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HALF-SPACE MACDONALD PROCESSES

Part of: Applications to specific types of physical systems Equilibrium statistical mechanics Miscellaneous topics - Partial differential equations Special processes Algebraic combinatorics Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  26 May 2020

GUILLAUME BARRAQUAND ,
ALEXEI BORODIN  and
IVAN CORWIN
GUILLAUME BARRAQUAND
Affiliation:
Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France; [email protected]
ALEXEI BORODIN
Affiliation:
Department of Mathematics, MIT, Cambridge, USA Institute for Information Transmission Problems, Moscow, Russia; [email protected]
IVAN CORWIN
Affiliation:
Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA; [email protected]

Abstract

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Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts.

We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.

MSC classification

Primary: 60B99: None of the above, but in this section
Secondary: 05E05: Symmetric functions and generalizations 82B23: Exactly solvable models; Bethe ansatz 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 82D60: Polymers 60K35: Interacting random processes; statistical mechanics type models; percolation theory 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
Probability
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e11
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Aggarwal, A., ‘Dynamical stochastic higher spin vertex models’, Selecta Math. (N.S.) 24(3) (2018), 2659–2735.CrossRefGoogle Scholar
Aggarwal, A. and Borodin, A., ‘Phase transitions in the ASEP and stochastic six-vertex model’, Ann. Probab. 47(2) (2019), 613–689.CrossRefGoogle Scholar
Alberts, T., Khanin, K. and Quastel, J., ‘The intermediate disorder regime for directed polymers in dimension 1 + 1’, Ann. Probab. 42(3) (2014), 1212–1256.CrossRefGoogle Scholar
Amir, G., Corwin, I. and Quastel, J., ‘Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions’, Comm. Pure Appl. Math. 64(4) (2011), 466–537.CrossRefGoogle Scholar
Andrews, G. E., Askey, R. and Roy, R., Special Functions, (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
Baik, J., Barraquand, G., Corwin, I. and Suidan, T., ‘Facilitated exclusion process’, inThe Abel Symposium: Computation and Combinatorics in Dynamics, Stochastics and Control (Springer International Publishing, 2018), 1–35.Google Scholar
Baik, J., Barraquand, G., Corwin, I. and Suidan, T., ‘Pfaffian schur processes and last passage percolation in a half-quadrant’, Ann. Probab. 46(6) (2018), 3015–3089.CrossRefGoogle Scholar
Baik, J. and Rains, E. M., ‘Algebraic aspects of increasing subsequences’, Duke Math. J. 109(1) (2001), 1–65.CrossRefGoogle Scholar
Baik, J. and Rains, E. M., ‘The asymptotics of monotone subsequences of involutions’, Duke Math. J. 109(2) (2001), 205–281.Google Scholar
Baik, J. and Rains, E. M., ‘Symmetrized random permutations’, inRandom Matrix Models and their Applications, Mathematical Sciences Research Institute Publications, 40 (Cambridge University Press, Cambridge, 2001), 1–19.Google Scholar
Barraquand, G., ‘A phase transition for q-TASEP with a few slower particles’, Stochastic Process. Appl. 125(7) (2015), 2674–2699.CrossRefGoogle Scholar
Barraquand, G., Borodin, A., Corwin, I. and Wheeler, M., ‘Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process’, Duke Math. J. 167(13) (2018), 2457–2529.CrossRefGoogle Scholar
