TASEP and generalizations: method for exact solution
Tóm tắt
The explicit biorthogonalization method, developed in [24] for continuous time TASEP, is generalized to a broad class of determinantal measures which describe the evolution of several interacting particle systems in the KPZ universality class. The method is applied to sequential and parallel update versions of each of the four variants of discrete time TASEP (with Bernoulli and geometric jumps, and with block and push dynamics) which have determinantal transition probabilities; to continuous time PushASEP; and to a version of TASEP with generalized update. In all cases, multipoint distribution functions are expressed in terms of a Fredholm determinant with an explicit kernel involving hitting times of certain random walks to a curve defined by the initial data of the system. The method is further applied to systems of interacting caterpillars, an extension of the discrete time TASEP models which generalizes sequential and parallel updates.
Tài liệu tham khảo
Arai, Y.: The KPZ fixed point for discrete time TASEPs. J. Phys. A 53(41), 415202, 33 (2020)
citation_journal_title=Ann. Inst. H. Poincaré Probab. Statist.; citation_title=Multiplicative functionals on ensembles of nonintersecting paths; citation_author=A Borodin, I Corwin, D Remenik; citation_volume=51; citation_issue=1; citation_publication_date=2015; citation_pages=28-58; citation_doi=10.1214/13-AIHP579; citation_id=CR2
citation_journal_title=I. PushASEP. Electron. J. Probab.; citation_title=Large time asymptotics of growth models on space-like paths; citation_author=A Borodin, PL Ferrari; citation_volume=13; citation_issue=50; citation_publication_date=2008; citation_pages=1380-1418; citation_id=CR3
citation_journal_title=Comm. Math. Phys.; citation_title=Anisotropic growth of random surfaces in 2 + 1 dimensions; citation_author=A Borodin, PL Ferrari; citation_volume=325; citation_issue=2; citation_publication_date=2014; citation_pages=603-684; citation_doi=10.1007/s00220-013-1823-x; citation_id=CR4
citation_journal_title=Markov Process. Related Fields; citation_title=Random tilings and Markov chains for interlacing particles; citation_author=A Borodin, PL Ferrari; citation_volume=24; citation_issue=3; citation_publication_date=2018; citation_pages=419-451; citation_id=CR5
Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process. Int. Math. Res. Pap. IMRP, Art. ID rpm002, 47 (2007)
citation_journal_title=J. Stat. Phys.; citation_title=Fluctuation properties of the TASEP with periodic initial configuration; citation_author=A Borodin, PL Ferrari, M Prähofer, T Sasamoto; citation_volume=129; citation_issue=5-6; citation_publication_date=2007; citation_pages=1055-1080; citation_doi=10.1007/s10955-007-9383-0; citation_id=CR7
citation_journal_title=Comm. Math. Phys.; citation_title=Large time asymptotics of growth models on space-like paths II PNG and parallel TASEP; citation_author=A Borodin, PL Ferrari, T Sasamoto; citation_volume=283; citation_issue=2; citation_publication_date=2008; citation_pages=417-449; citation_doi=10.1007/s00220-008-0515-4; citation_id=CR8
Borodin, A., Gorin, V.: Lectures on integrable probability. In: Probability and statistical physics in St. Petersburg. Vol. 91. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, pp. 155-214 (2016)
citation_journal_title=Phys. Rev. E (3); citation_title=Generalized determinant solution of the discrete-time totally asymmetric exclusion process and zero-range process; citation_author=JG Brankov, VB Priezzhev, RV Shelest; citation_volume=69; citation_issue=6; citation_publication_date=2004; citation_doi=10.1103/PhysRevE.69.066136; citation_id=CR10
