# Generalizations of TASEP in Discrete and Continuous Inhomogeneous Space

- 19 Downloads

## Abstract

We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson–Schensted–Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams. For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation results which put the systems into the Kardar–Parisi–Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy–Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions. A homogeneous version of a discrete space system we consider is a one-parameter deformation of the geometric last-passage percolation, and we obtain extensions of the limit shape parabola and the corresponding asymptotic fluctuation results. The exact solvability and asymptotic behavior results are powered by a new nontrivial connection to Schur measures and processes.

## Notes

### Acknowledgements

We are grateful to Guillaume Barraquand, Riddhipratim Basu, Alexei Borodin, Eric Cator, Francis Comets, Ivan Corwin, Patrik Ferrari, Vadim Gorin, Pavel Krapivsky, Alexander Povolotsky, Timo Seppäläinen, and Jon Warren for helpful discussions. A part of the work was completed when the authors attended the 2017 IAS PCMI Summer Session on Random Matrices, and we are grateful to the organizers for the hospitality and support. AK was partially supported by the NSF Grant DMS-1704186. LP was partially supported by the NSF grant DMS-1664617.

## Supplementary material

## References

- [Agg18]Aggarwal, A.: Current fluctuations of the stationary ASEP and six-vertex model. Duke Math J.
**167**(2), 269–384 (2018). arXiv:1608.04726 [math.PR]MathSciNetzbMATHGoogle Scholar - [And82]Andjel, E.: Invariant measures for the zero range process. Ann. Probab.
**10**(3), 525–547 (1982)MathSciNetzbMATHGoogle Scholar - [AG05]Andjel, E., Guiol, H.: Long-range exclusion processes, generator and invariant measures. Ann. Probab.
**33**(6), 2314–2354 (2005). arXiv:math/0411655 [math.PR]MathSciNetzbMATHGoogle Scholar - [AK84]Andjel, E., Kipnis, C.: Derivation of the hydrodynamical equation for the zero-range interaction process. Ann. Probab.
**12**(2), 325–334 (1984)MathSciNetzbMATHGoogle Scholar - [Bai06]Baik, J.: Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math J.
**133**(2), 205–235 (2006). arXiv:math/0504606 [math.PR]MathSciNetzbMATHGoogle Scholar - [BBP05]Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab.
**33**(5), 1643–1697 (2005). arXiv:math/0403022 [math.PR]MathSciNetzbMATHGoogle Scholar - [BDJ99]Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. AMS
**12**(4), 1119–1178 (1999). arXiv:math/9810105 [math.CO]MathSciNetzbMATHGoogle Scholar - [BKS12]Balász, M., Komjáthy, J., Seppäläinen, T.: Microscopic concavity and fluctuation bounds in a class of deposition processes. Ann. Inst. H. Poincaré B
**48**, 151–187 (2012)ADSMathSciNetzbMATHGoogle Scholar - [Bar15]Barraquand, G.: A phase transition for q-TASEP with a few slower particles. Stoch. Process. Appl.
**125**(7), 2674–2699 (2015). arXiv:1404.7409 [math.PR]MathSciNetzbMATHGoogle Scholar - [Bar01]Baryshnikov, Yu.: GUEs and queues. Probab. Theory Relat. Fields
**119**, 256–274 (2001)MathSciNetzbMATHGoogle Scholar - [BSS17]Basu, R., Sarkar, S., Sly, A.: Invariant measures for TASEP with a slow bond (2017). arXiv preprint arXiv:1704.07799
- [BSS14]Basu, R., Sidoravicius, V., Sly, A.: Last passage percolation with a defect line and the solution of the slow bond problem (2014). arXiv preprint arXiv:1408.3464 [math.PR]
- [BNKR94]Ben-Naim, E., Krapivsky, P., Redner, S.: Kinetics of clustering in traffic flows. Phys. Rev. E.
**50**(2), 822–829 (1994). arXiv:cond-mat/9402054 ADSGoogle Scholar - [Ben+99]Bengrine, M., Benyoussef, A., Ez-Zahraouy, H., Krug, J., Loulidi, M., Mhirech, F.: A simulation study of an asymmetric exclusion model with disorder. MJ Condens. Matter
**2**(1), 117–126 (1999)Google Scholar - [Bla11]Blank, M.: Exclusion-type spatially heterogeneous processes in continua. J. Stat. Mech.
