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.
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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
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]
Babbitt D., Gutkin E.: The Plancherel formula for the infinite XXZ Heisenberg spin chain. Lett. Math. Phys. 20, 91–99 (1990)
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)
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]
Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Courier Dover Publications, Mineola (2007)
Bertini L.N. Cancrini: The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78(5–6), 1377–1401 (1995)
Bertini L., Giacomin G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)
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)
Borodin, A., Corwin, I.: Discrete time q-TASEPs. Intern. Math. Res. Notices (2013). arXiv:1305.2972 [math.PR]. doi:10.1093/imrn/rnt206
Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields 158(1–2), 225–400 (2014). arXiv:1111.4408 [math.PR]
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]
Borodin, A., Corwin, I., Ferrari, P., Veto, B.: Height fluctuations for the stationary KPZ equation (2014). arXiv:1407.6977 [math.PR]
Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. (2014). arXiv:1407.6729 [math.PR]
Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: Observables of Macdonald processes (2013). arXiv:1306.0659 [math.PR]
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]
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]
Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. (2013, to appear). arXiv:1305.5501 [math.PR]
Borodin, A., Petrov, L.: Integrable probability: From representation theory to Macdonald processes. Probab. Surv. 11, 1–58 (2014). arXiv:1310.8007 [math.PR]
Calabrese, P., Caux, J.S.: Dynamics of the attractive 1D Bose gas: analytical treatment from integrability. J. Stat. Mech. P08032 (2007)
Calabrese P., Le Doussal P., Rosso A.: Free-energy distribution of the directed polymer at high temperature. Eur. Phys. Lett. 90(2), 20002 (2010)
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]
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
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]
Dotsenko, V.: Bethe ansatz derivation of the Tracy–Widom distribution for one-dimensional directed polymers. Eur. Phys. Lett. 90(20003) (2010)
Dotsenko, V.: Universal randomness. Physics-Uspekhi 54(3), 259–280, (2011). arXiv:1009.3116 [cond-mat.stat-mech]
Faddeev, L.: How Algebraic Bethe Ansatz works for integrable model. Les-Houches lectures (1996). arXiv:hep-th/9605187
Ferrari, P., Veto, B.: Tracy–Widom asymptotics for q-TASEP. Ann. Inst. Henri Poincar Probab. Stat. (2013, to appear). arXiv:1310.2515 [math.PR]
Gaudin M.: Boundary energy of a Bose gas in one dimension. Phys. Rev. A 4, 386–394 (1971)
Gutkin, E.: Heisenberg-Ising spin chain: Plancherel decomposition and Chebyshev polynomials. Calogero–Moser–Sutherland Models. CRM Series in Mathematical Physics, pp. 177–192 (2000)
Gwa L.-H.H. Spohn: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844–854 (1992)
Heckman G.J., Opdam E.M.: Yang’s system of particles and Hecke algebras. Ann. Math. 145(1), 139–173 (1997)
Helgason S.: An analogue of the Paley–Wiener theorem for the Fourier transform on certain symmetric spaces. Math. Ann. 165, 297–308 (1966)
Imamura T., Sasamoto T., Spohn H.: KPZ, ASEP and Delta-Bose gas. J. Phys. Conf. Ser. 297, 012–016 (2011)
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)
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]
Lieb E.H.: The residual entropy of square ice. Phys. Rev. 162, 162–172 (1967)
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)
Liggett T.: Interacting Particle Systems. Springer, New York (1985)
Macdonald, I.G.: Spherical functions of p-adic type. Publ. Ramanujan Inst. 2 (1971)
Macdonald I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)
Macdonald, I.G.: Orthogonal polynomials associated with root systems. Sem. Lothar. Combin. 45(B45a) (2000)
Matveev, K., Petrov, L.: q-randomized Robinson–Schensted–Knuth correspondences and random polymers (2015). arXiv:1504.00666 [math.PR]
Moreno Flores, G.R., Quastel, J., Remenik, D.: In preparation (2015)
O’Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40(2), 437–458 (2012). arXiv:0910.0069 [math.PR]
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]
O’Connell N., Yor M.: Brownian analogues of Burke’s theorem. Stoch. Proc. Appl. 96(2), 285–304 (2001)
Oxford, S.: The Hamiltonian of the quantized nonlinear Schrödinger equation. Ph.D. thesis, UCLA (1979)
Povolotsky, A.: On integrability of zero-range chipping models with factorized steady state. J. Phys. A 46(465205) (2013). arXiv:1308.3250 [math-ph]
Povolotsky, A., Priezzhev, V.: Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. 7, P07002 (2006)
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]
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]
Sasamoto T., Wadati M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31, 6057–6071 (1998)
Schütz G.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86, 12651287 (1997)
Schütz G.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88, 427–445 (1997)
Spitzer F.: Interaction of Markov processes. Adv. Math. 5(2), 246–290 (1970)
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]
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]
Tracy, C., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132(2), 291–300 (2008). arXiv:0804.1379 [math.PR]
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)]
Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009). arXiv:0807.1713 [math.PR]
Tracy, C., Widom, H.: On ASEP with step Bernoulli initial condition. J. Stat. Phys. 137, 825–838 (2009). arXiv:0907.5192 [math.PR]
Tracy, C., Widom, H.: Total current fluctuations in ASEP. J. Math. Phys. 50(9), 095–204 (2009). arXiv:0902.0821 [math.PR]
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)
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]
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]
Veto, B.: TracyWidom limit of q-Hahn TASEP (2014). arXiv:1407.2787 [math.PR]
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)
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)
Yang, C.N., Yang, C.P.: One dimensional chain of anisotropic spin-spin interaction. Phys. Rev. 150, 321–327, 327–339 (1966)
Yang C.N., Yang C.P.: One dimensional chain of anisotropic spin-spin interaction. Phys. Rev. 151, 258–264 (1966)
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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
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DOI: https://doi.org/10.1007/s00220-015-2424-7