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The Strict-Weak Lattice Polymer

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Abstract

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.

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References

  1. Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in \(1+1\) dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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

  4. Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields 158(1–2), 225–400 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borodin, A., Corwin, I., Ferrari, P.: Free energy fluctuations for directed polymers in random media in \(1+1\) dimension. Commun. Pure Appl. Math. 67(7), 1129–1214 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  9. Borodin, A., Corwin, I., Remenik, D.: Log-gamma polymer free energy fluctuations via a Fredholm determinant identity. Commun. Math. Phys. 324(1), 215–232 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Corwin, I., O’Connell, N., Seppäläinen, T., Zygouras, N.: Tropical combinatorics and Whittaker functions. Duke Math. J. 163(3), 513–563 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  13. O’Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40(2), 437–458 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. O’Connell, N., Ortmann, J.: Tracy–Widom asymptotics for a random polymer model with gamma-distributed weights. arXiv:1408.5326 (2014)

  15. Seppäläinen, T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40(1), 19–73 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Thiery, T., Le Doussal, P.: Log-gamma directed polymer with fixed endpoints via the replica bethe ansatz. arXiv:1406.5963 (2014)

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Acknowledgments

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.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Ivan Corwin

  2. Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, Providence, RI, 02903, USA

    Ivan Corwin

  3. Institute Henri Poincare, 11 Rue Pierre et Marie Curie, 75005, Paris, France

    Ivan Corwin

  4. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Ivan Corwin

  5. Department of Mathematics, University of Wisconsin-Madison, 425 Van Vleck Hall, Madison, WI, 53706-1388, USA

    Timo Seppäläinen

  6. Mathematics Department, University of Warwick, Coventry, CV4 7AL, UK

    Hao Shen

Authors
  1. Ivan Corwin
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  2. Timo Seppäläinen
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  3. Hao Shen
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Correspondence to Timo Seppäläinen.

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Corwin, I., Seppäläinen, T. & Shen, H. The Strict-Weak Lattice Polymer. J Stat Phys 160, 1027–1053 (2015). https://doi.org/10.1007/s10955-015-1267-0

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  • DOI: https://doi.org/10.1007/s10955-015-1267-0

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