Periodic TASEP with general initial conditions

Abstract

We consider the one-dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain, and obtain explicit formulas for the multi-point distributions in the space-time plane. The formulas are given in terms of an integral involving a Fredholm determinant. We then evaluate the large-time, large-period limit in the relaxation time scale, which is the scale such that the system size starts to affect the height fluctuations. The limit is obtained assuming certain conditions on the initial condition, which we show that the step, flat, and step-flat initial conditions satisfy. Hence, we obtain the limit theorem for these three initial conditions in the periodic model, extending the previous work on the step initial condition. We also consider uniform random and uniform-step random initial conditions.

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Notes

  1. 1.

    See [14, 18, 21,22,23, 33] for the earlier work for the other properties of periodic models.

  2. 2.

    We allow the possibility that the sequence \(N=N_L\) does not exist for some values of L. In that case, we take the limit \(L\rightarrow \infty \) in the set \(\{L: N_L \ \text {exists}\}\), which we assumed to be an infinite set. The flat initial condition (see the Definition 2.3) is an example of such a case where we take \(N= L/d\) for a positive integer d.

  3. 3.

    We only proved the flat case with \(\rho ^{-1}\in \{2,3,\ldots \}\). However, heuristically we expect this is true for any \(\rho \in (0,1)\).

References

  1. 1.

    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, 466–537 (2011)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bethe, H.: Zur Theorie der Metalle: Eigenwerte und Eigenfunktionen der linearen Atomkette. Zeitschrift für Physik 71, 205–226 (1931)

    Article  Google Scholar 

  3. 3.

    Borodin, A.: On a family of symmetric rational functions. Adv. Math. 306, 973–1018 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Böttcher, A.: On the determinant formulas by Borodin, Okounkov, Baik, Deift and Rains. In: Böttcher, A., et al. (eds.) Toeplitz Matrices and Singular Integral Equations (Pobershau, 2001), volume 135 of Operator Theory: Advances and Applications, pp. 91–99. Birkhäuser, Basel (2002)

    Google Scholar 

  5. 5.

    Baik, J., Barraquand, G., Corwin, I., Suidan, T.: Pfaffian Schur processes and last passage percolation in a half-quadrant. Ann. Probab. 46(6), 3015–3089 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Barraquand, G., Borodin, A., Corwin, I., Wheeler, M.: Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process. Duke Math. J. 167(13), 2457–2529 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

    Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Borodin, A., Ferrari, P.L., Prähofer, M., Sasamoto, T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129(5–6), 1055–1080 (2007)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Baik, J., Liu, Z.: Discrete Toeplitz/Hankel determinants and the width of nonintersecting processes. Int. Math. Res. Not. IMRN 20, 5737–5768 (2014)

    MathSciNet  Article  Google Scholar 

  11. 11.

    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 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

    Borodin, A., Okounkov, A.: A Fredholm determinant formula for Toeplitz determinants. Integral Equ. Oper. Theory 37(4), 386–396 (2000)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Brankov, J.G., Papoyan, V.B., Poghosyan, V.S., Priezzhev, V.B.: The totally asymmetric exclusion process on a ring: exact relaxation dynamics and associated model of clustering transition. Physica A 368(8), 471480 (2006)

    Google Scholar 

  15. 15.

    Baik, J., Rains, E.M.: The asymptotics of monotone subsequences of involutions. Duke Math. J. 109(2), 205–281 (2001)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Basor, E.L., Widom, H.: On a Toeplitz determinant identity of Borodin and Okounkov. Integral Equ. Oper. Theory 37(4), 397–401 (2000)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow decorrelation in KPZ growth. Ann. Inst. Henri Poincaré Probab. Stat. 48(1), 134–150 (2012)

    MathSciNet  Article  Google Scholar 

  18. 18.

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

    MathSciNet  Article  Google Scholar 

  19. 19.