citation_journal_title=J. Stat. Phys.; citation_title=Limit processes for TASEP with shocks and rarefaction fans; citation_author=I Corwin, PL Ferrari, S Péché; citation_volume=140; citation_issue=2; citation_publication_date=2010; citation_pages=232-267; citation_doi=10.1007/s10955-010-9995-7; citation_id=CR11
citation_journal_title=J. Stat. Mech. Theory Exp.; citation_title=The totally asymmetric exclusion process with generalized update; citation_author=AE Derbyshev, SS Poghosyan, AM Povolotsky, VB Priezzhev; citation_volume=5; citation_publication_date=2012; citation_pages=P05014, 13; citation_id=CR12
citation_journal_title=J. Phys. A; citation_title=Dynamics of an anchored Toom interface; citation_author=B Derrida, JL Lebowitz, ER Speer, H Spohn; citation_volume=24; citation_issue=20; citation_publication_date=1991; citation_pages=4805-4834; citation_doi=10.1088/0305-4470/24/20/015; citation_id=CR13
citation_journal_title=Ann. Inst. Henri Poincaré Probab. Stat.; citation_title=Determinantal transition kernels for some interacting particles on the line; citation_author=AB Dieker, J Warren; citation_volume=44; citation_issue=6; citation_publication_date=2008; citation_pages=1162-1172; citation_doi=10.1214/07-AIHP176; citation_id=CR14
citation_journal_title=I. Eigenval. correlat. J. Phys. A; citation_title=Matrices coupled in a chain; citation_author=B Eynard, ML Mehta; citation_volume=31; citation_issue=19; citation_publication_date=1998; citation_pages=4449-4456; citation_doi=10.1088/0305-4470/31/19/010; citation_id=CR15
citation_journal_title=Nuclear Phys. B; citation_title=Fluctuations of the one-dimensional polynuclear growth model with external sources; citation_author=T Imamura, T Sasamoto; citation_volume=699; citation_issue=3; citation_publication_date=2004; citation_pages=503-544; citation_doi=10.1016/j.nuclphysb.2004.07.030; citation_id=CR16
citation_journal_title=Comm. Math. Phys.; citation_title=Shape fluctuations and random matrices; citation_author=K Johansson; citation_volume=209; citation_issue=2; citation_publication_date=2000; citation_pages=437-476; citation_doi=10.1007/s002200050027; citation_id=CR17
citation_journal_title=Comm. Math. Phys.; citation_title=Discrete polynuclear growth and determinantal processes; citation_author=K Johansson; citation_volume=242; citation_issue=1-2; citation_publication_date=2003; citation_pages=277-329; citation_doi=10.1007/s00220-003-0945-y; citation_id=CR18
Johansson, K.: Random matrices and determinantal processes. In: Mathematical statistical physics. Elsevier B. V., Amsterdam, pp. 1-55 (2006)
citation_journal_title=Ark. Mat.; citation_title=A multi-dimensional Markov chain and the Meixner ensemble; citation_author=K Johansson; citation_volume=48; citation_issue=1; citation_publication_date=2010; citation_pages=79-95; citation_doi=10.1007/s11512-008-0089-6; citation_id=CR20
Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric orthogonal polynomials and their q-analogues. Springer Monographs in Mathematics. With a foreword by Tom H. Koornwinder. Springer-Verlag, Berlin, 2010, pp. xx+578
citation_journal_title=J. Stat. Phys.; citation_title=Scale invariance of the PNG droplet and the Airy process; citation_author=S Karlin, G McGregor; citation_volume=108; citation_issue=5-6; citation_publication_date=2002; citation_pages=1071-1106; citation_id=CR22
Liechty, K., Nguyen, G.B., Remenik, D.: Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy-Widom GOE distribution. (2020).
arXiv: 2009.07781
Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point. To appear in Acta Math. (2021).
