Abstract
We prove that the variance of the current across a characteristic is of order $t^{2/3}$ in a stationary asymmetric simple exclusion process, and that the diffusivity has order $t^{1/3}$. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle.
-
[aldo-diac]
D. Aldous and P. Diaconis, "Hammersley’s interacting particle process and longest increasing subsequences," Probab. Theory Related Fields, vol. 103, iss. 2, pp. 199-213, 1995.
@article {aldo-diac, MRKEY = {1355056},
AUTHOR = {Aldous, D. and Diaconis, P.},
TITLE = {Hammersley's interacting particle process and longest increasing subsequences},
JOURNAL = {Probab. Theory Related Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {103},
YEAR = {1995},
NUMBER = {2},
PAGES = {199--213},
ISSN = {0178-8051},
CODEN = {PTRFEU},
MRCLASS = {60C05 (60K35)},
MRNUMBER = {96k:60017},
MRREVIEWER = {Graham Brightwell},
DOI = {10.1007/BF01204214},
ZBLNUMBER = {0836.60107},
} -
[bdj]
J. Baik, P. Deift, and K. Johansson, "On the distribution of the length of the longest increasing subsequence of random permutations," J. Amer. Math. Soc., vol. 12, iss. 4, pp. 1119-1178, 1999.
@article {bdj, MRKEY = {1682248},
AUTHOR = {Baik, Jinho and Deift, Percy and Johansson, Kurt},
TITLE = {On the distribution of the length of the longest increasing subsequence of random permutations},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {12},
YEAR = {1999},
NUMBER = {4},
PAGES = {1119--1178},
ISSN = {0894-0347},
MRCLASS = {05A05 (33D45 45E05 60C05)},
MRNUMBER = {2000e:05006},
MRREVIEWER = {David J. Aldous},
DOI = {10.1090/S0894-0347-99-00307-0},
ZBLNUMBER = {0932.05001},
} -
[fluct] M. Balázs, "Growth fluctuations in a class of deposition models," Ann. Inst. H. Poincaré Probab. Statist., vol. 39, iss. 4, pp. 639-685, 2003.
@article {fluct, MRKEY = {1983174},
AUTHOR = {Bal{á}zs, M{á}rton},
TITLE = {Growth fluctuations in a class of deposition models},
JOURNAL = {Ann. Inst. H. Poincaré Probab. Statist.},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Probabilités et Statistiques},
VOLUME = {39},
YEAR = {2003},
NUMBER = {4},
PAGES = {639--685},
ISSN = {0246-0203},
CODEN = {AHPBAR},
MRCLASS = {60K35 (82C31)},
MRNUMBER = {2005j:60178a},
MRREVIEWER = {Wolfgang Freudenberg},
ZBLNUMBER = {1029.60075},
} -
[third]
M. Balázs, E. Cator, and T. Seppäläinen, "Cube root fluctuations for the corner growth model associated to the exclusion process," Electron. J. Probab., vol. 11, p. no. 42, 1094-1132, 2006.
@article {third, MRKEY = {2268539},
AUTHOR = {Bal{á}zs, M{á}rton and Cator, E. and Sepp{ä}l{ä}inen, T.},
TITLE = {Cube root fluctuations for the corner growth model associated to the exclusion process},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {11},
YEAR = {2006},
PAGES = {no. 42, 1094--1132},
ISSN = {1083-6489},
MRCLASS = {60K35 (82C43)},
MRNUMBER = {2007k:60312},
MRREVIEWER = {Glauco Valle},
URL = {http://www.math.washington.edu/~ejpecp/EjpVol11/paper42.abs.html},
ZBLNUMBER={1139.60046},
} -
[raprwre]
M. Balázs, F. Rassoul-Agha, and T. Seppäläinen, "The random average process and random walk in a space-time random environment in one dimension," Comm. Math. Phys., vol. 266, iss. 2, pp. 499-545, 2006.