    Dauvergne, D., Ortmann, J., Virág, B.: The directed landscape. arXiv:1812.00309 (2018)

  20. 20.

    Geronimo, J.S., Case, K.M.: Scattering theory and polynomials orthogonal on the unit circle. J. Math. Phys. 20(2), 299–310 (1979)

    MathSciNet  Article  Google Scholar 

  21. 21.

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

    MathSciNet  Article  Google Scholar 

  22. 22.

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

    MathSciNet  Article  Google Scholar 

  23. 23.

    Gwa, L.-H., Spohn, H.: Bethe solution for the dynamical-scaling exponent of the noisy burgers equation. Phys. Rev. A 46, 844–854 (1992)

    Article  Google Scholar 

  24. 24.

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

    MathSciNet  Article  Google Scholar 

  25. 25.

    Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242(1–2), 277–329 (2003)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Johansson, K.: Two time distribution in Brownian directed percolation. Commun. Math. Phys. 351(2), 441–492 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Johansson, K.: The two-time distribution in geometric last-passage percolation. Probab. Theory Relat. Fields 175, 849–895 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Johansson, K., Rahman, M.: Multi-time distribution in discrete polynuclear growth (2019). arXiv:1906.01053

  29. 29.

    Liu, Z.: Height fluctuations of stationary TASEP on a ring in relaxation time scale. Ann. Inst. Henri Poincaré Probab. Stat. 54(2), 1031–1057 (2018)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Liu, Z.: Multi-time distribution of TASEP (2019). arXiv:1907.09876

  31. 31.

    Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point. arXiv:1701.00018

  32. 32.

    Motegi, K., Sakai, K.: Vertex models, TASEP and Grothendieck polynomials. J. Phys. A Math. Theor. 46(35), 355201 (2013)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Priezzhev, V.B.: Exact nonstationary probabilities in the asymmetric exclusion process on a ring. Phys. Rev. Lett. 91(5), 050601 (2003)

    Article  Google Scholar 

  34. 34.

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

    Article  Google Scholar 

  35. 35.

    Rákos, A., Schütz, G.M.: Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118(3–4), 511–530 (2005)

    MathSciNet  Article  Google Scholar 

  36. 36.

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

    MathSciNet  Article  Google Scholar 

  37. 37.

    Sasamoto, T., Imamura, T.: Fluctuations of the one-dimensional polynuclear growth model in half-space. J. Stat. Phys. 115(3–4), 749–803 (2004)

    MathSciNet  Article  Google Scholar 

  38. 38.

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

    MathSciNet  Article  Google Scholar 

  39. 39.

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

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The work of Jinho Baik was supported in part by NSF grant DMS-1664531 and DMS-1664692. The work of Zhipeng Liu 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.

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Probabilistic argument for the step-flat case when \(L_s=O(L)\)

Probabilistic argument for the step-flat case when \(L_s=O(L)\)

In the step-flat initial condition, the parameters satisfy \(L=dN+L_s\) for \(0<L_s<N\). We evaluated the limit when \(L_s=O(\sqrt{L})\) in Sect. 7. In this section, we discuss the case when \(L_s=O(L)\) and provide a probabilistic argument that the large time limit should be same as the step initial condition. It should be possible to make this argument rigorous but we content to discuss heuristically since this is not a main part of this paper.

We discuss in terms of the periodic directed last passage percolation (DLPP) model which is well-known to be related to the TASEP. Assume that each lattice point \((i,j)\in {\mathbb {Z}}^2\) is assigned an exponential random variable w(ij) with parameter 1. These w(ij)’s are all independent except for the following periodicity condition

$$\begin{aligned} w(i,j) = w(i+L-N,j-N),\quad i,j\in {\mathbb {Z}}. \end{aligned}$$

