Abstract
We consider the stochastic higher spin six vertex (SHS6V) model introduced by Corwin and Petrov (Commun. Math. Phys., 343(2), 651–700 2016) with general integer spin parameters I, J. Starting from near stationary initial condition, we prove that the SHS6V model converges to the Kardar-Parisi-Zhang (KPZ) equation under weakly asymmetric scaling. This generalizes the result in Corwin et al. (2018, Theorem 1.1) from I = J = 1 to general I, J.
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References
Andrews, G.E., Askey, R., Roy, R.: Special Functions, vol. 71. Cambridge University Press (2000)
Aggarwal, A.: Current fluctuations of the stationary ASEP and six-vertex model. Duke Math. J. 167(2), 269–384 (2018)
Aggarwal, A.: Dynamical stochastic higher spin vertex models. Sel. Math., New Ser. 24(3), 2659–2735 (2018)
Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. Duke Math. J. 165(3), 563–624 (2016)
Borodin, A, Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz. Commun. Math. Phys. 339(3), 1167–1245 (2015)
Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Correction to: Spectral theory for interacting particle systems solvable by coordinate bethe ansatz. Commun. Math. Phys. 370(3), 1069–1072 (2019)
Bertini, L., Giacomin, G.: Stochastic burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)
Borodin, A., Gorin, V.: A stochastic telegraph equation from the six-vertex model. arXiv:1803.09137 (2018)
Billingsley, P: Convergence of Probability Measures. Wiley (2013)
Baik, J., Liu, Z.: Multipoint distribution of periodic TASEP. Journal of the American Mathematical Society (2019)
Borodin, A.: On a family of symmetric rational functions. Adv. Math. 306, 973–1018 (2017)
Borodin, A: Stochastic higher spin six vertex model and Macdonald measures. J. Math. Phys. 59(2), 023301 (2018)
Borodin, A., Petrov, L.: Higher spin six vertex model and symmetric rational functions. Sel. Math. New Ser. 24(2), 751–874 (2018)
Carinci, G., Giardinà, C., Redig, F., Sasamoto, T.: A generalized asymmetric exclusion process with Uq(sl2) stochastic duality. Probab. Theory Relat. Fields 166(3–4), 887–933 (2016)
Corwin, I., Ghosal, P., Shen, H., Tsai, L.-C.: Stochastic PDE limit of the six vertex model. arXiv:1803.08120 (2018)
Corwin, I: The Kardar–Parisi–Zhang equation and universality class. Random Matrices: Theory Appl. 1(01), 1130001 (2012)
Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343(2), 651–700 (2016)
Corwin, I., Petrov, L.: Correction to: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 371(1), 353–355 (2019)
Corwin, I., Shen, H.: Open ASEP in the weakly asymmetric regime. Commun. Pure Appl. Math. 71(10), 2065–2128 (2018)
Corwin, I., Shen, H., Tsai, L.-C.: ASEP(q, j) converges to the KPZ equation. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 54(2), 995–1012 (2018)
Corwin, I., Tsai, L.-C.: KPZ equation limit of higher-spin exclusion processes. Ann. Probab. 45(3), 1771–1798 (2017)
Dauvergne, D., Ortmann, J., Virág, B.: The directed landscape. arXiv:1812.00309 (2018)
Dembo, A., Tsai, L.-C.: Weakly asymmetric non-simple exclusion process and the Kardar–Parisi–Zhang equation. Commun. Math. Phys. 341(1), 219–261 (2016)
Ghosal, P.: Hall-Littlewood-PushTASEP and its KPZ limit. arXiv:1701.07308 (2017)
Gonçalves, P, Jara, M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212(2), 597–644 (2014)
Gubinelli, M., Perkowski, N.: KPZ reloaded. Commun. Math. Phys. 349 (1), 165–269 (2017)
Gubinelli, M., Perkowski, N.: Energy solutions of KPZ are unique. J. Am. Math. Soc. 31(2), 427–471 (2018)
Gwa, L.-H., Spohn, H.: Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68(6), 725 (1992)
