Integral Formulas of ASEP and q-TAZRP on a Ring

Abstract

In this paper, we obtain the transition probability formulas for the asymmetric simple exclusion process and the q-deformed totally asymmetric zero range process on the ring by applying the coordinate Bethe ansatz. We also compute the distribution function for a tagged particle with general initial condition.

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Fig. 1

Notes

  1. 1.

    When we say the coordinate Bethe ansatz is applied to the TASEP and other models, we mean that certain eigenstates for the linear evolution operator of the process may be solved exactly through a “guess and check” method, with the “check” part equivalent to the verification of a type of Yang-Baxter equation (i.e. k-particle interactions are equivalent to a sequence of 2-particle interactions, independent of the order of the sequence). In all these models on \(\mathbb {Z}\), there is no need of solving the Bethe equation, since the boundary condition is trivial.

  2. 2.

    For TASEP on \(\mathbb {Z}/L\mathbb {Z}\), the transition probability function is found in the similar method and similar form as the TASEP on \(\mathbb {Z}\). However, the Bethe equation was not solved in the usual sense. This feature is shared in our paper.

References

  1. [AAR99]

    Andrews, G.E., Askey, R., Roy, R.: Special Functions, Volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  2. [ACQ11]

    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)

    MathSciNet  Article  Google Scholar 

  3. [BC14]

    Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields 158(1–2), 225–400 (2014)

    MathSciNet  Article  Google Scholar 

  4. [BCPS15]

    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)

    ADS  MathSciNet  Article  Google Scholar 

  5. [BCS14]

    Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for \(q\)-TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014)

    MathSciNet  Article  Google Scholar 

  6. [Bet31]

    Bethe, H.: Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71(3), 205–226 (1931)

    ADS  Article  Google Scholar 

  7. [BL16]

    Baik, J., Liu, Z.: TASEP on a ring in sub-relaxation time scale. J. Stat. Phys. 165(6), 1051–1085 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  8. [BL18]

    Baik, J., Liu, Z.: Fluctuations of TASEP on a ring in relaxation time scale. Commun. Pure Appl. Math. 71(4), 747–813 (2018)

    MathSciNet  Article  Google Scholar 

  9. [BL19]

    Baik, J., Liu, Z.: Multipoint distribution of periodic TASEP. J. Am. Math. Soc. 32(3), 609–674 (2019)

    MathSciNet  Article  Google Scholar 

  10. [CLDR10]

    Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. EPL (Europhys. Lett.) 90(2), 20002 (2010)

    ADS  Article  Google Scholar 

  11. [DL98]

    Derrida, B., Lebowitz, J.L.: Exact large deviation function in the asymmetric exclusion process. Phys. Rev. Lett. 80(2), 209–213 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  12. [Dot10]

    Dotsenko, V.: Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers. EPL (Europhys. Lett.) 90(2), 20003 (2010)

    ADS  Article  Google Scholar 

  13. [FP19]

    Fehér, G.Z., Pozsgay, B.: The propagator of the finite XXZ spin-\(\tfrac{1}{2}\) chain. SciPost Phys. 6, 63 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  14. [Gau14]

    Gaudin, M.: The Bethe Wave Function. Cambridge University Press, New York (2014). Translated from the 1983 French original by Jean-Sébastien Caux

  15. [GM04]

    Golinelli, O., Mallick, K.: Bethe ansatz calculation of the spectral gap of the asymmetric exclusion process. J. Phys. A 37(10), 3321–3331 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  16. [GM05]

    Golinelli, O., Mallick, K.: Spectral gap of the totally asymmetric exclusion process at arbitrary filling. J. Phys. A 38(7), 1419–1425 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  17. [GMW81]

    Gaudin, M., McCoy, B.M., Wu, T.T.: Normalization sum for the Bethe’s hypothesis wave functions of the Heisenberg–Ising chain. Phys. Rev. D (3) 23(2), 417–419 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  18. [GS92]

    Gwa, L.-H., Spohn, H.: Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68(6), 725–728 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  19. [Joh00]

