Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

KPZ and Airy limits of Hall–Littlewood random plane partitions

Evgeni Dimitrov

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In this paper we consider a probability distribution $\mathbb{P}^{q,t}_{\mathrm{HL}}$ on plane partitions, which arises as a one-parameter generalization of the standard $q^{\mathrm{volume}}$ measure. This generalization is closely related to the classical multivariate Hall–Littlewood polynomials, and it was first introduced by Vuletić in (Trans. Am. Math. Soc. 361 (2009) 2789–2804).

We prove that as the plane partitions become large ($q$ goes to $1$, while the Hall–Littlewood parameter $t$ is fixed), the scaled bottom slice of the random plane partition converges to a deterministic limit shape, and that one-point fluctuations around the limit shape are asymptotically given by the GUE Tracy–Widom distribution. On the other hand, if $t$ simultaneously converges to its own critical value of $1$, the fluctuations instead converge to the one-dimensional Kardar–Parisi–Zhang (KPZ) equation with the so-called narrow wedge initial data.

The algebraic part of our arguments is closely related to the formalism of Macdonald processes (Probab. Theory Relat. Fields 158 (1) (2014) 225–400). The analytic part consists of detailed asymptotic analysis of the arising Fredholm determinants.


Dans cet article, nous considérons une distribution de probabilité $\mathbb{P}^{q,t}_{\mathrm{HL}}$ sur les partitions planes, qui apparaît comme une généralisation à un paramètre de la mesure standard $q^{\mathrm{volume}}$. Cette généralisation est étroitement reliée aux classiques polynômes multivariés de Hall–Littlewood, et a été introduite pour la première fois par Vuletić dans (Trans. Am. Math. Soc. 361 (2009) 2789–2804).

Nous montrons que lorsque la partition plane devient grande ($q$ tend vers $1$, alors que le paramètre de Hall–Littlewood $t$ est fixé), la partie inférieure proprement renormalisée de la partition plane converge vers une forme limite déterministe, et que les fluctuations à un point autour de la forme limite sont asymptotiquement données par la distribution du GUE Tracy–Widom. Par contre, si $t$ converge vers sa propre valeur critique $1$, les fluctuations convergent cette fois vers l’équation unidimensionnelle de Kardar–Parisi–Zhang (KPZ) avec les conditions initiales à courte bande (narrow wedge data).

La partie algébrique de notre argument est étroitement reliée au formalisme des processus de Macdonald (Probab. Theory Relat. Fields 158 (1) (2014) 225–400). La partie analytique consiste en une analyse asymptotique détaillée des déterminants de Fredholm associés.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 640-693.

Received: 18 November 2016
Accepted: 13 December 2016
First available in Project Euclid: 25 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 82B23: Exactly solvable models; Bethe ansatz

Hall–Littlewood polynomial Plane partition GUE Tracy–Widom distribution KPZ equation


Dimitrov, Evgeni. KPZ and Airy limits of Hall–Littlewood random plane partitions. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 640--693. doi:10.1214/16-AIHP817.