Hairer, M.: A theory of regularity structures. Inventiones Mathematicae 198 (2), 269–504 (2014)
Imamura, T., Mucciconi, M., Sasamoto, T.: Stationary Higher Spin Six Vertex Model and q-Whittaker measure. arXiv:1901.08381 (2019)
Krug, J., Meakin, P., Halpin-Healy, T.: Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A 45(2), 638 (1992)
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889 (1986)
Kirillov, A.N., Reshetikhin, N.Yu: Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum. J. Phys. A Math. Gen. 20(6), 1565 (1987)
Kuan, J.: A multi-species ASEP and-TAZRP with stochastic duality. Int. Math. Res. Not. 2018(17), 5378–5416 (2017)
Kuan, J.: An algebraic construction of duality functions for the stochastic \(U_{q} ({A_{n}^{1}})\) vertex model and its degenerations. Commun. Math. Phys. 359(1), 121–187 (2018)
Labbé, C.: Weakly asymmetric bridges and the KPZ equation. Commun. Math. Phys. 353(3), 1261–1298 (2017)
Liggett, T.M: Interacting Particle Systems, vol. 276. Springer Science & Business Media (2012)
Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, vol. 324. Springer Science & Business Media (2013)
Lin, Y.: Markov duality for stochastic six vertex model. Electron. Commun. Probab. 24(67), 1–17 (2019)
Mangazeev, V.V.: On the Yang–Baxter equation for the six-vertex model. Nucl. Phys. B 882, 70–96 (2014)
Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point. arXiv:1701.00018 (2016)
Mueller, C.: On the support of solutions to the heat equation with noise. Stochastics: Int. J. Probab. Stoch. Process. 37(4), 225–245 (1991)
Orr, D., Petrov, L.: Stochastic higher spin six vertex model and q-TASEPs. Adv. Math. 317, 473–525 (2017)
Parekh, S.: The KPZ limit of ASEP with boundary. Commun. Math. Phys. 365(2), 569–649 (2019)
Pauling, L.: The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57(12), 2680–2684 (1935)
Quastel, J: Introduction to KPZ. Curr. Develop. Math., 2011(1) (2011)
Spohn, H.: KPZ scaling theory and the semidiscrete directed polymer model. Random Matrix Theory, Interacting Particle Systems and Integrable Systems, 65 (2012)
Shen, H., Tsai, L.-C.: Stochastic telegraph equation limit for the stochastic six vertex model. Proc. Am. Math. Soc. 147(6), 2685–2705 (2019)
Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279(3), 815–844 (2008)
Acknowledgments
The author heartily thanks his advisor Ivan Corwin for suggesting this problem, his encouragement and helpful discussion, as well as reading part of the manuscript. We thank Jeffrey Kuan for the helpful discussion over the degeneration of [35, Theorem 4.10] to the stochastic higher spin six vertex model. We also wish to thank Promit Ghosal and Li-Cheng Tsai, who provided very useful discussions about the technical part of the paper. We are also grateful to two anonymous referees for their valuable suggestions. The author was partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program and also received summer support from Ivan Corwin’s NSF grant DMS:1811143.
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Appendices
Appendix A: Stationary Distribution of the SHS6V Model
In this section, we provide a one parameter family of stationary distribution for the unfused SHS6V model. It is worth to remark that in the recent work of [30], a translation-invariant Gibbs measure was obtained (using the idea from [2]) for the space-time inhomogeneous SHS6V model on the full lattice, see Proposition 4.5 of [30]. However, It is not obvious that the dynamic of SHS6V model under this Gibbs measure coincides with the one of the bi-infinite SHS6V model specified in Lemma 2.1. This being the case, we choose to proceed without relying on the result from [30].
We start with a well-known combinatoric lemma.