**2011**(06), P06016 (2011). arXiv:1105.4232 [math.DS]Google Scholar - [Bla12]Blank, M.: Discrete time TASEP in heterogeneous continuum. Markov Process. Relat. Fields
**18**(3), 531–552 (2012)MathSciNetzbMATHGoogle Scholar - [Bor10]Bornemann, Folkmar: On the numerical evaluation of Fredholm determinants. Math. Comput.
**79**(270), 871–915 (2010). arXiv:0804.2543 [math.NA]ADSMathSciNetzbMATHGoogle Scholar - [Bor11]Borodin, A.: Determinantal point processes. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011). arXiv:0911.1153 [math.PR]
- [Bor17]Borodin, A.: On a family of symmetric rational functions. Adv. Math.
**306**, 973–1018 (2017). arXiv:1410.0976 [math.CO]MathSciNetzbMATHGoogle Scholar - [BC14]Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields
**158**, 225–400 (2014). arXiv:1111.4408 [math.PR]MathSciNetzbMATHGoogle Scholar - [BCF14]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]MathSciNetzbMATHGoogle Scholar - [BD11]Borodin, A., Duits, M.: Limits of determinantal processes near a tacnode. Annales de l’institut Henri Poincaré (B)
**47**(1), 243–258 (2011). arXiv:0911.1980 [math.PR]ADSMathSciNetzbMATHGoogle Scholar - [BF14]Borodin, A., Ferrari, P.: Anisotropic growth of random surfaces in \(2+1\) dimensions. Commun. Math. Phys.
**325**, 603–684 (2014). arXiv:0804.3035 [math-ph]ADSMathSciNetzbMATHGoogle Scholar - [BFPS07]Borodin, A., Ferrari, P., Prähofer, M., Sasamoto, T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys.
**129**(5–6), 1055–1080 (2007). arXiv:math-ph/0608056 ADSMathSciNetzbMATHGoogle Scholar - [BFS09]Borodin, A., Ferrari, P., Sasamoto, T.: Two speed TASEP. J. Stat. Phys
**137**(5), 936–977 (2009). arXiv:0904.4655 [math-ph]ADSMathSciNetzbMATHGoogle Scholar - [BK08]Borodin, A., Kuan, J.: Asymptotics of Plancherel measures for the infinite-dimensional unitary group. Adv. Math.
**219**(3), 894–931 (2008). arXiv:0712.1848 [math.RT]MathSciNetzbMATHGoogle Scholar - [BOO00]Borodin, A., Okounkov, A., Olshanski, G.: Asymptotics of Plancherel measures for symmetric groups. J. AMS
**13**(3), 481–515 (2000). arXiv:math/9905032 [math.CO]MathSciNetzbMATHGoogle Scholar - [BO07]Borodin, A., Olshanski, G.: Asymptotics of Plancherel-type random partitions. J. Algebra
**313**(1), 40–60 (2007). arXiv:math/0610240 MathSciNetzbMATHGoogle Scholar - [BO16]Borodin, A., Olshanski, G.: Representations of the Infinite Symmetric Group, vol. 160. Cambridge University Press, Cambridge (2016)zbMATHGoogle Scholar
- [BO17]Borodin, A., Olshanski, G.: The ASEP and determinantal point processes. Commun. Math. Phys.
**353**(2), 853–903 (2017). arXiv:1608.01564 [math-ph]ADSMathSciNetzbMATHGoogle Scholar - [BP08]Borodin, A., Peche, S.: Airy kernel with two sets of parameters in directed percolation and random matrix theory. J. Stat. Phys.
**132**(2), 275–290 (2008). arXiv:0712.1086v3 [math-ph]ADSMathSciNetzbMATHGoogle Scholar - [BP16a]Borodin, A., Petrov, L.: Lectures on integrable probability: stochastic vertex models and symmetric functions. Lecture Notes of the Les Houches Summer School
**104**(2016). arXiv:1605.01349 [math.PR] - [BP16b]Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math.
**300**, 71–155 (2016). arXiv:1305.5501 [math.PR]MathSciNetzbMATHGoogle Scholar - [BP18a]Borodin, A., Petrov, L.: Higher spin six vertex model and symmetric rational functions. Selecta Math.
