Current fluctuations of the stationary ASEP and six-vertex model. (English) Zbl 1403.60081

In this paper, the author considers an asymmetric simple exclusion process (ASEP). Note that the ASEP is a particle system on \(\mathbb Z\), such that at most one particle occupies any site. Particles may jump to the left with exponential rate \(L\); jump to the right with exponential rate \(R\) and the ones that jump to occupied locations are suppressed and assumed that \(R>L\). The ASEP starts with the stationary initial data, i.e., from the Bernoulli product measure: each site of \(\mathbb Z\) is occupied independently with a given probability. By the condition \(R>L\), the particles in the ASEP on average jump in the right direction. This jumping is captured by a natural statistic called the current. The author is interested in the behavior of the particle system, through the behavior of its current, when time \(T\to +\infty\). The KPZ (Kardar-Parisi-Zhang) universality predicts that the fluctuations of the current of the ASEP are on scale \(T^{1/3}\), and moreover the distribution of these fluctuations becomes the Baik-Rains distribution as \(T\to +\infty\). The author proves these universality predictions for the ASEP. Moreover, the author is interested to integrability and fluctuations of the translation invariant stochastic six-vertex model. This is discrete time model which is related to the ASEP. A relation between translation invariant measures for the stochastic six-vertex model to Gibbs measures of the symmetric six-vertex model with arbitrary vertex weights is established.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
Full Text: DOI arXiv Euclid


[1] A. Aggarwal,Convergence of the stochastic six-vertex model to the ASEP, Math. Phys. Anal. Geom.20(2017), no. 3. · Zbl 1413.82005
[2] A. Aggarwal and A. Borodin,Phase transitions in the ASEP and stochastic six-vertex model, to appear in Ann. Probab., preprint,arXiv:1607.08684v1[math.PR]. · Zbl 1466.60191
[3] D. Babbitt and E. Gutkin,The Plancherel formula for the infinite \(XXZ\) Heisenberg spin chain, Lett. Math. Phys.20(1990), 91-99. · Zbl 0719.58047
[4] 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), 255-278.
[5] J. Baik, P. L. Ferrari, and S. Péché,Limit process of stationary TASEP near the characteristic line, Comm. Pure Appl. Math.63(2010), 1017-1070. · Zbl 1194.82067
[6] J. Baik, P. L. Ferrari, and S. Péché, “Convergence of the two-point function of the stationary TASEP” inSingular Phenomena and Scaling in Mathematical Models, Springer, Cham, 2014, 91-100. · Zbl 1355.82024
[7] J. Baik and E. M. Rains,Limiting distribution for a polynuclear growth model with external sources, J. Statist. Phys.100(2000), 523-541. · Zbl 0976.82043
[8] M. Balázs, J. Quastel, and T. Seppäläinen,Fluctuation exponent of the KPZ/stochastic Burgers equation, J. Amer. Math. Soc.24(2011), 683-708. · Zbl 1227.60083
[9] M. Balázs and T. Seppäläinen,Fluctuation bounds for the asymmetric simple exclusion process, ALEA Lat. Am. J. Probab. Math. Stat.6(2009), 1-24. · Zbl 1160.60333
[10] M. Balázs and T. Seppäläinen,Order of current variance and diffusivity in the asymmetric simple exclusion process, Ann. of Math. (2)171(2010), 1237-1265. · Zbl 1200.60083
[11] R. J. Baxter,Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1989. · Zbl 0723.60120
[12] G. Ben-Arous and I. Corwin,Current fluctuations for TASEP: A proof of the Prähofer-Spohn conjecture, Ann. Probab.39(2011), 104-138. · Zbl 1208.82036
[13] L. Bertini and G. Giacomin,Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys.183(1997), 571-607. · Zbl 0874.60059
[14] H. Bethe,Zur Theorie der Metalle, I: Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys.71(1931), 205-226. · Zbl 0002.37205
[15] P. Bleher and K. Liechty,Random Matrices and the Six-Vertex Model, CRM Monogr. Ser.32, Amer. Math. Soc., Providence, 2014. · Zbl 1279.82001
[16] A. Borodin, “Determinantal point processes” inThe Oxford Handbook of Random Matrix Theory, Oxford Univ. Press, Oxford, 2011, 231-249.
