahn orthogonal polynomials), we write down moment formulas that characterize the fixed time distribution of the q-Hahn TASEP with general initial data. The Bethe ansatz eigenfunctions of the q-Hahn system degenerate into eigenfunctions of other (not necessarily stochastic) interacting particle systems solvable by the coordinate Bethe ansatz. This includes the ASEP, the (asymmetric) six-vertex model, and the Heisenberg XXZ spin chain (all models are on the infinite lattice). In this way, each of the latter systems possesses a spectral theory, too. In particular, biorthogonality of the ASEP eigenfunctions, which follows from the corresponding q-Hahn statement, implies symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration) as corollaries. Another degeneration takes the q-Hahn system to the q-Boson particle system (dual to q-TASEP) studied in detail in our previous paper (2013). Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar–Parisi–Zhang equation/stochastic heat equation, namely, q-TASEP and ASEP.

This is a preview of subscription content, log in to check access.

Change history

  • 09 August 2019

    This is a correction to Theorems 7.3 and 8.12 in [1].

  • 09 August 2019

    This is a correction to Theorems 7.3 and 8.12 in [1].

References

  1. 1

    Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011). arXiv:1003.0443 [math.PR]

  2. 2

    Babbitt D., Gutkin E.: The Plancherel formula for the infinite XXZ Heisenberg spin chain. Lett. Math. Phys. 20, 91–99 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 3

    Babbitt D.L., Thomas.: Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. An explicit Plancherel formula. Commun. Math. Phys. 54, 255–278 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  4. 4

    Barraquand, G.: A short proof of a symmetry identity for the \({(q,\mu,\nu)}\) -deformed Binomial distribution. Electron. Commun. Probab. 19(50), 1–3 (2014). arXiv:1404.4265 [math.PR]

  5. 5

    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Courier Dover Publications, Mineola (2007)

    Google Scholar 

  6. 6

    Bertini L.N. Cancrini: The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78(5–6), 1377–1401 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. 7

    Bertini L., Giacomin G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)

    Article  MathSciNet  Google Scholar 

  8. 8

    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). Zeitschrift fur Physik 71, 205–226 (1931)

    Article  ADS  Google Scholar 

  9. 9

    Borodin, A., Corwin, I.: Discrete time q-TASEPs. Intern. Math. Res. Notices (2013). arXiv:1305.2972 [math.PR]. doi:10.1093/imrn/rnt206

  10. 10

    Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields 158(1–2), 225–400 (2014). arXiv:1111.4408 [math.PR]

  11. 11

    Borodin, A., Corwin, I., Ferrari, P.: Free energy fluctuations for directed polymers in random media in 1 + 1 dimension. Commun. Pure Appl. Math. 67(7), 1129–1214 (2014). arXiv:1204.1024 [math.PR]

  12. 12

    Borodin, A., Corwin, I., Ferrari, P., Veto, B.: Height fluctuations for the stationary KPZ equation (2014). arXiv:1407.6977 [math.PR]

  13. 13

    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. (2014). arXiv:1407.6729 [math.PR]

  14. 14

    Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: Observables of Macdonald processes (2013). arXiv:1306.0659 [math.PR]

  15. 15

    Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for the q-Boson particle system. Composit. Math. 151(1), 1–67 (2015). arXiv:1308.3475 [math-ph]

  16. 16

    Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014). arXiv:1207.5035 [math.PR]

  17. 17

    Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. (2013, to appear). arXiv:1305.5501 [math.PR]

  18. 18

    Borodin, A., Petrov, L.: Integrable probability: From representation theory to Macdonald processes. Probab. Surv. 11, 1–58 (2014). arXiv:1310.8007 [math.PR]

  19. 19

    Calabrese, P., Caux, J.S.: Dynamics of the attractive 1D Bose gas: analytical treatment from integrability. J. Stat. Mech. P08032 (2007)

  20. 20

    Calabrese P., Le Doussal P., Rosso A.: Free-energy distribution of the directed polymer at high temperature. Eur. Phys. Lett. 90(2), 20002 (2010)

    Article  ADS  Google Scholar 

  21. 21

    Carinci, G., Giardinà à, C., Redig, F., Sasamoto, T.: A generalized asymmetric exclusion process with \({U_q(\mathfrak{sl}_2)}\) stochastic duality (2014). arXiv:1407.3367 [math.PR]