Let \(\varLambda \) be a lattice path; \(\varLambda \) consists of connected unit horizontal or vertical line segments with vertices in \({\mathbb {Z}}^2\), and \(\varLambda \) does not intersect any line \(y-x=\)constant twice. We define the last passage time from \(\varLambda \) to a lattice point \(\mathrm{p}\) as

$$\begin{aligned} {\mathcal {L}}_\varLambda (\mathrm{p}):= \max _{\varPi :\varLambda \rightarrow \mathrm{p}} \sum _{(i,j)\in \varPi } w(i,j), \end{aligned}$$

where the maximum is over all possible up/right lattice path starting from any lattice point in \(\varLambda \) and ending at \(\mathrm{p}\). We assume that \({\mathcal {L}}_\varLambda (\mathrm{p})=-\infty \) if no such path exists. Similarly, one can define \({\mathcal {L}}_{\mathrm{q}}(\mathrm{p})\) if the lattice path is restricted to start from a given point \(\mathrm{q}\).

Now we consider the step-flat and step initial conditions of the \({{\,\mathrm{PTASEP}\,}}\). These two initial conditions in the language of periodic DLPP, correspond to the lattice paths

$$\begin{aligned} \begin{aligned} \varLambda _{\mathrm {sf}} =&\left( \{(i,j): -(d-1)j \le i \le -(d-1)(j-1), 1\le i\le N\}\right. \\&\left. \cup \{(i,0):0\le i\le L_s\}\right) + (N,-L+N){\mathbb {Z}}\end{aligned} \end{aligned}$$

and

$$\begin{aligned} \varLambda _{\mathrm{step}}=\left( \{(0, j): 0\le j\le N\} \cup \{(i,0): 0\le i\le L-N\}\right) + (N,-L+N){\mathbb {Z}}, \end{aligned}$$

respectively. From the well-known connection between the TASEP and the DLPP, the convergence of the \({{\,\mathrm{PTASEP}\,}}\) with the step-flat initial condition to the step flat initial condition in the large L limit with \(L_s=O(L)\) is translated into the following question on the periodic DLPP: Our goal is to show that when (1) \(\mathrm{p}\) is far enough, more precisely,

$$\begin{aligned} \mathrm{{dist}}\,\left( \mathrm{p},\varLambda _{\mathrm {sf}}\right) = O(L^{3/2}), \end{aligned}$$

which corresponds to the relaxation time scale in PTASEP, and (2) \(L_s= O(L)\), then

$$\begin{aligned} {\mathcal {L}}_{\varLambda _{\mathrm {sf}}} (\mathrm{p}) = {\mathcal {L}}_{\varLambda _{\mathrm{step}}} (\mathrm{p}) +o(L^{1/2}) \text { as }L\rightarrow \infty . \end{aligned}$$
(A.1)

It is known that in the relaxation time scale, both \({\mathcal {L}}_{\varLambda _{\mathrm {sf}}} (\mathrm{p})\) and \({\mathcal {L}}_{\varLambda _{\mathrm{step}}} (\mathrm{p})\) have \(O(\sqrt{L})\) fluctuations. The above estimate implies \({\mathcal {L}}_{\varLambda _{\mathrm {sf}}} (\mathrm{p})\) and \({\mathcal {L}}_{\varLambda _{\mathrm{step}}} (\mathrm{p})\) have the same limiting fluctuation. In other words, if \(L_s=O(L)\), these two initial conditions yield to the same limiting fluctuations in the relaxation time scale.