@article {raprwre, MRKEY = {2238887},
AUTHOR = {Bal{á}zs, M{á}rton and Rassoul-Agha, Firas and Sepp{ä}l{ä}inen, Timo},
TITLE = {The random average process and random walk in a space-time random environment in one dimension},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {266},
YEAR = {2006},
NUMBER = {2},
PAGES = {499--545},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {60K37 (60F05 82C22)},
MRNUMBER = {2007k:60334},
MRREVIEWER = {Francis Comets},
DOI = {10.1007/s00220-006-0036-y},
ZBLNUMBER = {1129.60097},
} -
[varj2nd]
M. Balázs and T. Seppäläinen, "Exact connections between current fluctuations and the second class particle in a class of deposition models," J. Stat. Phys., vol. 127, iss. 2, pp. 431-455, 2007.
@article {varj2nd, MRKEY = {2314355},
AUTHOR = {Bal{á}zs, M{á}rton and Sepp{ä}l{ä}inen, Timo},
TITLE = {Exact connections between current fluctuations and the second class particle in a class of deposition models},
JOURNAL = {J. Stat. Phys.},
FJOURNAL = {Journal of Statistical Physics},
VOLUME = {127},
YEAR = {2007},
NUMBER = {2},
PAGES = {431--455},
ISSN = {0022-4715},
MRCLASS = {82C22 (60K35 82C41)},
MRNUMBER = {2008e:82056},
MRREVIEWER = {Wolfgang Freudenberg},
DOI = {10.1007/s10955-007-9291-3},
ZBLNUMBER={1147.82348} , } -
[vBKS85]
H. van Beijeren, R. Kutner, and H. Spohn, "Excess noise for driven diffusive systems," Phys. Rev. Lett., vol. 54, iss. 18, pp. 2026-2029, 1985.
@article {vBKS85, MRKEY = {789756},
AUTHOR = {van Beijeren, H. and Kutner, R. and Spohn, H.},
TITLE = {Excess noise for driven diffusive systems},
JOURNAL = {Phys. Rev. Lett.},
FJOURNAL = {Physical Review Letters},
VOLUME = {54},
YEAR = {1985},
NUMBER = {18},
PAGES = {2026--2029},
ISSN = {0031-9007},
CODEN = {PRLTAO},
MRCLASS = {82A40 (82A31)},
MRNUMBER = {86e:82031},
DOI = {10.1103/PhysRevLett.54.2026},
} -
[cuberoot]
E. Cator and P. Groeneboom, "Second class particles and cube root asymptotics for Hammersley’s process," Ann. Probab., vol. 34, iss. 4, pp. 1273-1295, 2006.
@article {cuberoot, MRKEY = {2257647},
AUTHOR = {Cator, Eric and Groeneboom, Piet},
TITLE = {Second class particles and cube root asymptotics for {H}ammersley's process},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {34},
YEAR = {2006},
NUMBER = {4},
PAGES = {1273--1295},
ISSN = {0091-1798},
CODEN = {APBYAE},
MRCLASS = {60C05 (60F05 60K35)},
MRNUMBER = {2008e:60021},
MRREVIEWER = {Martin M{ö}hle},
DOI = {10.1214/009117906000000089},
ZBLNUMBER = {1101.60076},
} -
[dgl]
D. Dürr, S. Goldstein, and J. L. Lebowitz, "Asymptotics of particle trajectories in infinite one-dimensional systems with collisions," Comm. Pure Appl. Math., vol. 38, iss. 5, pp. 573-597, 1985.
@article {dgl, MRKEY = {803248},
AUTHOR = {D{ü}rr, Detlef and Goldstein, Sheldon and Lebowitz, Joel L.},
TITLE = {Asymptotics of particle trajectories in infinite one-dimensional systems with collisions},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {38},
YEAR = {1985},
NUMBER = {5},
PAGES = {573--597},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {60K35 (82A05)},
MRNUMBER = {87e:60166},
MRREVIEWER = {F. L. Spitzer},
DOI = {10.1002/cpa.3160380508},
ZBLNUMBER = {0578.60094},
} -
[evan] L. C. Evans, Partial Differential Equations, Providence, RI: Amer. Math. Soc., 1998.