  22. 22

    Corwin, I.: The q-Hahn Boson process and q-Hahn TASEP. Int. Math. Res. Notices (2014). arXiv:1401.3321 [math.PR]. doi:10.1093/imrn/rnu094

  23. 23

    Corwin, I., Quastel, J.: Crossover distributions at the edge of the rarefaction fan. Ann. Probab. 41(3A), 1243–1314 (2013). arXiv:1006.1338 [math.PR]

  24. 24

    Dotsenko, V.: Bethe ansatz derivation of the Tracy–Widom distribution for one-dimensional directed polymers. Eur. Phys. Lett. 90(20003) (2010)

  25. 25

    Dotsenko, V.: Universal randomness. Physics-Uspekhi 54(3), 259–280, (2011). arXiv:1009.3116 [cond-mat.stat-mech]

  26. 26

    Faddeev, L.: How Algebraic Bethe Ansatz works for integrable model. Les-Houches lectures (1996). arXiv:hep-th/9605187

  27. 27

    Ferrari, P., Veto, B.: Tracy–Widom asymptotics for q-TASEP. Ann. Inst. Henri Poincar Probab. Stat. (2013, to appear). arXiv:1310.2515 [math.PR]

  28. 28

    Gaudin M.: Boundary energy of a Bose gas in one dimension. Phys. Rev. A 4, 386–394 (1971)

    Article  ADS  Google Scholar 

  29. 29

    Gutkin, E.: Heisenberg-Ising spin chain: Plancherel decomposition and Chebyshev polynomials. Calogero–Moser–Sutherland Models. CRM Series in Mathematical Physics, pp. 177–192 (2000)

  30. 30

    Gwa L.-H.H. Spohn: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844–854 (1992)

    Article  ADS  Google Scholar 

  31. 31

    Heckman G.J., Opdam E.M.: Yang’s system of particles and Hecke algebras. Ann. Math. 145(1), 139–173 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. 32

    Helgason S.: An analogue of the Paley–Wiener theorem for the Fourier transform on certain symmetric spaces. Math. Ann. 165, 297–308 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  33. 33

    Imamura T., Sasamoto T., Spohn H.: KPZ, ASEP and Delta-Bose gas. J. Phys. Conf. Ser. 297, 012–016 (2011)

    Article  Google Scholar 

  34. 34

    Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Tech. report, Delft University of Technology and Free University of Amsterdam (1996)

  35. 35

    Korhonen, M., Lee, E.: The transition probability and the probability for the left-most particle’s position of the q-TAZRP. J. Math. Phys. 55, 013301 (2013). arXiv:1308.4769 [math-ph]

  36. 36

    Lieb E.H.: The residual entropy of square ice. Phys. Rev. 162, 162–172 (1967)

    Article  ADS  Google Scholar 

  37. 37

    Lieb E.H., Liniger W.: Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. Lett. 130, 1605–1616 (1963)

    ADS  MathSciNet  Google Scholar 

  38. 38

    Liggett T.: Interacting Particle Systems. Springer, New York (1985)

    Google Scholar 

  39. 39

    Macdonald, I.G.: Spherical functions of p-adic type. Publ. Ramanujan Inst. 2 (1971)

  40. 40

    Macdonald I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)

    Google Scholar 

  41. 41

    Macdonald, I.G.: Orthogonal polynomials associated with root systems. Sem. Lothar. Combin. 45(B45a) (2000)

  42. 42

    Matveev, K., Petrov, L.: q-randomized Robinson–Schensted–Knuth correspondences and random polymers (2015). arXiv:1504.00666 [math.PR]

  43. 43

    Moreno Flores, G.R., Quastel, J., Remenik, D.: In preparation (2015)

  44. 44

    O’Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40(2), 437–458 (2012). arXiv:0910.0069 [math.PR]

  45. 45

    O’Connell, N., Pei, Y.: A q-weighted version of the Robinson-Schensted algorithm. Electron. J. Probab. 18(95), 1–25 (2013). arXiv:1212.6716 [math.CO]

  46. 46

    O’Connell N., Yor M.: Brownian analogues of Burke’s theorem. Stoch. Proc. Appl. 96(2), 285–304 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  47. 47