Fig. 12
figure12

Illustration of periodic DLPP from \(\varLambda _{\mathrm {sf}}\) to \(\mathrm{p}\). In the figure on the left, the stair-shape path is \(\varLambda _{\mathrm{step}}\), the dotted line is \(\varLambda _{\mathrm{flat}}\) with particle density \(\rho \), and \(\varLambda _{\mathrm {sf}}\) lies between between \(\varLambda _{\mathrm{step}}\) and \(\varLambda _{\mathrm{flat}}\); It is the path between the white region above \(\varLambda _{\mathrm{flat}}\) and the black region below \(\varLambda _{\mathrm{step}}\). The figure on the right is a detailed view of the maximal path

See Fig. 12 for an illustration of \(\varLambda _{\mathrm{step}}\) and \(\varLambda _{\mathrm {sf}}\). From the definition, \(\varLambda _{\mathrm {sf}}\) lies on the lower left side of \(\varLambda _{\mathrm{step}}\). Hence, the last passage time from \(\varLambda _{\mathrm {sf}}\) is larger than or equal to the time from \(\varLambda _{\mathrm{step}}\). This implies that

$$\begin{aligned} {\mathcal {L}}_{\varLambda _{\mathrm {sf}}} (\mathrm{p}) \ge {\mathcal {L}}_{\varLambda _{\mathrm{step}}} (\mathrm{p}) . \end{aligned}$$

Thus, (A.1) follows is we show that

$$\begin{aligned} {\mathcal {L}}_{\varLambda _{\mathrm {sf}}} (\mathrm{p}) \le {\mathcal {L}}_{\varLambda _{\mathrm{step}}} (\mathrm{p}) +o(L^{1/2}) \text { as }L\rightarrow \infty . \end{aligned}$$
(A.2)

Let

$$\begin{aligned} \varLambda _{\mathrm{flat}}:=\{(x,y)\in {\mathbb {R}}^2: y=(1-\rho ^{-1}) x\}. \end{aligned}$$

We remark that \(\varLambda _{\mathrm{flat}}\) is not necessary a lattice path. See Fig. 12 for \(\varLambda _{\mathrm{flat}}\).

The inequality (A.2) heuristically follows from the slow decorrelation of (periodic) directed last passage percolation, which was discussed in [17]. We point out that although this paper was for the DLPP, the same argument extends to the periodic DLPP. The slow decorrelation implies, if the starting point \(\mathrm{q}\in \varLambda _{\mathrm {sf}}\) is not on \(\varLambda _{\mathrm{step}}\), say \(\mathrm{q}\) is between two corners A and B as shown in Fig. 12, then

$$\begin{aligned} {\mathcal {L}}_{\mathrm{q}} (\mathrm{p}) = -c\cdot \mathrm{{dist}}\,(\mathrm{q},{{\bar{\mathrm{q}}}}) + {\mathcal {L}}_{\bar{\mathrm{q}}} (\mathrm{p}) +o((\mathrm{{dist}}\,(\mathrm{p},\mathrm{q}))^{1/3})=-c\cdot \mathrm{{dist}}\,(\mathrm{q},{{\bar{\mathrm{q}}}}) + {\mathcal {L}}_{\bar{\mathrm{q}}} (\mathrm{p}) +o(L^{1/2}), \end{aligned}$$
(A.3)

where \(c>0\) is some constant independent of L, and \({\bar{\mathrm{q}}}\) is the intersection of the line \(\mathrm{pq}\) and \(\varLambda _{\mathrm{flat}}\).

Note that

$$\begin{aligned} {\mathcal {L}}_{\bar{\mathrm{q}}} (\mathrm{p})\le {\mathcal {L}}_{\varLambda _{\mathrm{flat}}}(\mathrm{p})={\mathcal {L}}_{\varLambda _{\mathrm{step}}}(\mathrm{p})+O(L^{1/2}), \end{aligned}$$