@book {evan, MRKEY = {1625845},
AUTHOR = {Evans, Lawrence C.},
TITLE = {Partial Differential Equations},
SERIES = {Grad. Stud. Math.},
NUMBER = {19},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1998},
PAGES = {xviii+662},
ISBN = {0-8218-0772-2},
MRCLASS = {35-01},
MRNUMBER = {99e:35001},
MRREVIEWER = {Luigi Rodino},
ZBLNUMBER={0902.35002},
} -
[se] P. A. Ferrari and L. R. G. Fontes, "Current fluctuations for the asymmetric simple exclusion process," Ann. Probab., vol. 22, iss. 2, pp. 820-832, 1994.
@article {se, MRKEY = {1288133},
AUTHOR = {Ferrari, P. A. and Fontes, L. R. G.},
TITLE = {Current fluctuations for the asymmetric simple exclusion process},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {22},
YEAR = {1994},
NUMBER = {2},
PAGES = {820--832},
ISSN = {0091-1798},
CODEN = {APBYAE},
MRCLASS = {60K35 (82C22 82C24 82C41)},
MRNUMBER = {95j:60162},
MRREVIEWER = {T. M. Liggett},
ZBLNUMBER = {0806.60099},
} -
[fks] P. A. Ferrari, C. Kipnis, and E. Saada, "Microscopic structure of travelling waves in the asymmetric simple exclusion process," Ann. Probab., vol. 19, iss. 1, pp. 226-244, 1991.
@article {fks, MRKEY = {1085334},
AUTHOR = {Ferrari, P. A. and Kipnis, C. and Saada, E.},
TITLE = {Microscopic structure of travelling waves in the asymmetric simple exclusion process},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {19},
YEAR = {1991},
NUMBER = {1},
PAGES = {226--244},
ISSN = {0091-1798},
CODEN = {APBYAE},
MRCLASS = {60K35 (35Q53)},
MRNUMBER = {92b:60099},
MRREVIEWER = {T. M. Liggett},
ZBLNUMBER = {0725.60113},
} -
[ferspohn] P. L. Ferrari and H. Spohn, "Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process," Comm. Math. Phys., vol. 265, iss. 1, pp. 1-44, 2006.
@article {ferspohn, MRKEY = {2217295},
AUTHOR = {Ferrari, Patrik L. and Spohn, Herbert},
TITLE = {Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {265},
YEAR = {2006},
NUMBER = {1},
PAGES = {1--44},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {82C22 (60K35 82B41)},
MRNUMBER = {2007g:82038a},
MRREVIEWER = {Ellen Saada},
ZBLNUMBER = {1118.82032},
} -
[1/3]
K. Johansson, "Shape fluctuations and random matrices," Comm. Math. Phys., vol. 209, iss. 2, pp. 437-476, 2000.
@article {1/3, MRKEY = {1737991},
AUTHOR = {Johansson, Kurt},
TITLE = {Shape fluctuations and random matrices},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {209},
YEAR = {2000},
NUMBER = {2},
PAGES = {437--476},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {60K35 (05A16 60C05 60F10 82B41)},
MRNUMBER = {2001h:60177},
MRREVIEWER = {Peter Eichelsbacher},
DOI = {10.1007/s002200050027},
ZBLNUMBER = {0969.15008},
} -
[kipn-land] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, New York: Springer-Verlag, 1999.
@book {kipn-land, MRKEY = {1707314},
AUTHOR = {Kipnis, Claude and Landim, Claudio},
TITLE = {Scaling Limits of Interacting Particle Systems},
SERIES = {Grundl. Math. Wissen.},
NUMBER = {320},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1999},
PAGES = {xvi+442},
ISBN = {3-540-64913-1},
MRCLASS = {60-02 (60F05 60F10 60K35 82B05 82C22)},
MRNUMBER = {2000i:60001},
MRREVIEWER = {Timo Sepp{ä}l{ä}inen},
ZBLNUMBER = {0927.60002},
} -
[lqsy]
C. Landim, J. Quastel, M. Salmhofer, and H. -T. Yau, "Superdiffusivity of asymmetric exclusion process in dimensions one and two," Comm. Math. Phys., vol. 244, iss. 3, pp. 455-481, 2004.
