## The Annals of Probability

### Ratios of partition functions for the log-gamma polymer

#### Abstract

We introduce a random walk in random environment associated to an underlying directed polymer model in $1+1$ dimensions. This walk is the positive temperature counterpart of the competition interface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of partition functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a family of ergodic invariant distributions for the random walk in random environment.

#### Article information

Source
Ann. Probab. Volume 43, Number 5 (2015), 2282-2331.

Dates
Revised: January 2014
First available in Project Euclid: 9 September 2015

Permanent link to this document
http://projecteuclid.org/euclid.aop/1441792286

Digital Object Identifier
doi:10.1214/14-AOP933

Mathematical Reviews number (MathSciNet)
MR3395462

Zentralblatt MATH identifier
06502675

#### Citation

Georgiou, Nicos; Rassoul-Agha, Firas; Seppäläinen, Timo; Yilmaz, Atilla. Ratios of partition functions for the log-gamma polymer. Ann. Probab. 43 (2015), no. 5, 2282--2331. doi:10.1214/14-AOP933. http://projecteuclid.org/euclid.aop/1441792286.

#### References

• [1] Bakhtin, Y., Cator, E. and Khanin, K. (2014). Space–time stationary solutions for the Burgers equation. J. Amer. Math. Soc. 27 193–238.
• [2] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225–400.
• [3] 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.
• [4] Cator, E. and Pimentel, L. P. R. (2012). Busemann functions and equilibrium measures in last passage percolation models. Probab. Theory Related Fields 154 89–125.
• [5] Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705–723.
• [6] 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.
• [7] 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.
• [8] Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746–1770.
• [9] Corwin, I. (2012). The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 1130001, 76.
• [10] Corwin, I., O’Connell, N., Seppäläinen, T. and Zygouras, N. (2014). Tropical combinatorics and Whittaker functions. Duke Math. J. 163 513–563.
• [11] Damron, M. and Hanson, J. (2014). Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Comm. Math. Phys. 325 917–963.
• [12] den Hollander, F. (2009). Random Polymers. Lecture Notes in Math. 1974. Springer, Berlin.
• [13] Ferrari, P. A., Martin, J. B. and Pimentel, L. P. R. (2009). A phase transition for competition interfaces. Ann. Appl. Probab. 19 281–317.
• [14] Ferrari, P. A. and Pimentel, L. P. R. (2005). Competition interfaces and second class particles. Ann. Probab. 33 1235–1254.
• [15] Georgiou, N. and Seppäläinen, T. (2013). Large deviation rate functions for the partition function in a log-gamma distributed random potential. Ann. Probab. 41 4248–4286.
• [16] Hoffman, C. (2005). Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 739–747.
• [17] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
• [18] Kosygina, E. and Varadhan, S. R. S. (2008). Homogenization of Hamilton–Jacobi–Bellman equations with respect to time–space shifts in a stationary ergodic medium. Comm. Pure Appl. Math. 61 816–847.
• [19] Krengel, U. (1985). Ergodic Theorems. De Gruyter Studies in Mathematics 6. Walter de Gruyter, Berlin. With a supplement by Antoine Brunel.
• [20] 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.
• [21] Newman, C. M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) 1017–1023. Birkhäuser, Basel.
• [22] O’Connell, N., Seppäläinen, T. and Zygouras, N. (2014). Geometric RSK correspondence, Whittaker functions and symmetrized random polymers. Invent. Math. 197 361–416.
• [23] O’Connell, N. and Yor, M. (2001). Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96 285–304.
• [24] Quastel, J. (2010). Weakly asymmetric exclusion and KPZ. In Proceedings of the International Congress of Mathematicians. Volume IV 2310–2324. Hindustan Book Agency, New Delhi.
• [25] Rassoul-Agha, F. and Seppäläinen, T. (2014). Quenched point-to-point free energy for random walks in random potentials. Probab. Theory Related Fields 158 711–750.
• [26] Rassoul-Agha, F., Seppäläinen, T. and Yilmaz, A. (2013). Quenched free energy and large deviations for random walks in random potentials. Comm. Pure Appl. Math. 66 202–244.
• [27] Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior. Die Grundlehren der mathematischen Wissenschaften 184 Springer, New York.
• [28] Seppäläinen, T. (2012). Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 19–73.
• [29] Spohn, H. (2012). Stochastic integrability and the KPZ equation. Available at arXiv:1204.2657.
• [30] Tracy, C. A. and Widom, H. (2002). Distribution functions for largest eigenvalues and their applications. In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) 587–596. Higher Ed. Press, Beijing.