TASEP fluctuations with soft-shock initial data
Annales Henri Lebesgue, Volume 3 (2020) , pp. 999-1021.

KeywordsBurgers equation, exclusion process, KPZ universality, shock fluctuations, Tracy–Widom law

### Abstract

We consider the totally asymmetric simple exclusion process with soft-shock initial particle density, which is a step function increasing in the direction of flow and the step size chosen small to admit KPZ scaling. The initial configuration is deterministic and the dynamics create a shock.

We prove that the fluctuations of a particle at the macroscopic position of the shock converge to the maximum of two independent GOE Tracy–Widom random variables, which establishes a conjecture of Ferrari and Nejjar. Furthermore, we show the joint fluctuations of particles near the shock are determined by the maximum of two lines described in terms of these two random variables. The microscopic position of the shock is then seen to be their difference.

Our proofs rely on determinantal formulae and a novel factorization of the associated kernels.

### References

[BB19] Borodin, Alexei; Bufetov, Alexey Color-position symmetry in interacting particle systems (2019) (https://arxiv.org/abs/1905.04692)

[BCR15] Borodin, Alexei; Corwin, Ivan; Remenik, Daniel Multiplicative functionals on ensembles of non-intersecting paths, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 1, pp. 28-58 | Article | Numdam | MR 3300963 | Zbl 1357.60012

[BFPS07] Borodin, Alexei; Ferrari, Patrick L.; Prahöfer, Michael; Sasamoto, Tomohiro Fluctuation properties of the TASEP with periodic initial configurations, J. Stat. Phys., Volume 129 (2007) no. 5-6, pp. 1055-1080 | Article | MR 2363389 | Zbl 1136.82028

[BFS09] Borodin, Alexei; Ferrari, Patrick L.; Sasamoto, Tomohiro Two speed TASEP, J. Stat. Phys., Volume 137 (2009) no. 5-6, pp. 936-977 | Article | MR 2570757 | Zbl 1183.82062

[BG16] Borodin, Alexei; Gorin, Vadim Lectures on integrable probability, Probability and Statistical Physics in St. Petersburg (Sidoravicious, V.; Smirnov, V., eds.) (Proceedings of Symposia in Pure Mathematics) Volume 91, American Mathematical Society, 2016, pp. 155-214 | MR 3526828 | Zbl 1388.60157

[C11] Ben Arous, Gérard; Corwin, Ivan Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture, Ann. Probab., Volume 29 (2011) no. 1, pp. 104-138 | Article | Zbl 1208.82036

[CFP10] Corwin, Ivan; Ferrari, Patrick L.; Péché, Sandrine Limit processes for TASEP with shocks and rarefaction fans, J. Stat. Phys., Volume 140 (2010) no. 2, pp. 232-267 | Article | MR 2659279 | Zbl 1197.82078

[CQR13] Corwin, Ivan; Quastel, Jeremy; Remenik, Daniel Continuum statistics of the ${Airy}_{2}$ process, Commun. Math. Phys., Volume 317 (2013) no. 2, pp. 347-362 | Article | MR 3010187 | Zbl 1257.82112

[CQR15] Corwin, Ivan; Quastel, Jeremy; Remenik, Daniel Renormalization fixed point of the KPZ universality class, J. Stat. Phys., Volume 160 (2015) no. 4, pp. 815-834 | Article | MR 3373642 | Zbl 1327.82064

[Fer94] Ferrari, Patrick A. Shocks in one-dimensional processes with drift, Probability and Phase Transition (Grimmett, Geoffrey, ed.) (NATO ASI Series. Series C. Mathematical and Physical Sciences) Volume 420, Kluwer Academic Publishers, 1994, pp. 35-48 | Article | MR 1283174 | Zbl 0827.60082

[FFV00] Ferrari, Patrick A.; Fontes, Luiz Renato G.; Vares, Maria Elalia The asymmetric simple exclusion process with multiple shocks, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 36 (2000) no. 2, pp. 109-126 | Article

[FGN19] Ferrari, Patrick L.; Ghosal, Pratik; Nejjar, Peter Limit law of a second class particle in TASEP with non-random initial condition, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 55 (2019) no. 3, pp. 1203-1225 | Article | MR 4010933 | Zbl 07133719

[FN15a] Ferrari, Patrick L.; Nejjar, Peter Anomalous shock fluctuations in TASEP and last passage percolation models, Probab. Theory Relat. Fields, Volume 161 (2015) no. 1-2, pp. 61-109 | Article | MR 3304747 | Zbl 1311.60116

