×

A Pfaffian representation for flat ASEP. (English) Zbl 1366.82024

The authors consider a 1-dimensional asymmetric simple exclusion process (ASEP) with flat initial data, i.e., initially even sites are occupied and odd sites are empty. The particles then perform nearest-neighbor asymmetric random walks, in continuous time, the only interaction being that jumps to already occupied sites are suppressed. The main result of the paper is a Fredholm Pfaffian formula for an appropriate generating function of the height function of the ASEP. Formal asymptotics give the correct fluctuations in the large-time limit, which in this case correspond to the GOE Tracy-Widom distribution [C. A. Tracy and H. Widom, Commun. Math. Phys. 177, No. 3, 727–754 (1996; Zbl 0851.60101)] in a new form.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
35Q82 PDEs in connection with statistical mechanics

Citations:

Zbl 0851.60101
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amir, Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions, Comm. Pure Appl. Math. 64 (4) pp 466– (2011) · Zbl 1222.82070 · doi:10.1002/cpa.20347
[2] Andréief, Note sur une relation entre les intégrales définies des produits des fonctions, Mém. de la Soc. Sci. Bordeaux 2 pp 1– (1883)
[3] Bertini, The stochastic heat equation: Feynman-Kac formula and intermittence, J. Statist. Phys. 78 (5-6) pp 1377– (1995) · Zbl 1080.60508 · doi:10.1007/BF02180136
[4] Bornemann, The Airy1 process is not the limit of the largest eigenvalue in GOE matrix diffusion, J. Stat. Phys. 133 (3) pp 405– (2008) · Zbl 1161.82340 · doi:10.1007/s10955-008-9621-0
[5] Borodin, Macdonald processes, Probab. Theory Related Fields 158 (1-2) pp 225– (2014) · Zbl 1291.82077 · doi:10.1007/s00440-013-0482-3
[6] Borodin, Free energy fluctuations for directed polymers in random media in 1+1 dimension, Comm. Pure Appl. Math. 67 (7) pp 1129– (2014) · Zbl 1295.82035 · doi:10.1002/cpa.21520
[7] Borodin, Height fluctuations for the stationary KPZ equation, Math. Phys. Anal. Geom. 18 (1) pp 95– (2015) · Zbl 1332.82068 · doi:10.1007/s11040-015-9189-2
[8] Borodin, From duality to determinants for q-TASEP and ASEP, Ann. Probab. 42 (6) pp 2314– (2014) · Zbl 1304.82048 · doi:10.1214/13-AOP868
[9] Borodin, Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys. 129 (5-6) pp 1055– (2007) · Zbl 1136.82028 · doi:10.1007/s10955-007-9383-0
[10] Calabrese, Interaction quench in a Lieb-Liniger model and the KPZ equation with flat initial conditions, J. Stat. Mech. Theory Exp. 2014 (5) pp P05004– · doi:10.1088/1742-5468/2014/05/P05004
[11] Calabrese, Free-energy distribution of the directed polymer at high temperature, EPL (Europhysics Letters) 90 (2) pp 20002– (2010) · doi:10.1209/0295-5075/90/20002
[12] Corwin, Crossover distributions at the edge of the rarefaction fan, Ann. Probab. 41 (3A) pp 1243– (2013) · Zbl 1285.82034 · doi:10.1214/11-AOP725
[13] Bruijn, On some multiple integrals involving determinants, J. Indian Math. Soc. (N.S.) 19 pp 133– (1955) · Zbl 0068.24904
[14] Nardis, Solution for an interaction quench in the Lieb-Liniger Bose gas, Phys. Rev. A 89 pp 033601– (2014) · doi:10.1103/PhysRevA.89.033601
[15] Deift, Random matrix theory: invariant ensembles and universality (2009) · Zbl 1171.15023 · doi:10.1090/cln/018
[16] Dotsenko, Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers, EPL (Europhysics Letters) 90 (2) pp 20003– (2010) · doi:10.1209/0295-5075/90/20003