[17] A. Borodin,On a family of symmetric rational functions, Adv. Math.306(2017), 973-1018. · Zbl 1355.05250
[18] A. Borodin and A. Bufetov,An irreversible local Markov chain that preserves the six vertex model on a torus, Ann. Inst. Henri Poincaré Probab. Stat.53(2017), 451-463. · Zbl 1361.60059
[19] A. Borodin, A. Bufetov, and I. Corwin,Directed random polymers via nested contour integrals, Ann. Physics368(2016), 191-247. · Zbl 1377.82050
[20] A. Borodin and I. Corwin,Macdonald processes, Probab. Theory Related Fields158(2014), 225-400. · Zbl 1291.82077
[21] A. Borodin, I. Corwin, P. Ferrari, and B. Vető,Height fluctuations for the stationary KPZ equation, Math. Phys. Anal. Geom.18(2015), no. 20. · Zbl 1332.82068
[22] A. Borodin, I. Corwin, and V. Gorin,Stochastic six-vertex model, Duke Math. J.165(2016), 563-624. · Zbl 1343.82013
[23] A. Borodin, I. Corwin, and T. Sasamoto,From duality to determinants for \(q\)-TASEP and ASEP, Ann. Probab.42(2014), 2314-2382. · Zbl 1304.82048
[24] A. Borodin and V. Gorin, “Lectures on integrable probability” inProbability and Statistical Physics in St. Petersburg, Proc. Sympos. Pure Math.91, Amer. Math. Soc., Providence, 2016, 155-214. · Zbl 1388.60157
[25] A. Borodin and L. Petrov,Integrable probability: From representation theory to Macdonald processes, Probab. Surv.11(2014), 1-58. · Zbl 1295.82023
[26] A. Borodin and L. Petrov,Higher spin six-vertex models and symmetric rational functions, Selecta Math. (N.S.), published electronically 20 December 2016. · Zbl 1405.60141
[27] A. Borodin and L. Petrov, “Integrable probability: Stochastic vertex models and symmetric functions” inStochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School,104, Oxford Univ. Press, Oxford, 2017. · Zbl 1397.82010
[28] D. J. Bukman and J. D. Shore,The conical point in the ferroelectric six-vertex model, J. Statist. Phys.78(1995), 1277-1309. · Zbl 1080.82535
[29] H. Cohn, R. Kenyon, and J. Propp,A variational principle for domino tilings, J. Amer. Math. Soc.14(2001), 297-346. · Zbl 1037.82016
[30] I. Corwin,The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl.1(2012), no. 1130001. · Zbl 1247.82040
[31] I. Corwin, “Macdonald processes, quantum integrable systems, and the Kardar-Parisi-Zhang universality class” inProceedings of the International Congress of Mathematicians (Seoul, Korea, 2014), 2014, 1007-1034. · Zbl 1373.82050
[32] I. Corwin, “Two ways to solve ASEP” inTopics in Percolative and Disordered Systems, Springer Proc. Math. Stat.69, Springer, New York, 2014, 1-13. · Zbl 1329.82077
[33] I. Corwin and L. Petrov,Stochastic higher spin vertex models on the line, Comm. Math. Phys.343(2016), 651-700. · Zbl 1348.82055
[34] I. Corwin and J. Quastel,Crossover distributions at the edge of the rarefaction fan, Ann. Probab.41(2013), 1243-1314. · Zbl 1285.82034
[35] E. Dimitrov,Six-vertex models and the GUE-corners process, preprint,arXiv:1610.06893v2[math.PR]. · Zbl 1439.82025
[36] G. Felder, V. Tarasov, and A. Varchenko, “Solutions of the elliptic qKZB equations and Bethe Ansatz, I” inTopics in Singularity Theory, Amer. Math. Soc. Transl. Ser. 2180, Amer. Math. Soc., Providence, 1998, 45-75. · Zbl 0884.65127
[37] P. A. Ferrari and L. R. G. Fontes,Current fluctuations for the asymmetric simple exclusion process, Ann. Probab.22(1994), 820-832. · Zbl 0806.60099
[38] P. L. Ferrari and H. Spohn,Domino tilings and the six-vertex model at its free-fermionic point, J. Phys. A39, no. 33 (2006), 10297-10306. · Zbl 1114.82006
[39] P. L. Ferrari and H. Spohn,Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process, Comm. Math. Phys.265(2006), 1-44. · Zbl 1118.82032
[40] L.-H. Gwa and H. Spohn,Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation, Phys. Rev. Lett.46, no. 2 (1992), 844-854.