@article {lqsy, MRKEY = {2034485},
AUTHOR = {Landim, C. and Quastel, J. and Salmhofer, M. and Yau, H.-T.},
TITLE = {Superdiffusivity of asymmetric exclusion process in dimensions one and two},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {244},
YEAR = {2004},
NUMBER = {3},
PAGES = {455--481},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {60J25 (60K35 82B41)},
MRNUMBER = {2005a:60118},
MRREVIEWER = {Christian Maes},
DOI = {10.1007/s00220-003-1020-4},
ZBLNUMBER = {1064.60164},
} -
[ligg85] T. M. Liggett, Interacting Particle Systems, New York: Springer-Verlag, 1985.
@book {ligg85, MRKEY = {776231},
AUTHOR = {Liggett, Thomas M.},
TITLE = {Interacting Particle Systems},
SERIES = {Grundl. Math. Wissen.},
NUMBER = {276},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1985},
PAGES = {xv+488},
ISBN = {0-387-96069-4},
MRCLASS = {60K35},
MRNUMBER = {86e:60089},
MRREVIEWER = {F. L. Spitzer},
ZBLNUMBER = {1103.82016},
} -
[stochi] T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, New York: Springer-Verlag, 1999.
@book {stochi, MRKEY = {1717346},
AUTHOR = {Liggett, Thomas M.},
TITLE = {Stochastic Interacting Systems: Contact, Voter and Exclusion Processes},
SERIES = {Grundl. Math. Wissen.},
NUMBER = {324},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1999},
PAGES = {xii+332},
ISBN = {3-540-65995-1},
MRCLASS = {60K35 (82B21 82C21 91B12)},
MRNUMBER = {2001g:60247},
MRREVIEWER = {Ellen Saada},
ZBLNUMBER = {0949.60006},
} -
[spohn] M. Prähofer and H. Spohn, "Current fluctuations for the totally asymmetric simple exclusion process," in In and Out of Equilibrium, Boston, MA: Birkhäuser, 2002, pp. 185-204.
@incollection {spohn, MRKEY = {1901953},
AUTHOR = {Pr{ä}hofer, Michael and Spohn, Herbert},
TITLE = {Current fluctuations for the totally asymmetric simple exclusion process},
BOOKTITLE = {In and Out of Equilibrium},
VENUE={{M}ambucaba, 2000},
SERIES = {Progr. Probab.},
NUMBER = {51},
PAGES = {185--204},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2002},
MRCLASS = {60K35 (82C22 82C31)},
MRNUMBER = {2003e:60224},
MRREVIEWER = {Timo Sepp{ä}l{ä}inen},
ZBLNUMBER={1015.60093},
} -
[quava]
J. Quastel and B. Valkó, "$t^{1/3}$ Superdiffusivity of finite-range asymmetric exclusion processes on $\mathbb{Z}$," Comm. Math. Phys., vol. 273, iss. 2, pp. 379-394, 2007.
@article {quava, MRKEY = {2318311},
AUTHOR = {Quastel, Jeremy and Valk{ó},
Benedek},
TITLE = {{$t\sp {1/3}$} {S}uperdiffusivity of finite-range asymmetric exclusion processes on {$\mathbb{Z}$}},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {273},
YEAR = {2007},
NUMBER = {2},
PAGES = {379--394},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {60K35 (60J25 82B41 82C22)},
MRNUMBER = {2008h:60414},
MRREVIEWER = {C{é}dric Bernardin},
DOI = {10.1007/s00220-007-0242-2},
ZBLNUMBER = {1127.60091},
} -
[quava2] J. Quastel and B. Valkó, "A note on the diffusivity of finite-range asymmetric exclusion processes on $\mathbb{Z}$," in In and Out of Equilibrium. 2, Basel: Birkhäuser, 2008, pp. 543-549.
@incollection {quava2, MRKEY = {2477398},
AUTHOR = {Quastel, Jeremy and Valk{ó},
Benedek},
TITLE = {A note on the diffusivity of finite-range asymmetric exclusion processes on {$\mathbb{Z}$}},
BOOKTITLE = {In and Out of Equilibrium. {\rm 2}},
SERIES = {Progr. Probab.},
NUMBER = {60},
PAGES = {543--549},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2008},
MRCLASS = {60K35 (82C22)},
MRNUMBER = {2477398},
ZBLNUMBER = {05363779},
} -
[reza]
F. Rezakhanlou, "Hydrodynamic limit for attractive particle systems on ${\bf Z}^d$," Comm. Math. Phys., vol. 140, iss. 3, pp. 417-448, 1991.