Barraquand, G. and Corwin, I., ‘Random-walk in beta-distributed random environment’, Probab. Theory Related Fields 167(3) (2017), 1057–1116.CrossRefGoogle Scholar
Baudoin, F. and O’Connell, N., ‘Exponential functionals of Brownian motion and class-one Whittaker functions’, Ann. Inst. Henri Poincaré 47(4) 1096–1120.CrossRefGoogle Scholar
Bertini, L. and Giacomin, G., ‘Stochastic Burgers and KPZ equations from particle systems’, Comm. Math. Phys. 183(3) (1997), 571–607.CrossRefGoogle Scholar
Betea, D., Bouttier, J., Nejjar, P. and Vuletić, M., ‘The free boundary Schur process and applications I’, Ann. Henri Poincaré 1 9 (2018), 3663–3742. Springer.CrossRefGoogle Scholar
Betea, D., Wheeler, M. and Zinn-Justin, P., ‘Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures’, J. Algebraic Combin. 42(2) (2015), 555–603.CrossRefGoogle Scholar
Bisi, E. and Zygouras, N., ‘Point-to-line polymers and orthogonal Whittaker functions’, Trans. Amer. Math. Soc. 371(12) (2019), 8339–8379.CrossRefGoogle Scholar
Borodin, A., ‘Limit Jordan normal form of large triangular matrices over a finite field’, Funct. Anal. Appl. 29(4) (1995), 279–281.CrossRefGoogle Scholar
Borodin, A., ‘The law of large numbers and the central limit theorem for the Jordan normal form of large triangular matrices over a finite field’, J. Math. Sci. 96(5) (1999), 3455–3471.Google Scholar
Borodin, A., ‘Schur dynamics of the Schur processes’, Adv. Math. 4(228) (2011), 2268–2291.CrossRefGoogle Scholar
Borodin, A., ‘Integrable probability’, In Proceedings of the ICM, Seoul (2014).Google Scholar
Borodin, A., ‘On a family of symmetric rational functions’, Adv. Math. 306 (2017), 973–1018.CrossRefGoogle Scholar
Borodin, A., ‘Stochastic higher spin six vertex model and Macdonald measures’, J. Math. Phys. 59(2) (2018), 023301.CrossRefGoogle Scholar
Borodin, A., Bufetov, A. and Corwin, I., ‘Directed random polymers via nested contour integrals’, Ann. Phys. 368 (2016), 191–247.CrossRefGoogle Scholar
Borodin, A., Bufetov, A. and Wheeler, M., ‘Between the stochastic six vertex model and Hall–Littlewood processes’, J. Combin. Theory Ser. A (2016), arXiv:1611.09486.Google Scholar
Borodin, A. and Corwin, I., ‘Discrete time q-taseps’, Int. Math. Res. Not. IMRN 2015(2) (2013), rnt206.Google Scholar
Borodin, A. and Corwin, I., ‘Macdonald processes’, Probab. Theory Related Fields 158(1–2) (2014), 225–400.CrossRefGoogle Scholar
Borodin, A. and Corwin, I., ‘Discrete time q-TASEPs’, Int. Math. Res. Not. IMRN (2) (2015), 499–537.CrossRefGoogle Scholar
Borodin, A., Corwin, I. and Ferrari, P., ‘Free energy fluctuations for directed polymers in random media in 1+ 1 dimension’, Comm. Pure Appl. Math. 67(7) (2014), 1129–1214.CrossRefGoogle Scholar
Borodin, A., Corwin, I., Ferrari, P. and Vető, B., ‘Height fluctuations for the stationary KPZ equation’, Math. Phys. Anal. Geom. 18(1) (2015), Art. 20, 95.CrossRefGoogle Scholar
Borodin, A., Corwin, I. and Gorin, V., ‘Stochastic six-vertex model’, Duke Math. J. 165(3) (2016), 563–624.CrossRefGoogle Scholar
Borodin, A., Corwin, I., Gorin, V. and Shakirov, S., ‘Observables of Macdonald processes’, Trans. Amer. Math. Soc. 368(3) (2016), 1517–1558.CrossRefGoogle Scholar
Borodin, A., Corwin, I., Petrov, L. and Sasamoto, T., ‘Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz’, Comm. Math. Phys. 339(3) (2015), 1167–1245.CrossRefGoogle Scholar
Borodin, A., Corwin, I., Petrov, L. and Sasamoto, T., ‘Spectral theory for the q-Boson particle system’, Compos. Math. 151 (2015), 1–67.CrossRefGoogle Scholar
