The Strict-Weak Lattice Polymer
- 122 Downloads
We introduce the strict-weak polymer model, and show the KPZ universality of the free energy fluctuation of this model for a certain range of parameters. Our proof relies on the observation that the discrete time geometric \(q\)-TASEP model, studied earlier by Borodin and Corwin, scales to this polymer model in the limit \(q\rightarrow 1\). This allows us to exploit the exact results for geometric \(q\)-TASEP to derive a Fredholm determinant formula for the strict-weak polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling and GUE Tracy–Widom limit for the free energy fluctuations. We also derive moments formulae for the polymer partition function directly by Bethe ansatz, and identify the limit of the free energy using a stationary version of the polymer model.
KeywordsDirected polymer Fredholm determinant Kardar–Parisi–Zhang q-TASEP Replica method Tracy–Widom distribution
I. Corwin was partially supported by the NSF grant DMS-1208998 as well as by Microsoft Research and MIT through the Schramm Memorial Fellowship, by the Clay Mathematics Institute through the Clay Research Fellowship and by the Institut Henri Poincaré through the Poincaré Chair. H. Shen would like to thank Prof. Martin Hairer for his support on a visit to MSRI in July 2014 where part of this work was done. T. Seppäläinen was partially supported by NSF grant DMS-1306777 and by the Wisconsin Alumni Research Foundation.
- 2.Barraquand, G., Corwin, I.: Random-walk in beta-distributed random environment. arXiv:1503.04117 (2015)
- 3.Borodin, A., Corwin, I.: Discrete time q-TASEPs. Int. Math. Res. Notices (2013). doi: 10.1093/imrn/rnt206
- 6.Borodin, A., Corwin, I., Ferrari, P., Veto, B.: Height fluctuations for the stationary kpz equation. arXiv:1407.6977 (2014)
- 7.Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for the q-boson particle system. Compos. Math. arXiv:1308.3475 (2013)
- 8.Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz. arXiv:1407.8534 (2014)
- 11.Matveev, K., Petrov, L.: \(q\)-Randomized Robinson–Schensted–Knuth correspondences and random polymers. arXiv:1504.00666 (2015)
- 12.O’Connell, N.: Directed percolation and tandem queues. Technical Report STP-99-12, DIAS (1999)Google Scholar
- 14.O’Connell, N., Ortmann, J.: Tracy–Widom asymptotics for a random polymer model with gamma-distributed weights. arXiv:1408.5326 (2014)
- 16.Thiery, T., Le Doussal, P.: Log-gamma directed polymer with fixed endpoints via the replica bethe ansatz. arXiv:1406.5963 (2014)