**24**(2), 751–874 (2018). arXiv:1601.05770 [math.PR]MathSciNetzbMATHGoogle Scholar - [BP18b]Borodin, A., Petrov, L.: Inhomogeneous exponential jump model. Probab. Theory Relat. Fields
**172**, 323–385 (2018). arXiv:1703.03857 [math.PR]MathSciNetzbMATHGoogle Scholar - [BM18]Bufetov, A., Matveev, K.: Hall–Littlewood RSK field. Selecta Math.
**24**(5), 4839–4884 (2018). arXiv:1705.07169 [math.PR]MathSciNetzbMATHGoogle Scholar - [BP17]Bufetov, A., Petrov, L.: Yang-Baxter field for spin Hall–Littlewood symmetric functions (2017). arXiv preprint arXiv:1712.04584 [math.PR]
- [Cal15]Calder, J.: Directed last passage percolation with discontinuous weights. J. Stat. Phys.
**158**(4), 903–949 (2015)ADSMathSciNetzbMATHGoogle Scholar - [CG18]Ciech, F., Georgiou, N.: Last passage percolation in an exponential environment with discontinuous rates (2018). arXiv preprint arXiv:1808.00917 [math.PR]
- [CEP96]Cohn, Henry, Elkies, Noam, Propp, James: Local statistics for random domino tilings of the Aztec diamond. Duke Math. J.
**85**(1), 117–166 (1996). https://doi.org/10.1215/S0012-7094-96-08506-3. arXiv:math/0008243 [math.CO]MathSciNetzbMATHGoogle Scholar - [Cor12]Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl.
**1**, 1130001 (2012). arXiv:1106.1596 [math.PR]MathSciNetzbMATHGoogle Scholar - [Cor16]Corwin, I.: Kardar–Parisi–Zhang Universality. Not. AMS
**63**(3), 230–239 (2016)MathSciNetzbMATHGoogle Scholar - [CP16]Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys.
**343**(2), 651–700 (2016). arXiv:1502.07374 [math.PR]ADSMathSciNetzbMATHGoogle Scholar - [CT18]Corwin, I., Tsai, L.-C.: SPDE Limit of Weakly Inhomogeneous ASEP (2018). arXiv preprint arXiv:1806.09682 [math.PR]
- [CLST13]Costin, O., Lebowitz, J., Speer, E., Troiani, A.: The blockage problem. Bull. Inst. Math. Acad. Sinica (New Series)
**8**(1), 47–72 (2013). arXiv:1207.6555 [math-ph]MathSciNetzbMATHGoogle Scholar - [DPPP12]Derbyshev, A., Poghosyan, S., Povolotsky, A., Priezzhev, V.: The totally asymmetric exclusion process with generalized update. J. Stat. Mech. (P05014) (2012). arXiv:1203.0902 [cond-mat.stat-mech]
- [DPP15]Derbyshev, A., Povolotsky, A., Priezzhev, V.: Emergence of jams in the generalized totally asymmetric simple exclusion process. Phys. Rev. E
**91**(2), 022125 (2015). arXiv:1410.2874 [math-ph]ADSGoogle Scholar - [DLSS91]Derrida, B., Lebowitz, J., Speer, E., Spohn, H.: Dynamics of an anchored Toom interface. J. Phys. A
**24**(20), 4805 (1991)ADSMathSciNetzbMATHGoogle Scholar - [DW08]Dieker, A.B., Warren, J.: Determinantal transition kernels for some interacting particles on the line. Annales de l’Institut Henri Poincaré
**44**(6), 1162–1172 (2008). arXiv:0707.1843 [math.PR]MathSciNetzbMATHGoogle Scholar - [DZS08]Dong, J., Zia, R., Schmittmann, B.: Understanding the edge effect in TASEP with mean-field theoretic approaches. J. Phys. A
**42**(1), 015002 (2008). arXiv:0809.1974 [cond-mat.stat-mech]ADSMathSciNetzbMATHGoogle Scholar - [Dui13]Duits, M.: The Gaussian free field in an interlacing particle system with two jump rates. Commun. Pure Appl. Math.
**66**(4), 600–643 (2013). arXiv:1105.4656 [math-ph]MathSciNetzbMATHGoogle Scholar - [Edr52]Edrei, A.: On the generating functions of totally positive sequences. II. J. Analyse Math.