Borodin, A., Corwin, I. and Remenik, D., ‘Log-gamma polymer free energy fluctuations via a Fredholm determinant identity’, Comm. Math. Phys. 324(1) (2013), 215–232.CrossRefGoogle Scholar
Borodin, A., Corwin, I. and Remenik, D., ‘A classical limit of Noumi’s q-integral operator’, SIGMA Symmetry Integrability Geom. Methods Appl. 11(0) (2015).Google Scholar
Borodin, A., Corwin, I. and Sasamoto, T., ‘From duality to determinants for q-TASEP and ASEP’, Ann. Probab. 42(6) (2014), 2314–2382.CrossRefGoogle Scholar
Borodin, A., Corwin, I. and Toninelli, F., ‘Stochastic heat equation limit of a (2 + 1)-d growth model’, Comm. Math. Phys. 350(3) (2017), 957–984.CrossRefGoogle Scholar
Borodin, A. and Ferrari, P. L., ‘Anisotropic growth of random surfaces in 2+ 1 dimensions’, Comm. Math. Phys. 325(2) (2014), 603–684.CrossRefGoogle Scholar
Borodin, A. and Ferrari, P. L., ‘Anisotropic growth of random surfaces in 2 + 1 dimensions’, Comm. Math. Phys. 325(2) (2014), 603–684.CrossRefGoogle Scholar
Borodin, A. and Gorin, V., Lectures on Integrable Probability, Lecture notes, (St. Petersburg School in Probability and Statistical Physics, 2012), arXiv:1212.3351.Google Scholar
Borodin, A. and Gorin, V., ‘General 𝛽-Jacobi corners process and the Gaussian free field’, Comm. Pure Appl. Math. 68(10) (2015), 1774–1844.CrossRefGoogle Scholar
Borodin, A. and Olshanski, G., ‘The ASEP and determinantal point processes’, Comm. Math. Phys. 353(2) (2017), 853–903.CrossRefGoogle Scholar
Borodin, A. and Petrov, L., ‘Integrable probability: From representation theory to macdonald processes’, Probab. Surv. 11 (2014), 1–58.CrossRefGoogle Scholar
Borodin, A. and Petrov, L., ‘Nearest neighbor Markov dynamics on Macdonald processes’, Adv. Math. 300 (2016), 71–155.CrossRefGoogle Scholar
Borodin, A. and Petrov, L., ‘Higher spin six vertex model and symmetric rational functions’, Selecta Math. (N.S.) 24(2) (2018), 751–874.CrossRefGoogle Scholar
Borodin, A. and Rains, E. M., ‘Eynard–Mehta theorem, Schur process, and their Pfaffian analogs’, J. Stat. Phys. 121(3–4) (2005), 291–317.CrossRefGoogle Scholar
Bufetov, A. and Matveev, K., ‘Hall–Littlewood RSK field’, Selecta Math. (N.S.) 24(5) (2018), 4839–4884.CrossRefGoogle Scholar
Bufetov, A. and Petrov, L., ‘Law of large numbers for infinite random matrices over a finite field’, Selecta Math. 21(4) (2015), 1271–1338.CrossRefGoogle Scholar
Calabrese, P., Le Doussal, P. and Rosso, A., ‘Free-energy distribution of the directed polymer at high temperature’, Europhys. Lett. 90(2)20002 (2010).CrossRefGoogle Scholar
Corwin, I., ‘The Kardar–Parisi–Zhang equation and universality class’, Random Matrices Theory Appl. 1(01) (2012).CrossRefGoogle Scholar
Corwin, I., ‘Macdonald processes, quantum integrable systems and the Kardar–Parisi–Zhang universality class’, Preprint, 2014, arXiv:1403.6877.Google Scholar
Corwin, I. and Dimitrov, E., ‘Transversal fluctuations of the ASEP stochastic six vertex model, and Hall–Littlewood Gibbsian line ensembles’, Comm. Math. Phys. 363(2) (2018), 435–501.CrossRefGoogle Scholar
Corwin, I. and Hammond, A., ‘Brownian Gibbs property for Airy line ensembles’, Invent. Math. 195(2) (2014), 441–508.CrossRefGoogle Scholar
Corwin, I. and Hammond, A., ‘KPZ line ensemble’, Probab. Theory Related Fields 166(1–2) (2016), 67–185.CrossRefGoogle Scholar
Corwin, I., O’Connell, N., Seppäläinen, T. and Zygouras, N., ‘Tropical combinatorics and Whittaker functions’, Duke Math. J. 163(3) (2014), 513–563.CrossRefGoogle Scholar