    Oxford, S.: The Hamiltonian of the quantized nonlinear Schrödinger equation. Ph.D. thesis, UCLA (1979)

  48. 48

    Povolotsky, A.: On integrability of zero-range chipping models with factorized steady state. J. Phys. A 46(465205) (2013). arXiv:1308.3250 [math-ph]

  49. 49

    Povolotsky, A., Priezzhev, V.: Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. 7, P07002 (2006)

  50. 50

    Prolhac, S., Spohn, H.: The propagator of the attractive delta-Bose gas in one dimension. J. Math. Phys. 52, 122106 (2011). arXiv:1109.3404 [math-ph]

  51. 51

    Reshetikhin, N.: Lectures on the integrability of the 6-vertex model. Lect. Notes Les Houches Summer School 89, 197–266 (2008). arXiv:1010.5031 [math-ph]

  52. 52

    Sasamoto T., Wadati M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31, 6057–6071 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. 53

    Schütz G.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86, 12651287 (1997)

    Article  Google Scholar 

  54. 54

    Schütz G.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88, 427–445 (1997)

    Article  Google Scholar 

  55. 55

    Spitzer F.: Interaction of Markov processes. Adv. Math. 5(2), 246–290 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  56. 56

    Takeyama, Y.: A deformation of affine Hecke algebra and integrable stochastic particle system. J. Phys. A 47(46), 465203 (2014). arXiv:1407.1960 [math-ph]

  57. 57

    Takeyama, Y.: A discrete analogoue of period delta Bose gas and affine Hecke algebra. Funkcialaj Ekvacioj 57(1), 107–118 (2014). arXiv:1209.2758 [math-ph]

  58. 58

    Tracy, C., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132(2), 291–300 (2008). arXiv:0804.1379 [math.PR]

  59. 59

    Tracy, C., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008). arXiv:0704.2633 [math.PR] [Erratum: Commun. Math. Phys. 304, 875–878 (2011)]

  60. 60

    Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009). arXiv:0807.1713 [math.PR]

  61. 61

    Tracy, C., Widom, H.: On ASEP with step Bernoulli initial condition. J. Stat. Phys. 137, 825–838 (2009). arXiv:0907.5192 [math.PR]

  62. 62

    Tracy, C., Widom, H.: Total current fluctuations in ASEP. J. Math. Phys. 50(9), 095–204 (2009). arXiv:0902.0821 [math.PR]

  63. 63

    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(1), 135–168 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  64. 64

    Van Diejen, J.F., Emsiz, E.: Diagonalization of the infinite q-Boson. J. Funct. Anal. 266(9), 5801–5817 (2014). arXiv:1308.2237 [math-ph]

  65. 65

    Van Diejen, J.F., Emsiz, E.: The semi-infinite q-Boson system with boundary interaction. Lett. Math. Phys. 104(1), 103–113 (2014). arXiv:1308.2242 [math-ph]

  66. 66

    Veto, B.: TracyWidom limit of q-Hahn TASEP (2014). arXiv:1407.2787 [math.PR]

  67. 67

    Yang C.N.: Some exact results for the many body problem in one dimension with repulsive delta function interaction. Phys. Rev. Lett. 19, 1312–1314 (1967)

    ADS  Google Scholar 

  68. 68

    Yang C.N.: S matrix for the one dimensional N-body problem with repulsive or attractive delta-function interaction. Phys. Rev. 168, 1920–1923 (1968)

    ADS  Google Scholar 

  69. 69

    Yang, C.N., Yang, C.P.: One dimensional chain of anisotropic spin-spin interaction. Phys. Rev. 150, 321–327, 327–339 (1966)

  70. 70

    Yang C.N., Yang C.P.: One dimensional chain of anisotropic spin-spin interaction. Phys. Rev. 151, 258–264 (1966)

    Article  ADS  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ivan Corwin.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Borodin, A., Corwin, I., Petrov, L. et al. Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz. Commun. Math. Phys. 339, 1167–1245 (2015). https://doi.org/10.1007/s00220-015-2424-7

Download citation

Keywords

  • Corwin
  • Laurent Polynomial
  • Bilinear Pairing
  • Interact Particle System
  • Totally Asymmetric Simple Exclusion Process