**2**, 104–109 (1952). ISSN: 0021-7670Google Scholar - [Emr16]Emrah, E.: Limit shapes for inhomogeneous corner growth models with exponential and geometric weights. Electron. Commun. Probab.
**21**(42), 16 (2016). arXiv:1502.06986 [math.PR]MathSciNetzbMATHGoogle Scholar - [Erd53]Erdélyi, A. (ed.): Higher Transcendental Functions. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
- [EM98]Eynard, B., Mehta, M.L.: Matrices coupled in a chain: I. Eigenvalue correlations. J. Phys. A
**31**, 4449–4456 (1998)ADSMathSciNetzbMATHGoogle Scholar - [Fer08]Ferrari, P.: The universal \(\text{Airy}_1\) and \(\text{ Airy }_2\) processes in the totally asymmetric simple exclusion process. In: Baik, J., Kriecherbauer, T., Li, L.-C., McLaughlin, K.T.-R., Tomei, C. (eds.) Integrable Systems and Random Matrices: In Honor of Percy Deift, Contemporary Mathematics, pp. 321–332. AMS (2008). arXiv:math-ph/0701021
- [FS11]Ferrari, P., Spohn, H.: Random growth models. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011). arXiv:1003.0881 [math.PR]
- [FNH99]Forrester, P.J., Nagao, T., Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl. Phys. B
**553**(3), 601–643 (1999). arXiv:cond-mat/9811142 [cond-mat.mes-hall]ADSMathSciNetzbMATHGoogle Scholar - [Ful97]Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, Cambridge (1997). ISBN: 0521567246Google Scholar
- [GKS10]Georgiou, N., Kumar, R., Seppäläinen, T.: TASEP with discontinuous jump rates. ALEA Lat. Am. J. Probab. Math. Stat.
**7**, 293–318 (2010). arXiv:1003.3218 [math.PR]MathSciNetzbMATHGoogle Scholar - [GTW02]Gravner, J., Tracy, C., Widom, H.: Fluctuations in the composite regime of a disordered growth model. Commun. Math. Phys.
**229**, 433–458 (2002)ADSMathSciNetzbMATHGoogle Scholar - [Gre74]Greene, C.: An extension of Schensted’s theorem. Adv. Math.
**14**(2), 254–265 (1974)MathSciNetzbMATHGoogle Scholar - [Gui97]Guiol, H.: Un résultat pour le processus d’exclusion à longue portée [A result for the long-range exclusion process]. In: Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol. 33(4), pp. 387–405 (1997)Google Scholar
- [HHT15]Halpin-Healy, T., Takeuchi, K.: A KPZ cocktail-shaken, not stirred. J. Stat. Phys
**160**(4), 794–814 (2015). arXiv:1505.01910 [cond-mat.stat-mech]ADSMathSciNetzbMATHGoogle Scholar - [Hel01]Helbing, D.: Traffic and related self-driven many-particle systems. Rev. Mod. Phys.
**73**(4), 1067–1141 (2001). arXiv:cond-mat/0012229 [cond-mat.stat-mech]ADSMathSciNetGoogle Scholar - [HKPV06]Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probab. Surv.
**3**, 206–229 (2006). arXiv:math/0503110 [math.PR]MathSciNetzbMATHGoogle Scholar - [IS05]Imamura, T., Sasamoto, T.: Polynuclear growth model with external source and random matrix model with deterministic source. Phys. Rev. E.
**71**(4), 041606 (2005)ADSGoogle Scholar - [IS07]Imamura, T., Sasamoto, T.: Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition. J. Stat. Phys.
**128**(4), 799–846 (2007). arXiv:math-ph/0702009 ADSMathSciNetzbMATHGoogle Scholar - [ITW01]Its, A., Tracy, C., Widom, H.: Random words, Toeplitz determinants and integrable systems. In: Bleher, P., Its, A. (eds.) Random Matrices and Their Applications, vol. 40. MSRI Publications, Providence (2001)Google Scholar
- [JL92]Janowsky, S., Lebowitz, J.: Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process. Phys. Rev. A
**45**(2), 618 (1992)ADSGoogle Scholar - [JPS98]Jockusch, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem (1998). arXiv preprint arXiv:math/9801068 [math.CO]
- [Joh00]Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys.