Lemma A.1 (q-binomial formula)
Set ν = q−Ias usual, the following identity holds for all\(q \in \mathbb {C}\),
Proof
According to q-binomial theorem [1],
When ν = q−I, (ν, q)n = 0 for n > I. Therefore,
□
Lemma A.2
Fix q > 1, ν = q−Iand ρ ∈ (0, I), define a probability measure πρon\(\{0, 1, \dots , I\}\):
where χis the unique negative real number satisfying
Furthermore, we have
Proof
We first show that πρ is indeed a probability measure. It is clear that \(\pi _{\rho } (i) \geqslant 0\) for all \(i \in \{0, 1, \dots , I\}\). By Lemma A.1,
Next, we compute the expectation and the variance of πρ. Using again Lemma A.1, the moment generating function is given by
It is clear that
Via (A.3), one has
Note that
combining this with (A.2) yields
which concludes the lemma. □
Theorem A.3
For ρ ∈ (0, I), the product measure\(\bigotimes \pi _{\rho }\)is stationary for the unfused SHS6V model\(\vec {\eta }(t)\) (Definition 2.3).
Proof
It suffices to show that if \(\vec {\eta }(t) \sim \bigotimes \pi _{\rho }\), then \(\vec {\eta }(t+1) \sim \bigotimes \pi _{\rho }\).
Recall that K(t, y) = N(t, y) − N(t + 1, y) records the number of particles (either zero or one) that move across location y at time t. We first show that \(K(t, y) \sim \text {Ber }(\frac {\alpha (t) \chi }{\alpha (t) \chi +1})\) (recall that α(t) = αqmod (t)). To this end, referring to (??),
Recalling from (??), \(B(t, z, \eta ) \sim \text {Ber }\big (\frac {\alpha (t)(1 - q^{\eta })}{1 +\alpha (t)}\big ), B^{\prime }(t, z, \eta ) \sim \text {Ber }\big (\frac {\alpha (t) + \nu q^{\eta }}{1 +\alpha (t)}\big )\). Since the random variables \(B, B^{\prime }\) are all independent,
Therefore, by tower property
As \(\eta _{y} (t) \sim \pi _{\rho }\), we obtain using Lemma A.1
Inserting the value of \(\mathbb {E}\big [q^{\eta _{y} (t)}\big ]\) into the RHS of (A.5) yields that
Since K(t, y) ∈{0, 1}, we conclude that
The next step is to show that the marginal of \(\vec {\eta }(t+1)\) is distributed as πρ for each coordinate. Referring to (A.4), it is straightforward that the following recursion holds
Therefore,
For the second equality above, we used K(t, y) = N(t, y) − N(t + 1, y). Therefore,
Due to (A.4), we see that \(K(t, y-1) \in \sigma \Big (B(t, z, \eta ), B^{\prime }(t, z, \eta ), \eta _{z}(t): z \leqslant y-1, \eta \in \{0, 1, \dots , I\} \Big )\). Note that we have assumed \(\vec {\eta }(t) \sim \bigotimes \pi _{\rho }\), which implies the independence between ηy(t) and ηz(t) for z≠y. Therefore, ηy(t) and K(t, y − 1) are independent. Using (A.8) we get
By \(K(t, y-1) \sim \text {Ber }(\frac {\alpha (t) \chi }{\alpha (t) \chi +1})\) and \(\eta _{y} (t) \sim \pi _{\rho }\), one readily has
To conclude Theorem A.3, it suffices to show the independence among ηy(t + 1) for different value of y. It is enough to show that
We need the following lemma.
Lemma A.4
For all\(y \in \mathbb {Z}\), ηy(t + 1) is independent with K(t, y).
Let us first see how this lemma leads to (A.9). We have via (A.4),
Combining this with (A.8),
Since ηi(t) are all independent for different i, one has
We achieve
Using Lemma A.4, we conclude
Therefore,
On the other hand, by (A.7) and (A.8), we conclude for all \(y \in \mathbb {Z}\)
Combining (A.10) and (A.11), we find that for all \(y \in \mathbb {Z}\)
which concludes (A.9). □
Proof of Lemma A.4
As K(t, y) ∈{0, 1}, it suffices to show that for all \(j \in \{0, 1, \dots , I\}\), one has
Due to (A.7),
Together with (A.8), we obtain that if K(t, y − 1) = 0,
If K(t, y − 1) = 1,
The discussion above yields (using the independence between ηy(t) and K(t, y − 1))
which concludes Lemma A.4. □
Remark A.5
Since \(\vec {g}(t) = \vec {\eta }(Jt)\), it is clear that for all ρ ∈ (0, I), \(\bigotimes \pi _{\rho }\) is also stationary for the fused SHS6V model \(\vec {g}(t)\).