@article {reza, MRKEY = {1130693},
AUTHOR = {Rezakhanlou, Fraydoun},
TITLE = {Hydrodynamic limit for attractive particle systems on {${\bf Z}\sp d$}},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {140},
YEAR = {1991},
NUMBER = {3},
PAGES = {417--448},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {82C22 (60K35)},
MRNUMBER = {93f:82035},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104248092},
ZBLNUMBER = {0738.60098},
} -
[sepp-ejp]
T. Seppäläinen, "A microscopic model for the Burgers equation and longest increasing subsequences," Electron. J. Probab., vol. 1, p. 51, 1996.
@article {sepp-ejp, MRKEY = {1386297},
AUTHOR = {Sepp{ä}l{ä}inen, Timo},
TITLE = {A microscopic model for the {B}urgers equation and longest increasing subsequences},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {1},
YEAR = {1996},
PAGES = {51 pp.},
ISSN = {1083-6489},
MRCLASS = {60K35 (35Q53 60C05 82C22)},
MRNUMBER = {97d:60162},
MRREVIEWER = {Shui Feng},
URL = {http://www.math.washington.edu/~ejpecp/EjpVol1/paper5.abs.html},
ZBLNUMBER={0891.60093},
} -
[hkl]
T. Seppäläinen, "Existence of hydrodynamics for the totally asymmetric simple $K$-exclusion process," Ann. Probab., vol. 27, iss. 1, pp. 361-415, 1999.
@article {hkl, MRKEY = {1681094},
AUTHOR = {Sepp{ä}l{ä}inen, Timo},
TITLE = {Existence of hydrodynamics for the totally asymmetric simple {$K$}-exclusion process},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {27},
YEAR = {1999},
NUMBER = {1},
PAGES = {361--415},
ISSN = {0091-1798},
CODEN = {APBYAE},
MRCLASS = {60K35 (82C22)},
MRNUMBER = {2000i:60116},
MRREVIEWER = {Francis Comets},
DOI = {10.1214/aop/1022677266},
ZBLNUMBER={0947.60088},
} -
[flucha]
T. Seppäläinen, "Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks," Ann. Probab., vol. 33, iss. 2, pp. 759-797, 2005.
@article {flucha, MRKEY = {2123209},
AUTHOR = {Sepp{ä}l{ä}inen, Timo},
TITLE = {Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {33},
YEAR = {2005},
NUMBER = {2},
PAGES = {759--797},
ISSN = {0091-1798},
CODEN = {APBYAE},
MRCLASS = {60K35 (82C22)},
MRNUMBER = {2006d:60153},
MRREVIEWER = {Ellen Saada},
DOI = {10.1214/009117904000000946},
ZBLNUMBER={1108.60083},
} -
[spit]
F. Spitzer, "Interaction of Markov processes," Advances in Math., vol. 5, pp. 246-290 (1970), 1970.
@article {spit, MRKEY = {0268959},
AUTHOR = {Spitzer, Frank},
TITLE = {Interaction of {M}arkov processes},
JOURNAL = {Advances in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {5},
YEAR = {1970},
PAGES = {246--290 (1970)},
ISSN = {0001-8708},
MRCLASS = {60.60},
MRNUMBER = {42 \#3856},
MRREVIEWER = {B. Weiss},
DOI = {10.1016/0001-8708(70)90034-4},
ZBLNUMBER = {0312.60060},
} -
[yau]
H. Yau, "$(\log t)^{2/3}$ law of the two dimensional asymmetric simple exclusion process," Ann. of Math., vol. 159, iss. 1, pp. 377-405, 2004.
@article {yau, MRKEY = {2052358},
AUTHOR = {Yau, Horng-Tzer},
TITLE = {{$(\log t)\sp {2/3}$} law of the two dimensional asymmetric simple exclusion process},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {159},
YEAR = {2004},
NUMBER = {1},
PAGES = {377--405},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {60K35 (60J25 60J35)},
MRNUMBER = {2005d:60156},
MRREVIEWER = {Laurent Miclo},
DOI = {10.4007/annals.2004.159.377},
ZBLNUMBER = {1060.60099},
}