**209**(2), 437–476 (2000). arXiv:math/9903134 [math.CO]ADSMathSciNetzbMATHGoogle Scholar - [Joh03]Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys.
**242**(1), 277–329 (2003). arXiv:math/0206208 [math.PR]ADSMathSciNetzbMATHGoogle Scholar - [Joh16]Johansson, K.: Two time distribution in Brownian directed percolation. Commun. Math. Phys.
**351**, 1–52 (2016). arXiv:1502.00941 [math-ph]MathSciNetGoogle Scholar - [Joh18]Johansson, K.: The two-time distribution in geometric last-passage percolation (2018). arXiv preprint arXiv:1802.00729 [math.PR]
- [Kru91]Krug, J.: Boundary-induced phase transitions in driven diffusive systems. Phys. Rev. Lett.
**67**(14), 1882–1885 (1991)ADSMathSciNetGoogle Scholar - [Kru00]Krug, J.: Phase separation in disordered exclusion models. Braz. J. Phys.
**30**(1), 97–104 (2000). arXiv:cond-mat/9912411 [cond-mat.stat-mech]ADSGoogle Scholar - [KF96]Krug, J., Ferrari, P.A.: Phase transitions in driven diffusive systems with random rates. J. Phys. A Math. Gen.
**29**(18), L465 (1996)ADSzbMATHGoogle Scholar - [Lan96]Landim, C.: Hydrodynamical limit for space inhomogeneous one-dimensional totally asymmetric zero-range processes. Ann. Probab.
**24**(2), 599–638 (1996)MathSciNetzbMATHGoogle Scholar - [Lig73]Liggett, T.: An infinite particle system with zero range interactions. Ann. Probab.
**1**(2), 240–253 (1973)MathSciNetzbMATHGoogle Scholar - [Lig76]Liggett, T.: Coupling the simple exclusion process. Ann. Probab.
**4**, 339–356 (1976)MathSciNetzbMATHGoogle Scholar - [Lig99]Liggett, T.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Volume 324. Grundlehren de mathematischen Wissenschaften. Springer, Berlin (1999)Google Scholar
- [Lig05]Liggett, T.: Interacting Particle Systems. Springer, Berlin (2005)zbMATHGoogle Scholar
- [MGP68]MacDonald, C., Gibbs, J., Pipkin, A.: Kinetics of biopolymerization on nucleic acid templates. Biopolymers
**6**(1), 1–25 (1968)Google Scholar - [Mac95]Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
- [Mac94]Macêdo, A.M.S.: Universal parametric correlations at the soft edge of the spectrum of random matrix ensembles. Europhys. Lett.
**26**(9), 641 (1994). arXiv:cond-mat/9404038 ADSGoogle Scholar - [Mar09]Martin, J.: Batch queues, reversibility and first-passage percolation. Queueing Syst.
**62**(4), 411–427 (2009). arXiv:0902.2026 [math.PR]MathSciNetzbMATHGoogle Scholar - [MQR17]Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point (2017). arXiv preprint arXiv:1701.00018 [math.PR]
- [O’C03a]O’Connell, N.: A path-transformation for random walks and the Robinson–Schensted correspondence. Trans. AMS
**355**(9), 3669–3697 (2003)MathSciNetzbMATHGoogle Scholar - [O’C03b]O’Connell, N.: Conditioned random walks and the RSK correspondence. J. Phys. A
**36**(12), 3049–3066 (2003)ADSMathSciNetzbMATHGoogle Scholar - [OP13]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]MathSciNetzbMATHGoogle Scholar - [Oko01]Okounkov, A.: Infinite wedge and random partitions. Selecta Math.
**7**(1), 57–81 (2001). arXiv:math/9907127 [math.RT]MathSciNetzbMATHGoogle Scholar - [OR03]Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. AMS
**16**(3), 581–603 (2003). arXiv:math/0107056 [math.CO]MathSciNetzbMATHGoogle Scholar - [OP17]Orr, D., Petrov, L.: Stochastic higher spin six vertex model and \(q\)-TASEPs. Adv. Math.
**317**, 473–525 (2017)MathSciNetzbMATHGoogle Scholar - [Pet19]Petrov, L.: PushTASEP in inhomogeneous space (2019). In preparationGoogle Scholar
- [Pov13]Povolotsky, A.: On integrability of zero-range chipping models with factorized steady state. J. Phys. A
**46**, 465205 (2013). arXiv:1308.3250 [math-ph]ADSMathSciNetzbMATHGoogle Scholar - [PS02]Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys.