Corwin, I. and Petrov, L., ‘The q-PushASEP: A new integrable model for traffic in 1+ 1 dimension’, J. Stat. Phys. 160(4) (2015), 1005–1026.CrossRefGoogle Scholar
Corwin, I., Seppäläinen, T. and Shen, H., ‘The strict-weak lattice polymer’, J. Stat. Phys. 160 (2015), 1027–1053.CrossRefGoogle Scholar
Corwin, I. and Shen, H., ‘Open ASEP in the weakly asymmetric regime’, Comm. Pure Appl. Math. 71(10) (2018), 2065–2128.CrossRefGoogle Scholar
Corwin, I. and Toninelli, F., ‘Stationary measure of the driven two-dimensional q-Whittaker particle system on the torus’, Electron. Commun. Probab. 21 (2016).CrossRefGoogle Scholar
Dembo, A. and Tsai, L., ‘Weakly asymmetric non-simple exclusion process and the Kardar–Parisi–Zhang equation’, Comm. Math. Phys. 341(1) (2016), 219–261.CrossRefGoogle Scholar
Diaconis, P. and Fill, J. A., ‘Strong stationary times via a new form of duality’, Ann. Probab. 18(4) (1990), 1483–1522.CrossRefGoogle Scholar
Diehl, J., Gubinelli, M. and Perkowski, N., ‘The Kardar–Parisi–Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions’, Comm. Math. Phys. 354(2) (2017), 549–589.CrossRefGoogle Scholar
Dimitrov, E., ‘KPZ and Airy limits of Hall–Littlewood random plane partitions’, Ann. Inst. Henri Poincaré 54(2) (2018), 640–693.CrossRefGoogle Scholar
Dotsenko, V., ‘Replica Bethe ansatz derivation of the Tracy–Widom distribution of the free energy fluctuations in one-dimensional directed polymers’, J. Stat. Mech. 2010(07) (2010), P07010.Google Scholar
Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J. and Yanagida, S., ‘A commutative algebra on degenerate CP1 and Macdonald polynomials’, J. Math. Phys. 50(9) (2009), 095215.CrossRefGoogle Scholar
Ferrari, P. L. and Vető, B., ‘Tracy–Widom asymptotics for q-TASEP’, Ann. Inst. Henri Poincaré Probab. Stat. 51(4) (2015), 1465–1485.CrossRefGoogle Scholar
Forrester, P. J., Nagao, T. and Rains, E. M., ‘Correlation functions for random involutions’, Int. Math. Res. Not. IMRN 2006 (2006), 89796.Google Scholar
Fulman, J., ‘Random matrix theory over finite fields’, Bull. Amer. Math. Soc. (N.S.) 39(1) (2002), 51–85.CrossRefGoogle Scholar
Georgiou, N., Rassoul-Agha, F. and Seppäläinen, T., ‘Variational formulas and cocycle solutions for directed polymer and percolation models’, Comm. Math. Phys. 346(2) (2016), 741–779.CrossRefGoogle Scholar
Georgiou, N., Rassoul-Agha, F., Seppäläinen, T. and Yilmaz, A., ‘Ratios of partition functions for the log-gamma polymer’, Ann. Probab. 43(5) (2015), 2282–2331.CrossRefGoogle Scholar
Georgiou, N. and Seppäläinen, T., ‘Large deviation rate functions for the partition function in a log-gamma distributed random potential’, Ann. Probab. 41(6) (2013), 4248–4286.CrossRefGoogle Scholar
Gerasimov, A., Lebedev, D. and Oblezin, S., ‘Baxter operator and Archimedean Hecke algebra’, Comm. Math. Phys. 284(3) (2008), 867–896.CrossRefGoogle Scholar
Gerasimov, A., Lebedev, D. and Oblezin, S., ‘On q-deformed glℓ+1 -whittaker function’, Comm. Math. Phys. 294(1) (2009), 97.CrossRefGoogle Scholar
Ghosal, P., ‘Correlation functions of the Pfaffian Schur process using Macdonald difference operators’, Preprint, 2017, arXiv:1705.05859.Google Scholar
Ghosal, P., ‘Hall–Littlewood-PushTASEP and its KPZ limit’, Preprint, 2017, arXiv:1701.07308.Google Scholar
Givental, A., ‘Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture’, Amer. Math. Soc. Transl. Ser. 2 180 (1997), 103–115.Google Scholar
Gnedin, A. and Olshanski, G., ‘A q-analogue of de Finetti’s theorem’, Electron. J. Combin. 16(1) (2009), Research Paper 78, 16.CrossRefGoogle Scholar