arXiv: 1701.00018
Nica, M., Quastel, J., Remenik, D.: One-sided reflected Brownian motions and the KPZ fixed point. Forum Math. Sigma 8 , Paper No. e63, 16 (2020)
citation_journal_title=Ann. Probab.; citation_title=Solution of the Kolmogorov equation for TASEP; citation_author=M Nica, J Quastel, D Remenik; citation_volume=48; citation_issue=5; citation_publication_date=2020; citation_pages=2344-2358; citation_doi=10.1214/20-AOP1425; citation_id=CR26
citation_journal_title=Probab. Theory Related Fields; citation_title=Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes; citation_author=L Petrov; citation_volume=160; citation_issue=3-4; citation_publication_date=2014; citation_pages=429-487; citation_doi=10.1007/s00440-013-0532-x; citation_id=CR27
citation_journal_title=J. Stat. Mech.: Theory Exp; citation_title=Determinant solution for the totally asymmetric exclusion process with parallel update; citation_author=AM Povolotsky, VB Priezzhev; citation_volume=2006; citation_issue=07; citation_publication_date=2006; citation_pages=P07002-P07002; citation_doi=10.1088/1742-5468/2006/07/P07002; citation_id=CR28
citation_journal_title=J. Stat. Mech.: Theory Exp; citation_title=Universal exit probabilities in the TASEP; citation_author=SS Poghosyan, AM Povolotsky, VB Priezzhev; citation_volume=2012; citation_issue=08; citation_publication_date=2012; citation_pages=P08013; citation_doi=10.1088/1742-5468/2012/08/P08013; citation_id=CR29
citation_journal_title=J. Stat. Phys.; citation_title=Generalized Green functions and current correlations in the TASEP; citation_author=AM Povolotsky, VB Priezzhev, GM Schütz; citation_volume=142; citation_issue=4; citation_publication_date=2011; citation_pages=754-791; citation_doi=10.1007/s10955-011-0133-y; citation_id=CR30
citation_journal_title=J. Stat. Phys.; citation_title=Scale invariance of the PNG droplet and the Airy process; citation_author=M Prähofer, H Spohn; citation_volume=108; citation_issue=5-6; citation_publication_date=2002; citation_pages=1071-1106; citation_doi=10.1023/A:1019791415147; citation_id=CR31
citation_journal_title=J. Statist. Phys.; citation_title=Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process; citation_author=A Rákos, GM Schütz; citation_volume=118; citation_issue=3; citation_publication_date=2005; citation_pages=511-530; citation_doi=10.1007/s10955-004-8819-z; citation_id=CR32
citation_journal_title=J. Phys. A Math. Gener.; citation_title=Spatial correlations of the 1D KPZ surface on a flat substrate; citation_author=T Sasamoto; citation_volume=38; citation_issue=33; citation_publication_date=2005; citation_pages=L549; citation_doi=10.1088/0305-4470/38/33/L01; citation_id=CR33
citation_journal_title=J. Statist. Phys.; citation_title=Exact solution of the master equation for the asymmetric exclusion process; citation_author=GM Schütz; citation_volume=88; citation_issue=1-2; citation_publication_date=1997; citation_pages=427-445; citation_doi=10.1007/BF02508478; citation_id=CR34
citation_journal_title=Comm. Math. Phys.; citation_title=Level-spacing distributions and the Airy kernel; citation_author=CA Tracy, H Widom; citation_volume=159; citation_issue=1; citation_publication_date=1994; citation_pages=151-174; citation_doi=10.1007/BF02100489; citation_id=CR35
citation_journal_title=Comm. Math. Phys.; citation_title=On orthogonal and symplectic matrix ensembles; citation_author=CA Tracy, H Widom; citation_volume=177; citation_issue=3; citation_publication_date=1996; citation_pages=727-754; citation_doi=10.1007/BF02099545; citation_id=CR36
citation_journal_title=Electron. J. Probab.; citation_title=Dyson’s Brownian motions, intertwining and interlacing; citation_author=J Warren; citation_volume=12; citation_issue=19; citation_publication_date=2007; citation_pages=573-590; citation_id=CR37
citation_journal_title=Electron. J. Probab.; citation_title=Some examples of dynamics for Gelfand-Tsetlin patterns; citation_author=J Warren, P Windridge; citation_volume=14; citation_issue=59; citation_publication_date=2009; citation_pages=1745-1769; citation_id=CR38