×

Scaling for a one-dimensional directed polymer with boundary conditions. (English) Zbl 1254.60098

Ann. Probab. 40, No. 1, 19-73 (2012); erratum ibid. 45, No. 3, 2056-2058 (2017).
A directed polymer in a random environment is a model which describes a polymer (that is to say a long chain of molecules) as a random walk path which interacts with a random environment. Here, one considers (and defines) a polymer model with boundaries which are assigned distinct weight distributions, and one derives upper and lower bounds for the model with boundary. One gives detailed results for the fluctuations of a path in the model with boundaries for a polymer with fixed endpoint but without boundaries and for a polymer with free endpoint.

MSC:

60K37 Processes in random environments
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82D60 Statistical mechanics of polymers
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramowitz, M. and Stegun, I. A. (1992). Handbook of Mathematical Functions with Formulas , Graphs , and Mathematical Tables . Dover, New York. · Zbl 0171.38503
[2] Amir, G., Corwin, I. and Quastel, J. (2011). Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Comm. Pure Appl. Math. 64 466-537. · Zbl 1222.82070 · doi:10.1002/cpa.20347
[3] Arous, G. B. and Corwin, I. (2011). Current fluctuations for TASEP: A proof of the Prähofer-Spohn conjecture. Ann. Probab. 39 104-138. · Zbl 1208.82036 · doi:10.1214/10-AOP550
[4] Artin, E. (1964). The Gamma Function . Holt, Rinehart and Winston, New York. · Zbl 0144.06802
[5] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119-1178. · Zbl 0932.05001 · doi:10.1090/S0894-0347-99-00307-0
[6] Balázs, M., Cator, E. and Seppäläinen, T. (2006). Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 1094-1132 (electronic). · Zbl 1139.60046
[7] Balázs, M., Quastel, J. and Seppäläinen, T. (2009). Scaling exponent for the Hopf-Cole solution of KPZ/stochastic Burgers. J. Amer. Math. Soc. To appear. Available at . · Zbl 1227.60083
[8] Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 571-607. · Zbl 0874.60059 · doi:10.1007/s002200050044
[9] Bezerra, S., Tindel, S. and Viens, F. (2008). Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab. 36 1642-1675. · Zbl 1149.82032 · doi:10.1214/07-AOP363
[10] Bolthausen, E. (1989). A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 529-534. · Zbl 0684.60013 · doi:10.1007/BF01218584
[11] Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19-42. · Zbl 0301.60035 · doi:10.1214/aop/1176997023
[12] Cator, E. and Groeneboom, P. (2005). Hammersley’s process with sources and sinks. Ann. Probab. 33 879-903. · Zbl 1066.60011 · doi:10.1214/009117905000000053
[13] Cator, E. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 1273-1295. · Zbl 1101.60076 · doi:10.1214/009117906000000089
[14] Comets, F. and Yoshida, N. (2005). Brownian directed polymers in random environment. Comm. Math. Phys. 254 257-287. · Zbl 1128.60089 · doi:10.1007/s00220-004-1203-7
[15] Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746-1770. · Zbl 1104.60061 · doi:10.1214/009117905000000828
[16] Ferrari, P. L. and Spohn, H. (2006). Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 1-44. · Zbl 1118.82032 · doi:10.1007/s00220-006-1549-0
[17] Huse, D. A. and Henley, C. L. (1985). Pinning and roughening of domain wall in Ising systems due to random impurities. Phys. Rev. Lett. 54 2708-2711.
[18] Imbrie, J. Z. and Spencer, T. (1988). Diffusion of directed polymers in a random environment. J. Stat. Phys. 52 609-626. · Zbl 1084.82595 · doi:10.1007/BF01019720
[19] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437-476. · Zbl 0969.15008 · doi:10.1007/s002200050027
[20] Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445-456. · Zbl 0960.60097 · doi:10.1007/s004409900039
[21] Kelly, F. P. (1979). Reversibility and Stochastic Networks . Wiley, Chichester. · Zbl 0422.60001
[22] Krug, J. and Spohn, H. (1992). Kinetic roughening of growing surfaces. In Solids Far from Equilibrium , Collection Aléa-Saclay : Monographs and Texts in Statistical Physics , 1 (C. Godrèche, ed.) 117-130. Cambridge Univ. Press, Cambridge.
[23] Lacoin, H. (2010). New bounds for the free energy of directed polymers in dimension 1+1 and 1+2. Comm. Math. Phys. 294 471-503. · Zbl 1227.82098 · doi:10.1007/s00220-009-0957-3
[24] Licea, C., Newman, C. M. and Piza, M. S. T. (1996). Superdiffusivity in first-passage percolation. Probab. Theory Related Fields 106 559-591. · Zbl 0870.60096 · doi:10.1007/s004400050075
[25] Lukacs, E. (1955). A characterization of the gamma distribution. Ann. Math. Statist. 26 319-324. · Zbl 0065.11103 · doi:10.1214/aoms/1177728549
[26] Mejane, O. (2004). Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 40 299-308. · Zbl 1041.60079 · doi:10.1016/S0246-0203(03)00072-4
[27] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977-1005. · Zbl 0835.60087 · doi:10.1214/aop/1176988171
[28] O’Connell, N. and Yor, M. (2001). Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96 285-304. · Zbl 1058.60078 · doi:10.1016/S0304-4149(01)00119-3
[29] Petermann, M. (2000). Superdiffusivity of directed polymers in random environment. Ph.D. thesis, Univ. Zürich.
[30] Piza, M. S. T. (1997). Directed polymers in a random environment: Some results on fluctuations. J. Stat. Phys. 89 581-603. · Zbl 0945.82527 · doi:10.1007/BF02765537
[31] Prähofer, M. and Spohn, H. (2002). Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium ( Mambucaba , 2000). Progress in Probability 51 185-204. Birkhäuser, Boston, MA. · Zbl 1015.60093
[32] Stromberg, K. R. (1981). Introduction to Classical Real Analysis . Wadsworth, Belmont, CA. · Zbl 0454.26001
[33] Tracy, C. A. and Widom, H. (2009). Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 129-154. · Zbl 1184.60036 · doi:10.1007/s00220-009-0761-0
[34] Wüthrich, M. V. (1998). Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. H. Poincaré Probab. Statist. 34 279-308. · Zbl 0909.60073 · doi:10.1016/S0246-0203(98)80013-7
[35] Wüthrich, M. V. (1998). Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26 1000-1015. · Zbl 0935.60099 · doi:10.1214/aop/1022855742
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.