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  • [1] T. Alberts, K. Khanin and J. Quastel. The intermediate disorder regime for directed polymers in dimension $1+1$. Ann. Probab. 42 (2014) 1212–1256.
  • [2] G. Amir, I. Corwin and J. Quastel. Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math. 64 (2011) 466–537.
  • [3] G. Andrews, R. Askey and R. Roy. Special Functions. Cambridge University Press, Cambridge, 2000.
  • [4] M. Balázs, J. Quastel and T. Seppäläinen. Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers. J. Amer. Math. Soc. 24 (2011) 683–708.
  • [5] G. Barraquand. A phase tansition for q-TASEP with a few slower particles. Stochastic Process. Appl. 125 (2015) 2674–2699.
  • [6] L. Bertini and G. Giacomin. Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 (1997) 571–607.
  • [7] V. Bogachev. Measure Theory Vol. I. Springer, Berlin, 2007.
  • [8] A. Borodin. Stochastic higher spin six vertex model and Macdonald measures, 2016. Preprint. Available at arXiv:1608.01553.
  • [9] A. Borodin, A. Bufetov and I. Corwin. Directed random polymers via nested contour integrals. Ann. Physics 368 (2016) 191–247.
  • [10] A. Borodin and I. Corwin. Macdonald processes. Probab. Theory Relat. Fields 158 (2014).
  • [11] A. Borodin and I. Corwin. Discrete time $q$-TASEPs. Int. Math. Res. Notices (2015).
  • [12] 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) 1129–1214.
  • [13] A. Borodin, I. Corwin, V. Gorin and S. Shakirov. Observables of Macdonald processes. Trans. Amer. Math. Soc. 368 (2016) 1517–1558.
  • [14] A. Borodin, I. Corwin and D. Remenik. Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity. Comm. Math. Phys. 324 (2013) 215–232.
  • [15] A. Borodin, I. Corwin and T. Sasamoto. From duality to determinants for $q$-TASEP and ASEP. Ann. Probab. 42 (2014) 2314–2382.
  • [16] A. Borodin and L. Petrov. Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. 300 (2016) 71–155.
  • [17] P. Calabrese, P. L. Doussal and A. Rosso. Free-energy distribution of the directed polymer at high temperature. Europhys. Lett. 90 (2010) 20002.
  • [18] R. Cerf and R. Kenyon. The low-temperature expansion of the Wulff crystal in the 3d Ising model. Comm. Math. Phys. 222 (2001) 147–179.
  • [19] I. Corwin. The Kardar–Parisi–Zhang equation and universality class. Random Matrices: Theory Appl. 1 (2012).
  • [20] I. Corwin. Macdonald processes, quantum integrable systems and the Kardar–Parisi–Zhang universality class Proceedings of the International Congress of Mathematicians, 2014. Preprint. Available at arXiv:1403.6877.
  • [21] I. Corwin and A. Hammond. Brownian Gibbs property for Airy line ensembles. Invent. Math. 195 (2014) 441–508.
  • [22] I. Corwin and A. Hammond. KPZ line ensemble. Probab. Theory Relat. Fields (2015). Advanced online publication.
  • [23] I. Corwin and L. Petrov. The $q$-PushASEP: A new integrable model for traffic in $1$ + $1$ dimension. J. Stat. Phys. 160 (2015) 1005–1026.
  • [24] E. Dimitrov. KPZ and Airy limits of Hall–Littlewood random plane partitions, 2016. Preprint. Available at arXiv:1602.00727.
  • [25] P. L. Ferrari. The universal Airy$_{1}$ and Airy$_{2}$ processes in the Totally Asymmetric Simple Exclusion Process, integrable systems and random matrices. In Honor of Percy Deift 321–332. Contemporary Mathematics Am. Math. Soc., Providence, 2008.
  • [26] P. L. Ferrari and B. Veto. Tracy–Widom asymptotics for q-TASEP. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 1465–1485.
  • [27] P. L. Ferrari and H. Spohn. Step fluctuations for a faceted crystal. J. Stat. Phys. 113 (2003) 1–46.
  • [28] D. Forster, D. R. Nelson and M. J. Stephen. Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A (3) 16 (1977) 732–749.
  • [29] O. Kallenberg. Foundations of Modern Probability. Springer, New York, 1997.
  • [30] M. Kardar, G. Parisi and Y. C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986) 889–892.
  • [31] R. Kenyon and A. Okounkov. Limit shapes and the complex Burgers equation. Acta Math. 199 (2007) 263–302.
  • [32] P. D. Lax. Functional Analysis. Wiley, New York, 2002.
  • [33] I. G. Macdonald. Symmetric Functions and Hall Polynomials, 2nd edition. Oxford University Press Inc., New York, 1995.
  • [34] C. Mueller. On the support of solutions to the heat equation with noise. Stochastics: An International Journal of Probability and Stochastic Processes 37 (1991) 225–245.
  • [35] C. Mueller and R. Nualart. Regularity of the density for the stochastic heat equation. Electron. J. Probab. 13 (2008) 2248–2258.
  • [36] 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.
  • [37] M. Prähofer and H. Spohn. Scale invariance of the PNG Droplet and the Airy process. J. Stat. Phys. 108 (2002) 1071–1106.
  • [38] J. Quastel. Introduction to KPZ. In Current Developments in Mathematics. Int. Press, Sommerville, MA, 2011. Available at
  • [39] J. Quastel and D. Remenik. Airy processes and variational problems, topics in Percolative and Disordered Systems. Springer Proc. Math. Stat. 69 (2014) 121–171.
  • [40] T. Sasamoto and H. Spohn. The $1$ + $1$-dimensional Kardar–Parisi–Zhang equation and its universality class. J. Stat. Mech.-Theory E. P11013 (2010).
  • [41] T. Sasamoto and H. Spohn. Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834 (2010) 523–542.
  • [42] B. Simon. Trace Ideals and Their Applications, 2nd edition. AMS, Providence, 2005.
  • [43] E. Stein and R. Shakarchi. Complex Analysis. Princeton University Press, Princeton, 2003.
  • [44] C. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994) 151–174.
  • [45] M. Vuletić. The shifted Schur process and asymptotics of large random strict plane partitions. Int. Math. Res. Notices 14 (2007).
  • [46] M. Vuletić. A generalization of MacMahon’s formula. Trans. Amer. Math. Soc. 361 (2009) 2789–2804.