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**The intermediate disorder regime for directed polymers in dimension \(1+1\).**
*(English)*
Zbl 1292.82014

Summary: We introduce a new disorder regime for directed polymers in dimension \(1+1\) that sits between the weak and strong disorder regimes. We call it the intermediate disorder regime. It is accessed by scaling the inverse temperature parameter \(\beta\) to zero as the polymer length \(n\) tends to infinity. The natural choice of scaling is \(\beta_{n}:=\beta n^{-1/4}\). We show that the polymer measure under this scaling has previously unseen behavior. While the fluctuation exponents of the polymer endpoint and the log partition function are identical to those for simple random walk \((\zeta=1/2,\, \chi=0)\), the fluctuations themselves are different. These fluctuations are still influenced by the random environment, and there is no self-averaging of the polymer measure. In particular, the random distribution of the polymer endpoint converges in law (under a diffusive scaling of space) to a random absolutely continuous measure on the real line. The randomness of the measure is inherited from a stationary process \(A_{\beta}\) that has the recently discovered crossover distributions as its one-point marginals, which for large \(\beta\) become the GUE Tracy-Widom distribution. We also prove existence of a limiting law for the four-parameter field of polymer transition probabilities that can be described by the stochastic heat equation. {

} In particular, in this weak noise limit, we obtain the convergence of the point-to-point free energy fluctuations to the GUE Tracy-Widom distribution. We emphasize that the scaling behaviour obtained is universal and does not depend on the law of the disorder.

} In particular, in this weak noise limit, we obtain the convergence of the point-to-point free energy fluctuations to the GUE Tracy-Widom distribution. We emphasize that the scaling behaviour obtained is universal and does not depend on the law of the disorder.

### MSC:

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

60F05 | Central limit and other weak theorems |

82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |

82D60 | Statistical mechanics of polymers |

35Q82 | PDEs in connection with statistical mechanics |

60K40 | Other physical applications of random processes |

82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

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\textit{T. Alberts} et al., Ann. Probab. 42, No. 3, 1212--1256 (2014; Zbl 1292.82014)

### References:

[1] | Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas , Graphs , and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55 . U.S. Government Printing Office, Washington, DC. · Zbl 0171.38503 |

[2] | Alberts, T., Khanin, K. and Quastel, J. (2014). The continuum directed random polymer. J. Stat. Phys. 154 305-326. · Zbl 1291.82143 |

[3] | Alberts, T., Khanin, K. and Quastel, J. (2010). Intermediate disorder regime for directed polymers in dimension \(1+1\). Phys. Rev. Lett. 105 090603. · Zbl 1292.82014 |

[4] | 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 |

[5] | Auffinger, A. and Damron, M. (2011). A simplified proof of the relation between scaling exponents in first-passage percolation. Available at [math.PR]. 1109.0523v1 |

[6] | Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 571-607. · Zbl 0874.60059 |

[7] | Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201 |

[8] | Biroli, G., Bouchaud, J. P. and Potters, M. (2007). On the top eigenvalue of heavy-tailed random matrices. Europhys. Lett. EPL 78 Art. 10001, 5. · Zbl 1244.82029 |

[9] | Biroli, G., Bouchaud, J.-P. and Potters, M. (2007). Extreme value problems in random matrix theory and other disordered systems. J. Stat. Mech. Theory Exp. 7 P07019, 15 pp. (electronic). |

[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 |

[11] | Calabrese, P. and Doussal, P. L. (2011). Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions. Phys. Rev. Lett. 106 250603. |

[12] | Calabrese, P., Le Doussal, P. and Rosso, A. (2010). Free-energy distribution of the directed polymer at high temperature. Europhys. Lett. EPL 90 20002. |

[13] | Carmona, P. and Hu, Y. (2002). On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 431-457. · Zbl 1015.60100 |

[14] | Chatterjee, S. (2013). The universal relation between scaling exponents in first-passage percolation. Ann. of Math. (2) 177 663-697. · Zbl 1271.60101 |

[15] | Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705-723. · Zbl 1042.60069 |

[16] | Comets, F., Shiga, T. and Yoshida, N. (2004). Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 115-142. Math. Soc. Japan, Tokyo. · Zbl 1114.82017 |