**108**, 1071–1106 (2002). arXiv:math.PR/0105240 MathSciNetzbMATHGoogle Scholar - [QS15]Quastel, J., Spohn, H.: The one-dimensional KPZ equation and its universality class. J. Stat. Phys.
**160**(4), 965–984 (2015). arXiv:1503.06185 [math-ph]ADSMathSciNetzbMATHGoogle Scholar - [Rez91]Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on \(Z^d\). Commun. Math. Phys.
**140**(3), 417–448 (1991)ADSzbMATHGoogle Scholar - [RT08]Rolla, L., Teixeira, A.: Last passage percolation in macroscopically inhomogeneous media. Electron. Commun. Probab.
**13**, 131–139 (2008)MathSciNetzbMATHGoogle Scholar - [Ros81]Rost, H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete
**58**(1), 41–53 (1981). https://doi.org/10.1007/BF00536194 ADSMathSciNetzbMATHGoogle Scholar - [Sag01]Sagan, B.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Springer, Berlin (2001). ISBN: 0387950672Google Scholar
- [Sep98]Seppäläinen, T.: Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Relat. Fields
**4**(1), 1–26 (1998)MathSciNetzbMATHGoogle Scholar - [Sep99]Seppäläinen, T.: Existence of hydrodynamics for the totally asymmetric simple k-exclusion process. Ann. Probab.
**27**(1), 361–415 (1999)MathSciNetzbMATHGoogle Scholar - [Sep01]Seppäläinen, T.: Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond. J. Stat. Phys.
**102**(1–2), 69–96 (2001). arXiv:math/0003049 [math.PR]MathSciNetzbMATHGoogle Scholar - [SK99]Seppäläinen, T., Krug, J.: Hydrodynamics and platoon formation for a totally asymmetric exclusion model with particlewise disorder. J. Stat. Phys.
**95**(3–4), 525–567 (1999)ADSMathSciNetzbMATHGoogle Scholar - [Sim05]Simon, B.: Trace Ideals and Their Applications, Second Edition, Mathematical Surveys and Monographs, vol. 120. AMS, Providence (2005)Google Scholar
- [Sos00]Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv.
**55**(5), 923–975 (2000). arXiv:math/0002099 [math.PR]MathSciNetzbMATHGoogle Scholar - [Spi70]Spitzer, F.: Interaction of Markov processes. Adv. Math.
**5**(2), 246–290 (1970)MathSciNetzbMATHGoogle Scholar - [Spo91]Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)zbMATHGoogle Scholar
- [Sta01]Stanley, R.: Enumerative Combinatorics. Volume 2. With a Foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin. Cambridge University Press, Cambridge (2001)Google Scholar
- [Tho64]Thoma, E.: Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Zeitschr
**85**, 40–61 (1964)zbMATHGoogle Scholar - [TTCB10]Thompson, A.G., Tailleur, J., Cates, M.E., Blythe, R.A.: Zero-range processes with saturated condensation: the steady state and dynamics. J. Stat. Mech.
**02**, P02013 (2010). arXiv:0912.3009 [cond-mat.stat-mech]Google Scholar - [TW94]Tracy, C., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys.
**159**(1), 151–174 (1994). arXiv:hep-th/9211141. ISSN: 0010-3616 - [TW09]Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys.
**290**, 129–154 (2009). arXiv:0807.1713 [math.PR]ADSMathSciNetzbMATHGoogle Scholar - [WW09]Warren, J., Windridge, P.: Some examples of dynamics for Gelfand–Tsetlin patterns. Electron. J. Probab.
**14**, 1745–1769 (2009). arXiv:0812.0022 [math.PR]MathSciNetzbMATHGoogle Scholar - [Woe05]Woelki, M.: Steady states of discrete mass transport models. PhD. thesis, Master thesis, University of Duisburg-Essen (2005)Google Scholar
- [ZDS11]Zia, R.K.P., Dong, J.J., Schmittmann, B.: Modeling translation in protein synthesis with TASEP: a tutorial and recent developments. J. Stat. Phys.
**144**, 405 (2011)ADSMathSciNetzbMATHGoogle Scholar