[41] L.-H. Gwa and H. Spohn,Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian, Phys. Rev. Lett.68, no. 6 (1992), 725-728. · Zbl 0969.82526
[42] M. Hairer,Solving the KPZ equation, Ann. of Math. (2)178(2013), 559-664. · Zbl 1281.60060
[43] M. Hairer,A theory of regularity structures, Invent. Math.198(2014), 269-504. · Zbl 1332.60093
[44] T. Imamura and T. Sasamoto,Stationary correlations for the 1D KPZ equation, J. Stat. Phys.150(2013), 908-939. · Zbl 1266.82045
[45] C. Jayaprakash and W. F. Saam,Thermal evolution of crystal shapes: The fcc crystal, Phys. Rev. B30(1984), 3916-3928.
[46] N. H. Jing,Vertex operators and Hall-Littlewood symmetric functions, Adv. Math.87(1991), 226-248. · Zbl 0742.16014
[47] M. Kardar, G. Parisi, and Y.-C. Zhang,Dynamic scaling of growing interfaces, Phys. Rev. Lett.56, no. 9 (1986), 889-892. · Zbl 1101.82329
[48] P. W. Kasteleyn, “Graph theory and crystal physics” inGraph Theory and Theoretical Physics, Academic Press, London, 1967, 43-110. · Zbl 0205.28402
[49] R. Kenyon, “Lectures on dimers” inStatistical Mechanics, IAS/Park City Math. Ser.16, Amer. Math. Soc., Providence, 2009, 191-230. · Zbl 1180.82001
[50] R. Kenyon, A. Okounkov, and S. Sheffield,Dimers and amoebae, Ann. of Math. (2)163(2006), 1019-1056. · Zbl 1154.82007
[51] A. N. Kirillov and N. Y. Reshetikhin,Exact solution of the integrable \(XXZ\) Heisenberg model with arbitrary spin, I: The ground state and the excitation spectrum, J. Phys. A.20, no. 6 (1987), 1565-1585.
[52] P. P. Kulish, N. Y. Reshetikhin, and E. K. Sklyanin,Yang-Baxter equation and representation theory, I, Lett. Math. Phys.5(1981), 393-403. · Zbl 0502.35074
[53] C. Landim, J. Quastel, M. Salmhofer, and H.-T. Yau,Superdiffusivity of asymmetric exclusion process in dimensions one and two, Comm. Math. Phys.244(2004), 455-481. · Zbl 1064.60164
[54] E. H. Lieb,Residual entropy of square ice, Phys. Rev. Lett.162(1) (1967), 162-172.
[55] T. M. Liggett,Coupling the simple exclusion process,Ann. Probab.4(1976), 339-356. · Zbl 0339.60091
[56] T. M. Liggett,Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Grundlehren Math. Wiss.324, Springer, Berlin, 1999. · Zbl 0949.60006
[57] J. MacDonald, J. Gibbs, and A. Pipkin,Kinetics of biopolymerization on nucleic acid templates, Biopolymers6(1968), 1-25.
[58] J. Neergard and M. den Nijs,Crossover scaling functions in one dimensional dynamic growth crystals, Phys. Rev. Lett.74(5) (1995), 730-733.
[59] I. M. Nolden,The asymmetric six-vertex model, J. Statist. Phys.67(1992), 155-201. · Zbl 0900.82026
[60] 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), 581-603. · Zbl 1009.05134
[61] L. Pauling,The structure and entropy of ice and of other crystals with some randomness of atomic arrangement, J. Am. Chem. Soc.57(1935), 2680-2684.