    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  20. [Kim95]

    Kim, D.: Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar-Parisi-Zhang-type growth model. Phys. Rev. E 52, 3512–3524 (1995)

    ADS  Article  Google Scholar 

  21. [KL14]

    Korhonen, M., Lee, E.: The transition probability and the probability for the left-most particle’s position of the \(q\)-totally asymmetric zero range process. J. Math. Phys. 55(1), 013301 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  22. [Kor82]

    Korepin, V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86(3), 391–418 (1982)

    ADS  MathSciNet  Article  Google Scholar 

  23. [LW19]

    Lee, E., Wang, D.: Distributions of a particle’s position and their asymptotics in the \(q\)-deformed totally asymmetric zero range process with site dependent jumping rates. Stoch. Process. Appl. 129(5), 1795–1828 (2019)

    MathSciNet  Article  Google Scholar 

  24. [MP18]

    Mallick, K., Prolhac, S.: Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume. J. Stat. Phys. 173(2), 322–361 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  25. [Pov04]

    Povolotsky, A.M.: Bethe ansatz solution of zero-range process with nonuniform stationary state. Phys. Rev. E (3) 69(6), 061109 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  26. [PP07]

    Povolotsky, A.M., Priezzhev, V.B.: Determinant solution for the totally asymmetric exclusion process with parallel update. II. Ring geometry. J. Stat. Mech. Theory Exp. 8, P08018 (2007)

    MathSciNet  MATH  Google Scholar 

  27. [Pro13]

    Prolhac, S.: Spectrum of the totally asymmetric simple exclusion process on a periodic lattice—bulk eigenvalues. J. Phys. A 46(41), 415001 (2013)

    MathSciNet  Article  Google Scholar 

  28. [Pro14]

    Prolhac, S.: Spectrum of the totally asymmetric simple exclusion process on a periodic lattice-first excited states. J. Phys. A 47(37), 375001 (2014)

    MathSciNet  Article  Google Scholar 

  29. [Pro15a]

    Prolhac, S.: Asymptotics for the norm of Bethe eigenstates in the periodic totally asymmetric exclusion process. J. Stat. Phys. 160(4), 926–964 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  30. [Pro15b]

    Prolhac, S.: Current fluctuations for totally asymmetric exclusion on the relaxation scale. J. Phys. A 48(6), 06FT02 (2015)

    Article  Google Scholar 

  31. [Pro16a]

    Prolhac, Sylvain: Extrapolation methods and Bethe ansatz for the asymmetric exclusion process. J. Phys. A 49(45), 454002 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  32. [Pro16b]

    Prolhac, S.: Finite-time fluctuations for the totally asymmetric exclusion process. Phys. Rev. Lett. 116, 090601 (2016)

    ADS  Article  Google Scholar 

  33. [Pro17]

    Prolhac, Sylvain: Perturbative solution for the spectral gap of the weakly asymmetric exclusion process. J. Phys. A 50(31), 315001 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  34. [Pro20]

    Prolhac, S.: Riemann surfaces for KPZ with periodic boundaries. SciPost Phys. 8, 8 (2020)

    ADS  Article  Google Scholar 

  35. [Sch93]

    Schütz, G.: Generalized Bethe ansatz solution of a one-dimensional asymmetric exclusion process on a ring with blockage. J. Stat. Phys. 71(3–4), 471–505 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  36. [Sch97]

    Schütz, G.M.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88(1–2), 427–445 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  37. [SS10]

    Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834(3), 523–542 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  38. [Sut04]

    Sutherland, B.: Beautiful Models. World Scientific Publishing, River Edge (2004). 70 years of exactly solved quantum many-body problems

    Google Scholar 

  39. [TW08]

    Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279(3), 815–844 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  40. [TW09]

    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290(1), 129–154 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  41. [TW11]

    Tracy, C.A., Widom, H.: Erratum to: Integral formulas for the asymmetric simple exclusion process [mr2386729]. Commun. Math. Phys. 304(3), 875–878 (2011)