Gorin, V., Kerov, S. and Vershik, A., ‘Finite traces and representations of the group of infinite matrices over a finite field’, Adv. Math. 254 (2014), 331–395.CrossRefGoogle Scholar
Gorin, V. and Shkolnikov, M., ‘Multilevel Dyson Brownian motions via Jack polynomials’, Probab. Theory Related Fields 163(3–4) (2015), 413–463.CrossRefGoogle Scholar
Gorin, V. and Zhang, L., ‘Interlacing adjacent levels of 𝛽-Jacobi corners processes’, Probab. Theory Related Fields 172(3–4) (2018), 915–981.CrossRefGoogle Scholar
Grange, P., ‘Log-gamma directed polymer with one free end via coordinate Bethe ansatz’, J. Stat. Mech. Theory Exp. 2017(7) (2017), 073102.CrossRefGoogle Scholar
Gueudré, T. and Le Doussal, P., ‘Directed polymer near a hard wall and KPZ equation in the half-space’, Europhys. Lett. 100(2) (2012), 26006.CrossRefGoogle Scholar
Gwa, L. and Spohn, H., ‘Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian’, Phys. Rev. Lett. 68(6) (1992), 725.CrossRefGoogle Scholar
Hairer, M. and Quastel, J., ‘A class of growth models rescaling to KPZ’, Forum Math., Pi 6 (2018).CrossRefGoogle Scholar
Hashizume, M., ‘Whittaker functions on semisimple Lie groups’, Hiroshima Math. J. 12(2) (1982), 259–293.CrossRefGoogle Scholar
Ishii, T. and Stade, E., ‘New formulas for Whittaker functions on GL(n, ℝ)’, J. Funct. Anal. 244(1) (2007), 289–314.CrossRefGoogle Scholar
Jorgensen, B., Statistical Properties of the Generalized Inverse Gaussian Distribution, Lecure Notes in Statistics, 9, (Springer, 1982).CrossRefGoogle Scholar
Kardar, M., Parisi, G. and Zhang, Y., ‘Dynamic scaling of growing interfaces’, Phys. Rev. Lett. 56 (1986), 889–892.CrossRefGoogle ScholarPubMed
Knuth, D., ‘Permutations, matrices, and generalized young tableaux’, Pacific J. Math. 34(3) (1970), 709–727.CrossRefGoogle Scholar
Krishnan, A. and Quastel, J., ‘Tracy–Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder’, Ann. Appl. Probab. 28(6) (2018), 3736–3764.CrossRefGoogle Scholar
Macdonald, I. G., Symmetric Functions and Hall Polynomials, vol. 354, (Clarendon Press, Oxford, 1995).Google Scholar
Matveev, K., ‘Macdonald-positive specializations of the algebra of symmetric functions: proof of the Kerov conjecture’, Ann. of Math. (2) 189(1) (2019), 277–316.CrossRefGoogle Scholar
Matveev, K. and Petrov, L., ‘q-randomized Robinson–Schensted–Knuth correspondences and random polymers’, Ann. Inst. H. Poincaré D 4(1) (2017), 1–123.Google Scholar
Nguyen, V. and Zygouras, N., ‘Variants of geometric RSK, geometric PNG, and the multipoint distribution of the log-gamma polymer’, Int. Math. Res. Not. IMRN 2017(15) (2017), 4732–4795.Google Scholar
Noumi, M. and Sano, A., An infinite family of higher-order difference operators that commute with Ruijsenaars operators of type A, unpublished.Google Scholar
Noumi, M. and Shiraishi, J..Google Scholar
Nteka, I., (2016), ‘Positive temperature dynamics on Gelfand–Tsetlin patterns restricted by wall’, PhD Thesis, University of Warwick.Google Scholar
O’Connell, N., ‘Directed polymers and the quantum Toda lattice’, Ann. Probab. 40(2) (2012), 437–458.CrossRefGoogle Scholar
O’Connell, N., Whittaker Functions and Related Stochastic Processes, (MSRI Publications, 2014), arXiv:1201.4849 65.Google Scholar
O’Connell, N. and Ortmann, J., ‘Tracy–Widom asymptotics for a random polymer model with gamma-distributed weights’, Electron. J. Probab. 20(25) (2015), 1–18.Google Scholar
O’Connell, N. and Pei, Y., ‘A q-weighted version of the Robinson–Schensted algorithm’, Electron. J. Probab. 18 (2013).Google Scholar
