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Large deviations and wandering exponent for random walk in a dynamic beta environment. (English) Zbl 1466.60196

Summary: Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The Doob-transformed environment is correlated in time, i.i.d. in space and its marginal density function is a product of a beta density and a hypergeometric function. Under its averaged distribution, the transformed walk obeys the wandering exponent \(2/3\) that agrees with Kardar-Parisi-Zhang universality. The harmonic function in the Doob transform comes from a Busemann-type limit and appears as an extremal in a variational problem for the quenched large deviation rate function.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
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[1] Abramowitz, M. and Stegun, I. A., eds. (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York. · Zbl 0171.38503
[2] Alberts, T., Khanin, K. and Quastel, J. (2014). The intermediate disorder regime for directed polymers in dimension \(1+1\). Ann. Probab.42 1212-1256. · Zbl 1292.82014
[3] Avena, L., Chino, Y., da Costa, C. and den Hollander, F. (2019). Random walk in cooling random environment: Ergodic limits and concentration inequalities. Electron. J. Probab.24 paper no. 38, 35. · Zbl 1466.60038
[4] Balázs, M., Rassoul-Agha, F. and Seppäläinen, T. (2006). The random average process and random walk in a space-time random environment in one dimension. Comm. Math. Phys.266 499-545. · Zbl 1129.60097
[5] Balázs, M., Rassoul-Agha, F. and Seppäläinen, T. (2018). Wandering exponent for random walk in a dynamic beta environment. Extended version. Preprint. Available at arXiv:1801.08070v1.
[6] Barraquand, G. and Corwin, I. (2017). Random-walk in beta-distributed random environment. Probab. Theory Related Fields167 1057-1116. · Zbl 1382.60125
[7] Blondel, O., Hilario, M. R., dos Santos, R. S., Sidoravicius, V. and Teixeira, A. (2017). Random walk on random walks: Higher dimensions. Preprint. Available at arXiv:1709.01253. · Zbl 1467.60084
[8] Blondel, O., Hilario, M. R., dos Santos, R. S., Sidoravicius, V. and Teixeira, A. (2017). Random walk on random walks: Low densities. Preprint. Available at arXiv:1709.01257. · Zbl 1467.60084
[9] Chaumont, H. and Noack, C. (2018). Fluctuation exponents for stationary exactly solvable lattice polymer models via a Mellin transform framework. ALEA Lat. Am. J. Probab. Math. Stat.15 509-547. · Zbl 1390.60344
[10] Comets, F., Gantert, N. and Zeitouni, O. (2000). Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields118 65-114. Erratum: Probab. Theory Related Fields125 42-44 (2003). · Zbl 0965.60098
[11] Comets, F. and Vargas, V. (2006). Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat.2 267-277. · Zbl 1105.60074
[12] Corwin, I. and Gu, Y. (2017). Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments. J. Stat. Phys.166 150-168. · Zbl 1364.35365
[13] Dufresne, D. (2010). The beta product distribution with complex parameters. Comm. Statist. Theory Methods39 837-854. · Zbl 1187.62031
[14] Georgiou, N., Rassoul-Agha, F. and Seppäläinen, T. (2016). Variational formulas and cocycle solutions for directed polymer and percolation models. Comm. Math. Phys.346 741-779. · Zbl 1355.82064
[15] Georgiou, N., Rassoul-Agha, F. and Seppäläinen, T. (2017). Stationary cocycles and Busemann functions for the corner growth model. Probab. Theory Related Fields169 177-222. · Zbl 1407.60122
[16] Georgiou, N., Rassoul-Agha, F., Seppäläinen, T. and Yilmaz, A. (2015). Ratios of partition functions for the log-gamma polymer. Ann. Probab.43 2282-2331. · Zbl 1357.60110
[17] Greven, A. and den Hollander, F. (1994). Large deviations for a random walk in random environment. Ann. Probab.22 1381-1428. · Zbl 0820.60054
[18] 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
[19] Olver, F. W. J. (1997). Asymptotics and Special Functions. A K Peters, Ltd., Wellesley, MA. · Zbl 0982.41018
[20] Pham-Gia, T. (2000). Distributions of the ratios of independent beta variables and applications. Comm. Statist. Theory Methods29 2693-2715. · Zbl 1107.62309
[21] Rassoul-Agha, F. and Seppäläinen, T. (2005). An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Related Fields133 299-314. · Zbl 1088.60094
[22] Rassoul-Agha, F. and Seppäläinen, T. (2014). Quenched point-to-point free energy for random walks in random potentials. Probab. Theory Related Fields158 711-750. · Zbl 1291.60051
[23] Rassoul-Agha, F. and Seppäläinen, T. (2015). A Course on Large Deviations with an Introduction to Gibbs Measures. Graduate Studies in Mathematics162. Amer. Math. Soc., Providence, RI.
[24] Rassoul-Agha, F., Seppäläinen, T. and Yilmaz, A. (2017). Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment. Electron. J. Probab.22 Paper No. 57, 47. · Zbl 1368.60028
[25] Redig, F. and Völlering, F. (2013). Random walks in dynamic random environments: A transference principle. Ann. Probab.41 3157-3180. · Zbl 1277.82051
[26] Rosenbluth, J. M. (2006). Quenched Large Deviation for Multidimensional Random Walk in Random Environment: A Variational Formula. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)—New York Univ.
[27] Sabot, C. and Tournier, L. (2017). Random walks in Dirichlet environment: An overview. Ann. Fac. Sci. Toulouse Math. (6) 26 463-509. · Zbl 1369.60074
[28] Seppäläinen, T. (2012). Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab.40 19-73. Corrected version available at arXiv:0911.2446. · Zbl 1254.60098
[29] Thiery, T. and Le Doussal, P. (2017). Exact solution for a random walk in a time-dependent 1D random environment: The point-to-point beta polymer. J. Phys. A50 045001, 44. · Zbl 1360.82100
[30] Yilmaz, A. (2009). Quenched large deviations for random walk in a random environment. Comm. Pure Appl. Math.62 1033-1075. · Zbl 1168.60370
[31] Yilmaz, A. and Zeitouni, O. (2010). Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three. Comm. Math. Phys.300 243-271. · Zbl 1202.60163
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