    ADS  Article  Google Scholar 

  42. [WW16]

    Wang, D., Waugh, D.: The transition probability of the \(q\)-TAZRP (\(q\)-bosons) with inhomogeneous jump rates. SIGMA Symmetry Integr. Geom. Methods Appl. 12, Paper No. 037, 16 (2016)

Download references

Acknowledgements

We are grateful to Jinho Baik, Ivan Corwin, Leonid Petrov, and Craig Tracy for helpful discussions. The authors would like to thank the organizer of the Integrable Probability Focused Research Group, funded by NSF Grants DMS-1664531, 1664617, 1664619, 1664650, for organizing stimulating events and the Park City Mathematics Institute (PCMI) for organizing “The \(27^{th}\) Annual Summer Session, Random Matrices,” funded by NSF Grant DMS-1441467. Z.L. was supported by the University of Kansas Start Up Grant, the University of Kansas New Faculty General Research Fund, and Simons Collaboration Grant No. 637861. A.S. was partially supported by NSF Grants DMS-1664617. D.W. was partially supported by the Singapore AcRF Tier 1 Grant R-146-000-217-112 and the Chinese NSFC Grant 11871425.

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Communicated by H. Spohn

A. Convergence Lemmas for ASEP and q-TAZRP on a Ring

A. Convergence Lemmas for ASEP and q-TAZRP on a Ring

We need to make sure that the right side of (1.16) and (1.26) are well-defined before we prove Theorems 1 and 3. That is, we need to show that the sum of \(\Lambda ^{\mathbb {L}}_Y(X; t; \sigma )\) over the infinite lattice \(\mathbb {Z}^N(0)\) converges. Also, we need to show that the integrands of the z-integral on the right side of (1.17) and (1.27) converge in Theorems 2 and 4. For these integrands, the problem is more serious since we need to deal with convergence of other series in the proofs of Theorems 2 and 4. In this appendix, we show that all the required convergences hold.

We establish some notation and a basic estimate to facilitate the upcoming proofs in this section. Recall the notation set up in (4.9). It is straightforward to see that given any k and n, there are only finitely many \(\mathbb {L}\in \mathbb {Z}^N(n)\) such that \(\max (\mathbb {L}) < k\). We have a very rough estimate.

Lemma A.1

The number of \(\mathbb {L}\in \mathbb {Z}^N(n)\) with \(\max (\mathbb {L}) < k\) is no more than \(N^N (k - \lfloor n/N \rfloor )^N\) if \(n < Nk\), and there is no \(\mathbb {L}\in \mathbb {Z}^N(n)\) with \(\max (\mathbb {L}) < k\) if \(n \ge Nk\).

Proof

The statement for \(n \ge Nk\) is obvious. For \(n < Nk\), we assume that \(n = lN\) with l an integer without loss of generality. Note that, if \(\mathbb {L}= (\ell _1, \dotsc , \ell _N) \in \mathbb {Z}^N(n)\), then \(\mathbb {L}' (\ell _1 - l, \ell _2 - l, \dotsc , \ell _N - l) \in \mathbb {Z}^N(0)\). Then, the inequality \(\max (\mathbb {L}) < k\) is equivalent to \(\max (\mathbb {L}') < k - l\). Hence, the lemma is reduced to the \(n = 0\) case.

Suppose \(n = 0\) and \(\mathbb {L}= (\ell _1, \dotsc , \ell _N) \in \mathbb {Z}^N(0)\) satisfies \(\max (\mathbb {L}) < k\). Then, if there is an \(\ell _i\) with \(\ell _i < (1 - N)k\), the sum of the other \((N - 1)\) components is greater than \((N - 1)k\), and at least one among them is greater than k, a contradiction. So, the value of each \(\ell _i\) is less than k and no less than \((1 - N)k\). Hence, each component \(\ell _i\) has no more than Nk possible values and \(\mathbb {L}\) has no more than \((Nk)^N\) possible choices. \(\quad \square \)