[FN15b] Ferrari, Patrick L.; Nejjar, Peter Shock fluctuations in flat TASEP under critical scaling, J. Stat. Phys., Volume 160 (2015) no. 4, pp. 985-1004 | Article | MR 3373648 | Zbl 1323.82030

[FN17] Ferrari, Patrick L.; Nejjar, Peter Fluctuations of the competition interface in presence of shocks, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 14 (2017) no. 1, pp. 299-325 | Article | MR 3637438 | Zbl 1361.60090

[FS05] Ferrari, Patrick L.; Spohn, Herbert A determinantal formula for the GOE Tracy–Widom distribution, J. Phys. A, Math. Gen., Volume 38 (2005) no. 33, p. L557-L561 | Article | MR 2165698

[Joh03] Johansson, Kurt Discrete polynuclear growth and determinantal processes, Commun. Math. Phys., Volume 242 (2003) no. 1-2, pp. 277-295 | Article | MR 2018275 | Zbl 1031.60084

[Lig99] Liggett, Thomas M. Stochastic interacting system: contact, voter and exclusion processes, Grundlehren der Mathematischen Wissenschaften, Volume 324, Springer, 1999 | MR 1717346 | Zbl 0949.60006

[MQR18] Matetski, Konstantin; Quastel, Jeremy; Remenik, Daniel The KPZ fixed point (2018) (https://arxiv.org/abs/1701.00018) | Zbl 07172193

[Nej18] Nejjar, Peter Transition to shocks in TASEP and decoupling of Last Passage Times, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 15 (2018) no. 2, pp. 1311-1334 | Article | MR 3867208 | Zbl 1414.60081

[Nej19] Nejjar, Peter GUE $×$ GUE limit law at hard shocks in ASEP (2019) (https://arxiv.org/abs/1906.07711)

[OODL + 17] Olver, Frank W. J.; Olde Daalhuis, Adri B.; Lozier, Daniel W.; Schneider, Barry I.; Boisvert, Ronald F.; Clark, Charles W.; Miller, Bruce R.; Saunders, Bonita V. NIST. Digital Library of Mathematical Functions, 2017 (http://dlmf.nist.gov/,Release 1.0.16)

[Pim17] Pimentel, Leandro P. R. Local behavior of Airy processes (2017) (https://arxiv.org/abs/1704.01903) | Zbl 1405.82029

[PS02] Prahöfer, Michael; Spohn, Herbert Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys., Volume 108 (2002) no. 5-6, pp. 1071-1106 | Article | MR 1933446 | Zbl 1025.82010

[QR13] Quastel, Jeremy; Remenik, Daniel Local behavior and hitting probabilities of the ${\text{Airy}}_{1}$ process, Probab. Theory Relat. Fields, Volume 157 (2013) no. 3-4, pp. 605-634 | Article | MR 3129799 | Zbl 1285.60095

[QR19] Quastel, Jeremy; Remenik, Daniel How flat is flat in random interface growth?, Trans. Am. Math. Soc., Volume 371 (2019) no. 9, pp. 6047-6085 | Article | MR 3937318 | Zbl 07050804

[Qua13] Quastel, Jeremy Introduction to KPZ, Current Developments in Mathematics, Volume 2011, International Press, 2013, pp. 125-194 | Zbl 1316.60019

[Rez91] Rezakhanlou, Fraydoun Hydrodynamic limit for attractive particle systems on ${\mathbf{Z}}^{d}$, Commun. Math. Phys., Volume 140 (1991) no. 3, pp. 417-448 | MR 1130693 | Zbl 0738.60098

[Sas05] Sasamoto, Tomohiro Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A, Math. Gen., Volume 38 (2005) no. 33, p. L.549-L.556 | Article | MR 2165697

[Sep98] Seppäläinen, Timo Hydrodynamic scaling, convex duality and asymptotic shapes of growth models, Markov Process. Relat. Fields, Volume 4 (1998) no. 1, pp. 1-26 | MR 1625007 | Zbl 0906.60082

[Sim05] Simon, Barry Trace ideals and their applications, Mathematical Surveys and Monographs, Volume 120, American Mathematical Society, 2005 | MR 2154153 | Zbl 1074.47001

[TW96] Tracy, Craig; Widom, Harold On orthogonal and symplectic matrix ensembles, Commun. Math. Phys., Volume 177 (1996) no. 3, pp. 727-754 | Article | MR 1385083 | Zbl 0851.60101