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Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity

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Abstract

We prove that under n 1/3 scaling, the limiting distribution as n → ∞ of the free energy of Seppäläinen’s log-Gamma discrete directed polymer is GUE Tracy-Widom. The main technical innovation we provide is a general identity between a class of n-fold contour integrals and a class of Fredholm determinants. Applying this identity to the integral formula proved in Corwin et al. (Tropical combinatorics and Whittaker functions. http://arxiv.org/abs/1110.3489v3 [math.PR], 2012) for the Laplace transform of the log-Gamma polymer partition function, we arrive at a Fredholm determinant which lends itself to asymptotic analysis (and thus yields the free energy limit theorem). The Fredholm determinant was anticipated in Borodin and Corwin (Macdonald processes. http://arxiv.org/abs/1111.4408v3 [math.PR], 2012) via the formalism of Macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach.

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References

  1. Abramowitz M., Stegun I. A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Volume 55. National Bureau of Standards Applied Mathematics Series, Washington, DC: Nat. Bur. of Standards, 1964

  2. Alberts, T., Khanin, K., Quastel, J.: Intermediate disorder regime for 1+1 dimensional directed polymers. http://arxiv.org/abs/1202.4398v2 [math.PR], 2012

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Andréief C.: Note sur une relation entre les intégrales définies des produits des fonctions. Mém. de la Soc. Sci., Bordeaux 2, 1–14 (1883)

    Google Scholar 

  5. Auffinger, A., Baik, J., Corwin, I.: Universality for directed polymers in thin rectangles. http://arxiv.org/abs/1204.4445v2 [math.PR], 2012

  6. Baik J., Ben Arous G., Peché S.: Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33, 1643–1697 (2006)

    Article  Google Scholar 

  7. Borodin, A., Corwin, I.: Macdonald processes. Probab. Theor. Rel. Fields. http://arxiv.org/abs/1111.4408v3 [math.PR], (2012, to appear)

  8. Borodin, A., Corwin, I., Ferrari, P.: Free energy fluctuations for directed polymers in random media in 1+1 dimension. Commun. Pure Appl. Math. http://arxiv.org/abs/1204.1024v1 [math.PR], (2012, to appear)

  9. Borodin A., Péché S.: Airy kernel with two sets of parameters in directed percolation and random matrix theory. J. Stat. Phys. 132, 275–290 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Corwin I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1, 1130001 (2012)

    Article  MathSciNet  Google Scholar 

  11. Corwin, I., O’Connell, N., Seppäläinen, T., Zygouras, N.: Tropical combinatorics and Whittaker functions. http://arxiv.org/abs/1110.3489v3 [math.PR], 2012

  12. Corwin I., Quastel J.: Universal distribution of fluctuations at the edge of the rarefaction fan. Ann. Probab. 41(3), 1243–1314 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kardar K., Parisi G., Zhang Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)

    Article  ADS  MATH  Google Scholar 

  14. Moreno Flores, G., Quastel, J., Remenik, D.: Scaling to KPZ for discrete polymers and q-TASEP. In preparation

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Tracy C., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Tracy C., Widom H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

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

    Alexei Borodin

  2. Microsoft Research, New England, 1 Memorial Drive, Cambridge, MA, 02142, USA

    Ivan Corwin

  3. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Daniel Remenik

  4. Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile

    Daniel Remenik

Authors
  1. Alexei Borodin
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  2. Ivan Corwin
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  3. Daniel Remenik
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Correspondence to Alexei Borodin.

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Communicated by H. Spohn

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Borodin, A., Corwin, I. & Remenik, D. Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity. Commun. Math. Phys. 324, 215–232 (2013). https://doi.org/10.1007/s00220-013-1750-x

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  • DOI: https://doi.org/10.1007/s00220-013-1750-x

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