A.1 q-TAZRP case

In this subsection, we assume \(C = \{ |z |= R \}\) is a circular contour with positive orientation and R a large constant. Recall that \(b_{[1]}, \cdots , b_{[L]}\) are positive constants defined in Sect. 1.1.2. We assume \(g(w_1, \dotsc , w_N)\) is an analytic function on \(\mathbb {C}\) and

$$\begin{aligned} \tilde{g}(w_1, \dotsc , w_N) = g(w_1, \dotsc , w_N) \prod ^N_{j = 1} \prod ^L_{i = 1} (b_{[i]} - w_j)^{-m} \end{aligned}$$
(8.29)

for some \(m > 0\). In this subsection, we take \(D_{\mathbb {L}}\) as defined by (1.23).

Lemma A.2

Denote

(A.1)

Then \(\Lambda ^{\mathbb {L}} = 0\) if \(\max (\mathbb {L}) > m\).

Proof

Consider the integral over \(w_{m(\mathbb {L})}\). We have that the contour C encircles no poles with respect to \(w_{m(\mathbb {L})}\). Then, the integral of \(w_{m(\mathbb {L})}\) over \(C_{m(\mathbb {L})}\) vanishes, which implies that \(\Lambda ^{\mathbb {L}}\) vanishes. \(\quad \square \)

Next, we consider the special case with \(b_{[1]} = \cdots = b_{[N]} = 1\). Then, the function \(\tilde{g}(w_1, \dotsc , w_N)\) defined in (A.1) becomes \(g(w_1, \dotsc , w_N) \prod ^N_{j = 1} (1 - w_j)^{-mL} \). Furthermore, we assume that the radius R of the contour C is bigger than N. We denote

$$\begin{aligned} M_g = \max _{w_i \in C,\ i = 1, \dotsc , N} |g(w_1, \dotsc , w_N) |. \end{aligned}$$
(A.2)

Lemma A.3

Denote

(A.3)

and \(\tilde{\Lambda }_s = \sum _{\mathbb {L}\in \mathbb {Z}^N(s)} |\tilde{\Lambda }^{\mathbb {L}} |\) for some \(s \in \mathbb {Z}\). Furthermore, assume the radius R of the contour C satisfies

$$\begin{aligned} \frac{R^{2LN}}{(R - 1)^{L(2N - 1)}} \left( \frac{1 + q}{1 - q} \right) ^{2N^2} < 1. \end{aligned}$$
(A.4)

Then, \(\tilde{\Lambda }_s < C M_g\) for some constant C that depends on ms but not g.

Proof

We partition \(\mathbb {Z}^N(s)\) into disjoint subsets \(Z_0 \cup Z_1 \cup Z_2 \cup \cdots \), with

$$\begin{aligned} Z_0&= \{ \mathbb {L}\in \mathbb {Z}^N(s) \mid L \max (\mathbb {L})< m \}\nonumber \\ Z_n&= \{ \mathbb {L}\in \mathbb {Z}^N(s) \mid m + (n - 1)L \le L\max (\mathbb {L}) < m + nL \} \end{aligned}$$
(A.5)

for \(n = 1, 2, \dotsc \). By Lemma A.1, each \(Z_n\) is a finite set and \(|Z_n |< N^N (n + \lceil m/L \rceil - \lfloor s/N \rfloor )^N\) if \(n + m/L - s/N > 0\). Without loss of generality, we assume \(m = s = 0\) below for notational convenience. Then, in this case, \(Z_0 = \emptyset \) and we only need to consider \(n \ge 1\).