[62] A. M. Povolotsky,On integrability of zero-range chipping models with factorized steady state, J. Phys. A46, no. 46 (2013), art. ID 465205. · Zbl 1290.82022
[63] M. Prähofer and H. Spohn, “Current fluctuations for the totally asymmetric simple exclusion process” inIn and Out of Equilibrium (Mambucaba, 2000), Progr. Probab.51, Birkhäuser, Boston, 2002, 185-204. · Zbl 1015.60093
[64] J. Quastel, “Introduction to KPZ” inCurrent Developments in Mathematics, 2011, Int. Press, Somerville, Mass., 2012, 125-194. · Zbl 1316.60019
[65] J. Quastel and B. Valkó, “A note on the diffusivity of finite-range asymmetric exclusion processes on \(\mathbb{Z} \)” inIn and Out of Equilibrium, 2, Progr. Probab.60, Birkhäuser, Basel, 2008, 543-549. · Zbl 1173.82341
[66] J. Quastel and B. Valkó,\(t^{1/3}\) superdiffusivity of finite-range asymmetric exclusion processes on \(\mathbb{Z} \), Comm. Math. Phys.273(2007), 379-394. · Zbl 1127.60091
[67] N. Reshetikhin, “Lectures on the integrability of the six-vertex model” inExact Methods in Low-Dimensional Statistical Physics and Quantum Computing, Oxford Univ. Press, Oxford, 2010, 197-266. · Zbl 1202.82022
[68] N. Reshetikhin and K. Palamarchuk, “The 6-vertex model with fixed boundary conditions” inProceedings of Bethe Ansatz: 75 Years Later, Proc. of Sci., Trieste, 2006, no. 12.
[69] N. Reshetikhin and A. Sridhar,Limit shapes of the stochastic six-vertex model, preprint,arXiv:1609.01756v1[math-ph]. · Zbl 1377.82022
[70] S. Sheffield,Random Surfaces, Astérisque304, Soc. Math. France, Paris, 2005. · Zbl 1104.60002
[71] J. Shore and D. J. Bukman,Coexistence point in the six-vertex model and the crystal shape of FCC materials, Phys. Rev. Lett.72(5) (1994), 604-607.
[72] J. C. Slater,Theory of transition in \(\text{KH}_2\text{PO}_4 \), J. Chem. Phys.9(1941), 16-33.
[73] F. Spitzer,Interaction of Markov processes, Adv. Math.5(1970), 246-290. · Zbl 0312.60060
[74] H. Spohn,Large Scale Dynamics of Interacting Particles, Springer, Berlin, 1991. · Zbl 0742.76002
[75] B. Sutherland, C. N. Yang, and C. P. Yang,Exact solution of a model of two-dimensional ferroelectrics in an arbitrary external electric field, Phys. Rev. Lett.19(10) (1967), 588-591.
[76] C. A. Tracy and H. Widom,A Fredholm determinant representation in ASEP, J. Stat. Phys.132(2008), 291-300. · Zbl 1144.82045
[77] C. A. Tracy and H. Widom,Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys.279(2008), 815-844. · Zbl 1148.60080
[78] C. A. Tracy and H. Widom,Asymptotics in ASEP with step initial condition, Comm. Math. Phys.290(2009), 129-154. · Zbl 1184.60036
[79] C. A. Tracy and H. Widom,On ASEP with step Bernoulli initial condition, J. Stat. Phys.137(2009), 825-838. · Zbl 1188.82043
[80] C. A. Tracy and H. Widom,Formulas for ASEP with two-sided Bernoulli initial condition, J. Stat. Phys.140(2010), 619-634. · Zbl 1197.82079
[81] H. van Beijern, R. Kutner, and H. Spohn,Excess noise for driven diffusive systems, Phys. Rev. Lett.54(18) (1985), 2026-2029.
[82] J. F. van Deijen,On the Plancherel formula for the (discrete) Laplacian in a Weyl chamber with repulsive boundary conditions at the walls, Ann. Henri Poincaré5(2004), 135-168. · Zbl 1062.81050
[83] P. Zinn-Justin,Six-Vertex, Loop and Tiling Models: Integrability and Combinatorics, Lambert Academic, 2010. · Zbl 1202.82024
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