Communications in Mathematical Physics

, Volume 343, Issue 2, pp 651–700 | Cite as

Stochastic Higher Spin Vertex Models on the Line

  • Ivan CorwinEmail author
  • Leonid Petrov


We introduce a four-parameter family of interacting particle systems on the line, which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain Markov dualities. Using this, for the systems started in step initial data, we write down nested contour integral formulas for moments and Fredholm determinant formulas for Laplace-type transforms. Taking various choices or limits of parameters, this family degenerates to many of the known exactly solvable models in the Kardar–Parisi–Zhang universality class, as well as leads to many new examples of such models. In particular, asymmetric simple exclusion process, the stochastic six-vertex model, q-totally asymmetric simple exclusion process and various directed polymer models all arise in this manner. Our systems are constructed from stochastic versions of the R-matrix related to the six-vertex model. One of the key tools used here is the fusion of R-matrices and we provide a probabilistic proof of this procedure.


High Spin Exclusion Process Corwin Totally Asymmetric Simple Exclusion Process Vertex Weight 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. ACQ11.
    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. Bar14.
    Barraquand, G.: A phase transition for q-TASEP with a few slower particles. Stoch. Proc. Appl. 125, 2674–2699 (2015). arXiv:1404.7409 [math.PR]
  3. BC13.
    Borodin, A., Corwin, I.: Discrete time q-TASEPs. Intern. Math. Res. Not. (2013). arXiv:1305.2972 [math.PR]. doi: 10.1093/imrn/rnt206
  4. BC14.
    Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields 158, 225–400 (2014). arXiv:1111.4408 [math.PR]
  5. BC15.
    Barraquand, G., Corwin, I.: The q-Hahn asymmetric exclusion process (2015). arXiv:1501.03445 [math.PR]
  6. BCF12.
    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
  7. BCFV14.
    Borodin, A., Corwin, I., Ferrari, P., Veto, B.: Height fluctuations for the stationary KPZ equation (2014). arXiv:1407.6977 [math.PR]
  8. BCG14.
    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model (2014). arXiv:1407.6729 [math.PR]
  9. BCPS14.
    Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz (2014). arXiv:1407.8534 [math-ph]
  10. BCR12.
    Borodin, A., Corwin, I., Remenik, D.: Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity. Commun. Math. Phys. 324(1), 215–232 (2013). arXiv:1206.4573
  11. BCS12.
    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
  12. BG.
    Bertini L., Giacomin G.: Stochastic Burgers and KP2 equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. Bor14.
    Borodin, A.: On a family of symmetric rational functions (2014). arXiv:1410.0976 [math.CO]
  14. BP13.
    Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. (2013). arXiv:1305.5501 [math.PR]
  15. CGRS14.
    Carinci, G., Giardina, C., Redig, F., Sasamoto, T.: A generalized asymmetric exclusion process with \({U_q(\mathfrak{sl}_2)}\) stochastic duality (2014). arXiv:1407.3367 [math.PR]
  16. Cor14.
    Corwin, I.: The q-Hahn Boson process and q-Hahn TASEP. Intern. Math. Res. Not. (2014). arXiv:1401.3321 [math.PR]
  17. COSZ14.
    Corwin, I., O’Connell, N., Seppäläinen, T., Zygouras, N.: Tropical combinatorics and Whittaker functions. Duke J. Math. 163(3), 513–563 (2014). arXiv:1110.3489 [math.PR]
  18. CP15.
    Corwin, I., Petrov, L.: The q-pushASEP: a new integrable model for traffic in 1 + 1 dimension. J. Stat. Phys. 160(4), 1005–1026 (2015). arXiv:1308.3124 [math.PR]
  19. CSS14.
    Corwin, I., Seppäläinen, T., Shen, H.: The strict-weak lattice polymer (2014). arXiv:1409.1794 [math.PR]
  20. Fad96.
    Faddeev, L.D.: How algebraic Bethe Ansatz works for integrable model. In: Les-Houches Lecture Notes (1996). arXiv:1407.3367 [math.PR]
  21. FV13.
    Ferrari, P., Veto, B.: Tracy–Widom asymptotics for q-TASEP. Ann. Inst. Hen. Poin. (2013). arXiv:1310.2515 [math.PR]
  22. GS92.
    Gwa L-H., Spohn H.: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844–854 (1992)ADSCrossRefGoogle Scholar
  23. IS11.
    Imamura T., Sasamoto T.: Current moments of 1D ASEP by duality. J. Stat. Phys. 142, 919–930 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. KR87.
    Kirillov A.N., Reshetikhin N.Y.: Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum. J. Phys. A 20(6), 1565–1585 (1987)ADSMathSciNetCrossRefGoogle Scholar
  25. KS96.
    Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. In: Technical Report, Delft University of Technology and Free University of Amsterdam (1996)Google Scholar
  26. Lie67.
    Lieb E.H.: The residual entropy of square ice. Phys. Rev. 162, 162–172 (1967)ADSCrossRefGoogle Scholar
  27. Man14.
    Mangazeev, V: On the Yang–Baxter equation for the six-vertex model. Nucl. Phys. B 882, 70–96 (2014). arXiv:1401.6494
  28. MFRQ15.
    Moreno Flores, G., Remenik, D., Quastel, J.: (2015, in preparation)Google Scholar
  29. O’C12.
    O’Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40(2), 437–458 (2012). arXiv:0910.0069 [math.PR]
  30. OO14.
    O’Connell, N., Ortmann, J.: Tracy–Widom asymptotics for a random polymer model with gamma-distributed weights (2014). arXiv:1408.5326 [math.PR]
  31. OY01.
    O’Connell N., Yor M.: Brownian analogues of Burke’s theorem. Stoch. Proc. Appl. 96(2), 285–304 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Pov13.
    Povolotsky A.: On integrability of zero-range chipping models with factorized steady state. J. Phys. A Math. Theor. 46, 465205 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. Res08.
    Reshetikhin, N.: Lectures on the integrability of the 6-vertex model. In: Les-Houches Lecture Notes (2008). arXiv:1010.5031 [math.PR]
  34. RP81.
    Rogers L.C.G., Pitman J.W.: Markov functions. Ann. Probab. 9(4), 573–582 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sch97.
    Schütz G.M.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86, 1265–1287 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. Sep12.
    Seppäläinen T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40(1), 19–73 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. SS10.
    Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834(3), 523–542 (2010) arXiv:1002.1879 [cond-mat.stat-mech]
  38. SS14.
    Sasamoto, T., Spohn, H.: Point-interacting Brownian motions in the KPZ universality class (2014). arXiv:1411.3142 [math.PH]
  39. SW98.
    Sasamoto T., Wadati M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31, 6057–6071 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. TLD14.
    Thimothée T., Le Doussal P.: Log-gamma directed polymer with fixed endpoints via the replica Bethe Ansatz. J. Stat. Mech. 2014(10), P10018 (2014)CrossRefGoogle Scholar
  41. TW08.
    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)]
  42. TW09.
    Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009). arXiv:0807.1713 [math.PR]
  43. Vet14.
    Veto, B.: Tracy–Widom limit of q-Hahn TASEP (2014). arXiv:1407.2787 [math.PR]

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Clay Mathematics InstituteProvidenceUSA
  3. 3.Institut Henri PoincaréParisFrance
  4. 4.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  5. 5.Institute for Information Transmission ProblemsMoscowRussia

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