For \(\mathbb {L}= (\ell _1, \dotsc , \ell _N) \in Z_n\), we have that the integrand in (A.4) has no pole with respect to \(w_{m(\mathbb {L})}\) within C and has a zero of order at least \((n-1)L\) at 1. So, by the identity

$$\begin{aligned} \frac{1}{w} = \frac{1}{w(1 - w)^{(n-1)L}} - \sum ^{(n-1)L}_{k = 1} \frac{1}{(1 - w)^k}, \end{aligned}$$
(A.6)

we have

(A.7)

Also, we have \(\sum ^N_{i = 1} |\ell _i |< 2nN\) since \(\max ^N_{i = 1} (\ell _i) < n\) and \(\sum ^N_{i = 1} \ell _i = 0\). Then, it is not hard to see that

$$\begin{aligned} |D_{\mathbb {L}}(w_1, \dotsc , w_N) |< \left( \left( \frac{R}{R - 1} \right) ^L \left( \frac{1 + q}{1 - q} \right) ^N \right) ^{2nN}, \end{aligned}$$
(A.8)

if \(w_i \in C\) for all \(i = 1, \dotsc , N\). Additionally, we have

$$\begin{aligned} \left|\frac{ g(w_1, \dotsc , w_N)}{(1 - w_{m(\mathbb {L})})^{(n-1)L}} \prod _{1 \le i< j \le N} (qw_i - w_j)^{-1} \right|< \frac{M_g}{(R - 1)^{(n-1)L}} ((1 + q)R)^{N(N - 1)/2}. \end{aligned}$$
(A.9)

Thus, we obtain the following estimate,

$$\begin{aligned} \sum _{\mathbb {L}\in Z_n} |\tilde{\Lambda }^{\mathbb {L}} |< (N n)^N M_g (R-1) ((1 + q)R)^{N(N - 1)/2} \left( \frac{R^{2N L}}{(R - 1)^{(2N+1) L}} \left( \frac{1 + q}{1 - q} \right) ^{2N^2} \right) ^n, \end{aligned}$$
(A.10)

by combining the previous two estimates. Hence, we obtain the result by summing the last estimate over \(n>1\). \(\quad \square \)

A.2 ASEP case

Later in this subsection, we assume that \(C = \{ |z |= p^2 \}\) is a circular contour with positive orientation. Let \(f(\xi _1, \dotsc , \xi _N)\) be a meromorphic function such that it is analytic on the multi-cylinder \(\{ |\xi _i |\le p^2 \mid i = 1, \dotsc , N \}\) with \(\max _{|\xi _i |= r,\ i = 1, \dotsc , N} |f(\xi _1, \dotsc , \xi _N) |= M_f\). Let \(t \in \mathbb {R}_{>0}\), \(m \in \mathbb {Z}_{>0}\) and \(s \in \mathbb {Z}\) be constants. Then, we have the following estimate.

Lemma A.4

Denote

(A.11)

We have \(\Lambda _s < C_1 M_f\) for a constant \(C_1\) that depends on ms but not on \(f(\xi _1, \dotsc , \xi _N)\).

For the proof of Lemma A.4, we need to estimate \(\Lambda ^{\mathbb {L}}\) for each \(\mathbb {L}\). We decompose the exponential factor \(e^{\epsilon (\xi _{m(\mathbb {L})})t}\) into the sum of two terms: one is the partial sum of the Taylor expansion of the exponential function, and the other is the remainder term. To be precise, we write

$$\begin{aligned} e^z = P_n(z) + E_n(z), \quad \text {so that} \quad P_n(z) = \sum ^n_{k = 0} \frac{z^k}{k!}, \quad E_n(z) = \sum ^{\infty }_{k = n + 1} \frac{z^k}{k!}. \end{aligned}$$
(A.12)

Then, we write

$$\begin{aligned} \Lambda ^{\mathbb {L}} = \Lambda ^{\mathbb {L}; P}(n) + \Lambda ^{\mathbb {L}; E}(n), \end{aligned}$$
(A.13)

for any \(n \ge 0\) with

(A.14)

for \(\square = P\) or E.

Lemma A.5

Suppose \(\max (\mathbb {L})L - m = M \ge 0\). Then

  1. 1.

    \(\Lambda ^{\mathbb {L}; P}(M) = 0\).

  2. 2.