[17] | 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 |

[18] | Comets, F. and Yoshida, N. (2005). Brownian directed polymers in random environment. Comm. Math. Phys. 254 257-287. · Zbl 1128.60089 |

[19] | Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746-1770. · Zbl 1104.60061 |

[20] | Durrett, R. (2010). Probability : Theory and Examples , 4th ed. Cambridge Univ. Press, Cambridge. · Zbl 1202.60001 |

[21] | Dynkin, E. B. and Mandelbaum, A. (1983). Symmetric statistics, Poisson point processes, and multiple Wiener integrals. Ann. Statist. 11 739-745. · Zbl 0518.60050 |

[22] | Feng, Z. S. (2012). Diffusivity of rescaled random polymer in random environment in dimensions 1 and 2. Available at [math.PR]. 1201.6215v1 |

[23] | Garsia, A. M. (1972). Continuity properties of Gaussian processes with multidimensional time parameter. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability ( Univ. California , Berkeley , CA , 1970 / 1971), Vol. II : Probability Theory 369-374. Univ. California Press, Berkeley, CA. · Zbl 0272.60034 |

[24] | Gine, E. (1997). Decoupling and limit theorems for \(U\)-statistics and \(U\)-processes. In Lectures on Probability Theory and Statistics ( Saint-Flour , 1996). Lecture Notes in Math. 1665 1-35. Springer, Berlin. · Zbl 0882.62016 |

[25] | Halpin-Healy, T. and Zhang, Y.-C. (1995). Kinetic roughening phenomena, stochastic growth, directed polymers and all that aspects of multidisciplinary statistical mechanics. Phys. Rep. 254 215-414. |

[26] | Huse, D. A. and Henley, C. L. (1985). Pinning and roughening of domain walls in ising systems due to random impurities. Phys. Rev. Lett. 54 2708-2711. |

[27] | Imbrie, J. Z. and Spencer, T. (1988). Diffusion of directed polymers in a random environment. J. Stat. Phys. 52 609-626. · Zbl 1084.82595 |

[28] | Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129 . Cambridge Univ. Press, Cambridge. · Zbl 0887.60009 |

[29] | Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889-892. · Zbl 1101.82329 |

[30] | 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 |

[31] | Mejane, O. (2004). Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. Henri Poincaré Probab. Stat. 40 299-308. · Zbl 1041.60079 |

[32] | Nolan, D. and Pollard, D. (1987). \(U\)-processes: Rates of convergence. Ann. Statist. 15 780-799. · Zbl 0624.60048 |

[33] | Nolan, D. and Pollard, D. (1988). Functional limit theorems for \(U\)-processes. Ann. Probab. 16 1291-1298. · Zbl 0665.60037 |

[34] | Petermann, M. (2000). Superdiffusivity of directed polymers in random environment. Ph.D. thesis, Univ. Zürich. |

[35] | Petrov, V. V. (1975). Sums of Independent Random Variables . Springer, New York. · Zbl 0322.60043 |

[36] | Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071-1106. · Zbl 1025.82010 |

[37] | Sasamoto, T. and Spohn, H. (2010). Universality of the one-dimensional KPZ equation. Available at [cond-mat.stat-mech]. 1002.1883v2 · Zbl 1204.35137 |

[38] | Sasamoto, T. and Spohn, H. (2010). Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834 523-542. · Zbl 1204.35137 |

[39] | Seppäläinen, T. (2012). Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 19-73. · Zbl 1254.60098 |

[40] | Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174. · Zbl 0789.35152 |

[41] | Tracy, C. A. and Widom, H. (2008). A Fredholm determinant representation in ASEP. J. Stat. Phys. 132 291-300. · Zbl 1144.82045 |

[42] | Tracy, C. A. and Widom, H. (2008). Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 815-844. · Zbl 1148.60080 |

[43] | Tracy, C. A. and Widom, H. (2009). Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 129-154. · Zbl 1184.60036 |

[44] | Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour , XIV- 1984. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060 |

[45] | Wüthrich, M. V. (1998). Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. Henri Poincaré Probab. Stat. 34 279-308. · Zbl 0909.60073 |

[46] | Wüthrich, M. V. (1998). Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26 1000-1015. · Zbl 0935.60099 |

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