Abstract
We study the free energy fluctuations for a mixture of directed polymers, which was first introduced by Borodin et al. (2015) to investigate the limiting distribution of a stationary Kardar-Parisi-Zhang (KPZ) equation. The main results consist of both the law of large numbers and the asymptotic fluctuation for the free energy as the model size tends to infinity. In particular, we find the explicit values (which may depend on model parameters) of normalizing constants in the fluctuation. This shows that such a mixture model is in the KPZ university class.
Similar content being viewed by others
References
Borodin A, Corwin I. Macdonald processes. Probab Theory Related Fields, 2014, 158: 225–400
Borodin A, Corwin I, Ferrari P. Free energy fluctuations for directed polymers in random media in 1 + 1 dimension. Comm Pure Appl Math, 2014, 67: 1129–1214
Borodin A, Corwin I, Ferrari P L, et al. Height fluctuations for the stationary KPZ equation. Math Phys Anal Geom, 2015, 18: Art 20, 95pp
Borodin A, Corwin I, Remenik D. Log-Gamma polymer free energy fluctuations via a Fredholm determinant, identity. Comm Math Phys, 2013, 324: 215–232
Borodin A, Petrov L. Integrable probability: From representation theory to Macdonald processes. Probab Surv, 2014, 11: 1–58
Corwin I. The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl, 2012, 1: 1130001
Georgiou N, Seppäläinen T. Large deviation rate functions for the partition function in a log-gamma distributed random potential. Ann Probab, 2013, 41: 4248–4286
Janjigian C. Large deviations of the free energy in the O’Connell-Yor polymer. J Stat Phys, 2015, 160: 1054–1080
Moriarty J, O’Connell N. On the free energy of a directed polymer in a Brownian environment. Markov Process Relat Fields, 2007, 13: 251–266
O’Connell N, Yor M. Brownian analogues of Burke’s theorem. Stochastic Process Appl, 2001, 96: 285–304
Seppäläinen T. Scaling for a one-dimensional directed polymer with boundary conditions. Ann Probab, 2012, 40: 19–73
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11371317) and the Fundamental Research Funds for the Central Universities.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Su, Z. Free energy fluctuations for a mixture of directed polymers. Sci. China Math. 60, 511–528 (2017). https://doi.org/10.1007/s11425-016-0060-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-0060-0
Keywords
- free energy
- KPZ universality class
- log-Gamma directed polymer
- O’Connell-Yor semi-discrete directed polymer
- Tracy-Widom law