    If \(\mathbb {L}\in \mathbb {Z}^N(s)\), we have \(|\Lambda ^{\mathbb {L}; E}(M) |< C_2 M_f e^{c_3 M + c'_3 L} /M!\) for some constant \(C_2, c_3, c'_3 > 0\) depending on s, t, m but independent of f and L.

Proof

The proof of statement 1 is similar to the proof of Lemma A.2. We first note that the contour C encloses no pole with respect to \(\xi _{m(\mathbb {L})}\). In particular, there is no pole at \(\xi _{m(\mathbb {L})} =0\) due to the assumption that \(\max (\mathbb {L}) = \ell _{m(\mathbb {L})} \ge m\). Hence, the integral with respect to \(\xi _{m(\mathbb {L})}\) over \(C_{m(\mathbb {L})}\) vanishes for the multiple integral that defines \(\Lambda ^{\mathbb {L}; P}_Y(X; t; \sigma ; M)\), and so does \(\Lambda ^{\mathbb {L}; P}_Y(X; t; \sigma ; M)\).

For statement 2, we note that \(|\xi _i |= p^2\) if \(\xi _i\) is on the contour C. Then,

$$\begin{aligned} \left|\prod ^N_{j = 1} \xi ^{-m}_j \prod _{j \ne m(\mathbb {L})} \xi ^{L \ell _j}_j \right|= p^{2(-mN + L(s - m(\mathbb {L})L))} \end{aligned}$$
(A.15)

for all \(\mathbb {L}\in \mathbb {Z}^N(s)\). Also, we have

$$\begin{aligned} \left|e^{t(q \xi _{m(\mathbb {L})} - 1)} \prod _{j = 1, \dotsc , N, \, j \ne m(\mathbb {L})} e^{\epsilon (\xi _j)t} \right|\le e^{N c_4 t} \end{aligned}$$
(A.16)

with \(c_4 = (1 + p^2)(p + p^{-2}q)\) since \(\mathfrak {R}\epsilon (\xi _j) \le c_1\) and \(\mathfrak {R}(q \xi _j - 1) \le c_4\) if \(|\xi _j |= p^2\). Next, we have that

$$\begin{aligned} \prod ^N_{j = 1} \prod ^N_{k = 1} \left|\frac{p + q\xi _k \xi _j - \xi _k}{p + q\xi _k \xi _j - \xi _j} \right|^{\ell _j} \le \prod ^N_{j = 1} c^{|\ell _j |}_5 \le c^{N \max (\mathbb {L})}_5 = c^{N(M + m)}_5 \end{aligned}$$
(A.17)

with \(c_5 = (1 + p + p^3q)/(q - p^3q)\) for all \(i = 1, \dotsc , N\) since

$$\begin{aligned} p(q - p^3q) \le |p + q\xi _i \xi _j - \xi _i |\le p(1 + p + p^3q). \end{aligned}$$
(A.18)

At last, we have an adequate upper bound of the integrand in (A.15) for \(\Lambda ^{\mathbb {L}; E}_Y(X; t; \sigma ; M)\) by using the estimate

$$\begin{aligned} |E_M(tp \xi ^{-1}_{m(\mathbb {L})}) |\le \frac{2}{(M + 1)!} (tp^{-1})^{M + 1}, \end{aligned}$$
(A.19)

which corresponds to the estimate of the error term in a Taylor expansion. Then, we have

$$\begin{aligned} |\Lambda ^{\mathbb {L}; E}(M) |\le M_f p^{2(-m(N+1) + Ls)} c^{N(M + m)}_5 \frac{2}{(M + 1)!} (tp^{-1})^{M + 2} e^{N c_4 t} \end{aligned}$$
(A.20)

after evaluating the integral. This implies statement 2 of the lemma. \(\quad \square \)

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Liu, Z., Saenz, A. & Wang, D. Integral Formulas of ASEP and q-TAZRP on a Ring. Commun. Math. Phys. 379, 261–325 (2020). https://doi.org/10.1007/s00220-020-03837-7

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