O’Connell, N., Seppäläinen, T. and Zygouras, N., ‘Geometric RSK correspondence, Whittaker functions and symmetrized random polymers’, Invent. Math. 197(2) (2014), 361–416.CrossRefGoogle Scholar
O’Connell, N. and Yor, M., ‘Brownian analogues of Burke’s theorem’, Stochastic Process. Appl. 96(2) (2001), 285–304.CrossRefGoogle Scholar
Okounkov, A., ‘Infinite wedge and random partitions’, Selecta Math. (N.S.) 7(1) (2001), 57–81.CrossRefGoogle Scholar
Okounkov, A. and Reshetikhin, N., ‘Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram’, J. Amer. Math. Soc. 16(3) (2003), 581–603.CrossRefGoogle Scholar
Orr, D. and Petrov, L., ‘Stochastic higher spin six vertex model and q-TASEPs’, Adv. Math. 317 (2017), 473–525.CrossRefGoogle Scholar
Ortmann, J., Quastel, J. and Remenik, D., ‘A Pfaffian representation for flat ASEP’, Comm. Pure Appl. Math. 70(1) (2017), 3–89.CrossRefGoogle Scholar
Parekh, S., ‘The KPZ limit of ASEP with boundary’, Comm. Math. Phys. 365(2) (2019), 569–649.CrossRefGoogle Scholar
Povolotsky, A. M., ‘On the integrability of zero-range chipping models with factorized steady states’, J. Phys. A 46(46) (2013), 465205.Google Scholar
Quastel, J., Introduction to KPZ, available online at www.math.toronto.edu/quastel/survey.pdf, 2012.CrossRefGoogle Scholar
Rains, E. M., ‘Correlation functions for symmetrized increasing subsequences’, Preprint, 2000, arXiv:math/0006097.Google Scholar
Rains, E. M., ‘Multivariate quadratic transformations and the interpolation kernel’, SIGMA 14(0) (2018), 19–69.Google Scholar
Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, Vol. 293, (Springer Science & Business Media, 2013).Google Scholar
Sasamoto, T. and Imamura, T., ‘Fluctuations of the one-dimensional polynuclear growth model in half-space’, J. Stat. Phys. 115(3–4) (2004), 749–803.CrossRefGoogle Scholar
Sasamoto, T. and Spohn, H., ‘Exact height distributions for the KPZ equation with narrow wedge initial condition’, Nuclear Phys. B 834(3) (2010), 523–542.CrossRefGoogle Scholar
Semenov-Tian-Shansky, M. A., ‘Quantization of open Toda lattices’, inDynamical Systems VII: Integrable Systems Nonholonomic Dynamical Systems, Vol. 16 (Springer, 1993), 226.Google Scholar
Seppäläinen, T., ‘Scaling for a one-dimensional directed polymer with boundary conditions’, Ann. Probab. 40(1) (2012), 19–73.CrossRefGoogle Scholar
Stade, E., ‘Mellin transforms of GL(n, ℝ) Whittaker functions’, Amer. J. Math. 123(1) (2001), 121–161.CrossRefGoogle Scholar
Thiery, T. and Le Doussal, P., ‘Log-gamma directed polymer with fixed endpoints via the Bethe ansatz replica’, J. Stat. Mech. 2014(10) (2014), P10018.CrossRefGoogle Scholar
Tracy, C. A. and Widom, H., ‘Asymptotics in ASEP with step initial condition’, Comm. Math. Phys. 290(1) (2009), 129–154.CrossRefGoogle Scholar
Tracy, C. A. and Widom, H., ‘The asymmetric simple exclusion process with an open boundary’, J. Math. Phys. 54(10) (2013), 103301.CrossRefGoogle Scholar
Tracy, C. A. and Widom, H., ‘The Bose gas and asymmetric simple exclusion process on the half-line’, J. Stat. Phys. 150(1) (2013), 1–12.CrossRefGoogle Scholar
Venkateswaran, V., ‘Symmetric and nonsymmetric Hall–Littlewood polynomials of type BC’, J. Algebraic Comb. 42 (2015), 331–364.CrossRefGoogle Scholar
Wheeler, M. and Zinn-Justin, P., ‘Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons’, Adv. Math. 299 (2016), 543–600.CrossRefGoogle Scholar
Wu, X., ‘Intermediate disorder regime for half-space directed polymers’, Preprint, 2018, arXiv:1804.09815.Google Scholar