Appendix B: KPZ Scaling Theory
The KPZ scaling theory has been developed in a landmark contribution by [31]. The scaling theory is a physics approach which makes prediction for the non-universal coefficients of the KPZ equation. In this appendix, we show how the coefficients of the KPZ (??) arise from the microscopic observables of the fused SHS6V model using the KPZ scaling theory.
Recall that Theorem 1.6 reads
Here, \(N^{\mathsf {f}}_{\epsilon } (t, x)\) is the fused height function and \({\mathscr{H}}(t, x)\) solves the KPZ equation
where
The first step in the KPZ scaling theory is to derive the stationary distribution of the fused SHS6V model, which is exactly what we did in Appendix A (see Remark A.5). Under stationary distribution \(\bigotimes \pi _{\rho }\), we proceed to define two natural quantities of the models:
The average steady state currentj(ρ) is defined as
$$ j(\rho) = \epsilon^{-\frac{1}{2}} \big(\big\langle N^{\mathsf{f}}(t, x) - N^{\mathsf{f}}(t, x+1) \big\rangle_{\rho} - \rho \mu_{\epsilon}\big), $$(B.1)where 〈⋅〉ρ means that we are taking the expectation under stationary distribution \(\bigotimes \pi _{\rho }\) and μ is given in (??). Note that under stationary distribution, the average steady state current j(ρ) depends neither on space or time. Let us explain the meaning of (B.1). Note that Nf(t, x) − Nf(t + 1, x) records the number of particles in the fused SHS6V model that move across location x at time t, we subtract ρμ𝜖 here because we are in a reference frame that moves to right with speed ρμ𝜖.
The integrated covariance is defined as
$$ \begin{array}{@{}rcl@{}} &A(\rho):= \lim\limits_{r \to \infty} \frac{1}{2 r} \bigg\langle N^{\mathsf{f}} (t, x+r) - N^{\mathsf{f}} (t, x - r) - \big\langle N^{\mathsf{f}} (t, x+r) - N^{\mathsf{f}} (t, x - r) \big\rangle_{\rho} \bigg\rangle_{\!\rho}. \end{array} $$
The KPZ scaling theory (equation (12) and (15) of [31]) predicts that
A𝜖(ρ) and j𝜖(ρ) depend on 𝜖 under weakly asymmetry scaling (??).
Let us first verify (ii), note that under stationary distribution, N𝜖f(t, x + r) − N𝜖f(t, x − r) is the sum of 2r i.i.d. random variables with the same distribution πρ, hence A𝜖(ρ) = Var[πρ]. By Lemma A.2, we know that
where χ is the unique negative solution of
Under weakly asymmetric scaling, one has \(q = e^{\sqrt {\epsilon }}\), which yields \(\lim _{\epsilon \downarrow 0} \chi _{\epsilon } = \frac {\rho }{\rho - I}\). Therefore,
This matches with the value of \(\frac {\alpha _{3}}{\alpha _{1}}\).
We proceed to verify (i). First, note that by Nf(t, x) = N(Jt, x),
where K(s, x) = N(s, x) − N(s + 1, x). We have shown in (A.6) that \(K(s, x) \sim \text {Ber }(\frac {\alpha (s) \chi }{1 + \alpha (s) \chi })\), where α(s) = αqmod (s). Therefore,
which yields
We proceed to taylor expand j𝜖(ρ) around 𝜖 = 0. Note that χ is implicitly defined through (B.2), we expand χ𝜖 around 𝜖 = 0
Note that α depends on 𝜖 through \(\alpha _{\epsilon } = \frac {1 - b}{b - e^{\sqrt {\epsilon }}}\). Via straightforward calculation, one has
which implies
Referring to the expression of μ in (??), one has the asymptotic expansion
Consequently,
We have
which coincides with the value of α2.
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Lin, Y. KPZ Equation Limit of Stochastic Higher Spin Six Vertex Model. Math Phys Anal Geom 23, 1 (2020). https://doi.org/10.1007/s11040-019-9325-5
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DOI: https://doi.org/10.1007/s11040-019-9325-5