[17] Ferrari, Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues, Comm. Math. Phys. 252 (1-3) pp 77– (2004) · Zbl 1124.82316 · doi:10.1007/s00220-004-1204-6
[18] Ferrari, A determinantal formula for the GOE Tracy-Widom distribution, J. Phys. A 38 (33) pp L557– (2005) · doi:10.1088/0305-4470/38/33/L02
[19] Ferrari, Tracy-Widom asymptotics for q-TASEP, Ann. Inst. Henri Poincaré Probab. Stat. 51 (4) pp 1465– (2015) · Zbl 1376.60080 · doi:10.1214/14-AIHP614
[20] Frenkel, Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993) (1995)
[21] Gasper, Basic hypergeometric series, 2. ed. (2004) · Zbl 1129.33005 · doi:10.1017/CBO9780511526251
[22] Gohberg, Traces and determinants of linear operators (2000) · Zbl 0946.47013 · doi:10.1007/978-3-0348-8401-3
[23] Hairer, Solving the KPZ equation, Ann. of Math. (2) 178 (2) pp 559– (2013) · Zbl 1281.60060 · doi:10.4007/annals.2013.178.2.4
[24] Hairer, A theory of regularity structures, Invent. Math. 198 (2) pp 269– (2014) · Zbl 1332.60093 · doi:10.1007/s00222-014-0505-4
[25] Imamura, Replica approach to the KPZ equation with the half Brownian motion initial condition, J. Phys. A 44 (38) pp 29– (2011) · Zbl 1227.82057 · doi:10.1088/1751-8113/44/38/385001
[26] Imamura, Stationary correlations for the 1D KPZ equation, J. Stat. Phys. 150 (5) pp 908– (2013) · Zbl 1266.82045 · doi:10.1007/s10955-013-0710-3
[27] Doussal, The KPZ equation with flat initial condition and the directed polymer with one free end, J. Stat. Mech. 2012 (06) pp P06001– (2012) · doi:10.1088/1742-5468/2012/06/P06001
[28] Lee, Distribution of a particle’s position in the ASEP with the alternating initial condition, J. Stat. Phys. 140 (4) pp 635– (2010) · Zbl 1198.82044 · doi:10.1007/s10955-010-0014-9
[29] Moreno Flores , G. Quastel , J. Remenik , D. Intermediate disorder limits for directed polymers with boundary conditions · Zbl 1257.82117
[30] Nelson, On the zeros of the q-analogue exponential function, J. Phys. A 27 (11) pp 3857– (1994) · Zbl 0841.05008 · doi:10.1088/0305-4470/27/11/034
[31] Ortmann , J. Quastel , J. Remenik , D. Exact formulas for random growth with half-flat initial data 2014 · Zbl 1334.60212
[32] Quastel, Topics in percolative and disordered systems (2014) · Zbl 1329.82059 · doi:10.1007/978-1-4939-0339-9_5
[33] Rains , E. M. Correlation functions for symmetrized increasing subsequences 2000
[34] Sasamoto, Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A 38 (33) pp L549– (2005) · doi:10.1088/0305-4470/38/33/L01
[35] Simon, Trace ideals and their applications, 2. ed. (2005) · Zbl 1074.47001
[36] Tracy, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (3) pp 727– (1996) · Zbl 0851.60101 · doi:10.1007/BF02099545
[37] Tracy, A Fredholm determinant representation in ASEP, J. Stat. Phys. 132 (2) pp 291– (2008) · Zbl 1144.82045 · doi:10.1007/s10955-008-9562-7
[38] Tracy, Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (3) pp 815– (2008) · Zbl 1148.60080 · doi:10.1007/s00220-008-0443-3
[39] Tracy, Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (1) pp 129– (2009) · Zbl 1184.60036 · doi:10.1007/s00220-009-0761-0
[40] Tracy, On ASEP with step Bernoulli initial condition, J. Stat. Phys. 137 (5-6) pp 825– (2009) · Zbl 1188.82043 · doi:10.1007/s10955-009-9867-1
[41] Vallée, An integral representation for the product of Airy functions, Z. Angew. Math. Phys. 48 (1) pp 156– (1997) · Zbl 0879.33002 · doi:10.1007/PL00001464
[42] Widder, The Laplace transform (1941) · Zbl 0060.24801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.