where the last equality follows from Theorem 6.4 (the one point distribution case) and the case we proved for step and flat initial conditions (Theorem 7.1)Footnote 3. We assume that \({{\bar{\mathrm{q}}}}\) is within an O(L) interval where the maximum path from \(\varLambda _{\mathrm{flat}}\) to \(\mathrm{p}\) is obtained. Otherwise, \({\mathcal {L}}_{\bar{\mathrm{q}}} (\mathrm{p})<{\mathcal {L}}_{\varLambda _{\mathrm{step}}}(\mathrm{p})\) for far enough \({{\bar{\mathrm{q}}}}\). And \({\mathcal {L}}_{\mathrm{q}} (\mathrm{p})<{\mathcal {L}}_{\bar{\mathrm{q}}} (\mathrm{p}) +o(L^{1/2})<{\mathcal {L}}_{\varLambda _{\mathrm{step}}} (\mathrm{p}) +o(L^{1/2})\) holds trivially. This assumption means that \({\mathcal {L}}_{\bar{\mathrm{q}}} (\mathrm{p})\) and \({\mathcal {L}}_{B} (\mathrm{p})\) have the same deterministic order terms and they only differ from the fluctuation terms, which is of \(O(L^{1/2})\).

We let C be the other intersection point of the line \(B\mathrm{q}\) with \(\varLambda _{\mathrm{step}}\). It also lies on \(\varLambda _{\mathrm {sf}}\). By the definition, \(\mathrm{{dist}}\,(A,C)=L_s\) has the same order as \(\mathrm{{dist}}\,(A,B)\). Hence \(\mathrm{{dist}}\,(\mathrm{q},{{\bar{\mathrm{q}}}})\) has the same order as \(\mathrm{{dist}}\,(B,{{\bar{\mathrm{q}}}})\).

Now we consider two situations. If \(\mathrm{{dist}}\,(\mathrm{q},{\bar{\mathrm{q}}})\ll O(L)\), then \(\mathrm{{dist}}\,(B,{{\bar{\mathrm{q}}}})\ll O(L)\). In this case, \({\mathcal {L}}_{{\bar{\mathrm{q}}}}(\mathrm{p})\) is asymptotically identical to \({\mathcal {L}}_{B}(\mathrm{p})\) since the two points B and \({{\bar{\mathrm{q}}}}\) are closer than the correlation length O(L). More precisely, we have

$$\begin{aligned} {\mathcal {L}}_{{\bar{\mathrm{q}}}}(\mathrm{p})={\mathcal {L}}_{B}(\mathrm{p})+o(L^{1/2})\le {\mathcal {L}}_{\varLambda _{\mathrm{step}}}(\mathrm{p})+o(L^{1/2}). \end{aligned}$$

Together with (A.3) we obtain

$$\begin{aligned} {\mathcal {L}}_{\mathrm{q}} (\mathrm{p})\le {\mathcal {L}}_{\varLambda _{\mathrm{step}}}(\mathrm{p})+o(L^{1/2}). \end{aligned}$$
(A.4)

The second situation is that \(\mathrm{{dist}}\,(\mathrm{q},{{\bar{\mathrm{q}}}})\gg O(L^{1/2})\). In this case, we use the trivial estimate

$$\begin{aligned} {\mathcal {L}}_{{\bar{\mathrm{q}}}}(\mathrm{p}) \le {\mathcal {L}}_{B}(\mathrm{p})+O(L^{1/2})\le {\mathcal {L}}_{\varLambda _{\mathrm{step}}}(\mathrm{p})+O(L^{1/2}), \end{aligned}$$

while the term \(O(L^{1/2})\) in this estimate is always dominated by \(-c\cdot \mathrm{{dist}}\,(\mathrm{q},{{\bar{\mathrm{q}}}})\) in (A.3). Hence we still have (A.4).

Note that \(\mathrm{q}\) is an arbitrary point on \(\varLambda _{\mathrm {sf}}\), the above estimates imply (A.2).

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Baik, J., Liu, Z. Periodic TASEP with general initial conditions. Probab. Theory Relat. Fields 179, 1047–1144 (2021). https://doi.org/10.1007/s00440-020-01004-6

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Keywords

  • Periodic TASEP
  • Multi-point distribution
  • General initial condition
  • Kardar–Parisi–Zhang Universality

Mathematics Subject Classification

  • 60K35
  • 82C22
  • 60K37