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Universal edge scaling in random partitions

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Abstract

We establish the universal edge scaling limit of random partitions with the infinite-parameter distribution called the Schur measure. We explore the asymptotic behavior of the wave function, which is a building block of the corresponding kernel, based on the Schrödinger-type differential equation. We show that the wave function is in general asymptotic to the Airy function and its higher-order analogs in the edge scaling limit. We construct the corresponding higher-order Airy kernel and the Tracy–Widom distribution from the wave function in the scaling limit and discuss its implication to the multicritical phase transition in the large-size matrix model. We also discuss the limit shape of random partitions through the semi-classical analysis of the wave function.

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Notes

  1. Not to be confused with the Schur measure parameter \({\mathsf {X}} = ({\mathsf {x}}_i)_{i \in {\mathbb {N}}}\).

References

  1. Mehta, M.L.: Random Matrices. Vol 142 of Pure and Applied Mathematics, 3rd edn. Academic Press, Cambridge (2004)

    Google Scholar 

  2. Forrester, P.J.: Log-gases and Random Matrices. Princeton Univ Press, Princeton (2010)

    Book  MATH  Google Scholar 

  3. Kuijlaars, A.: The Oxford Handbook of Random Matrix Theory, ch. Universality, pp. 103–134. Oxford University Press, Oxford (2011)

    Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bowick, M.J., Brézin, E.: Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett. B 268, 21–28 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  6. Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709–728 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Nagao, T., Wadati, M.: Eigenvalue distribution of random matrices at the spectrum edge. J. Phys. Soc. Jap. 62(11), 3845–3856 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12, 1119–1178 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Gross, D.J., Witten, E.: Possible third order phase transition in the large \(N\) Lattice Gauge Theory. Phys. Rev. D 21, 446–453 (1980)

    ADS  Google Scholar 

  11. Wadia, S.R.: \(N = \infty \) Phase transition in a class of exactly soluble model Lattice Gauge Theories. Phys. Lett. 93B, 403–410 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  12. Douglas, M.R., Kazakov, V.A.: Large \(N\) phase transition in continuum QCD in two-dimensions. Phys. Lett. B 319, 219–230 (1993)

    ADS  Google Scholar 

  13. Majumdar, S.N., Schehr, G.: Top eigenvalue of a random matrix: large deviations and third order phase transition. J. Stat. Mech. 2014, P01012 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zahabi, A.: New phase transitions in Chern-Simons matter theory. Nucl. Phys. B 903, 78–103 (2016)

    Article  ADS  MATH  Google Scholar 

  15. Saeedian, M., Zahabi, A.: Phase structure of XX0 spin chain and nonintersecting Brownian motion. J. Stat. Mech. 1801(1), 013104 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Saeedian, M., Zahabi, A.: Exact solvability and asymptotic aspects of geddneralized XX0 spin chainsd. Physica A 549, 124406 (2020)

    Article  MathSciNet  Google Scholar 

  17. Prähofer, M., Spohn, H.: Universal distributions for growth processes in \(1+1\) dimensions and random matrices. Phys. Rev. Lett. 84(21), 4882–4885 (2000)

    ADS  Google Scholar 

  18. Takeuchi, K.A., Sano, M.: Universal fluctuations of growing interfaces: evidence in Turbulent liquid crystals. Phys. Rev. Lett. 104(23), 230601 (2010)

    Article  ADS  Google Scholar 

  19. Takeuchi, K.A., Sano, M., Sasamoto, T., Spohn, H.: Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep. 1(1), 34 (2011)

    Article  ADS  Google Scholar 

  20. Kerov, S.V.: Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis, Vol 219of Translations of Mathematical Monographs. American Mathematical Society, Providence (2003)

    Google Scholar 

  21. Olshanski, G.: The Oxford Handbook of Random Matrix Theory, ch. Random Permutations and Related Topics, pp. 510–533. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  22. Okounkov, A.: Infinite wedge and random partitions. Sel. Math. 7(1), 57–81 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Borodin, A., Okounkov, A.: A Fredholm determinant formula for Toeplitz determinants. Int. Eq. Op. Th. 37(4), 386–396 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Periwal, V., Shevitz, D.: Exactly solvable unitary matrix models: multicritical potentials and correlations. Nucl. Phys. B 344, 731–746 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  25. Claeys, T., Krasovsky, I., Its, A.: Higher-order analogues of the Tracy-Widom distribution and the Painlevé II hierarchy. Commun. Pure Appl. Math. 63, 362–412 (2009)

    MATH  Google Scholar 

  26. Le Doussal, P., Majumdar, S.N., Schehr, G.: Multicritical edge statistics for the momenta of Fermions in nonharmonic traps. Phys. Rev. Lett. 121(3), 030603 (2018)

    Article  Google Scholar 

  27. Cafasso, M., Claeys, T., Girotti, M.: Fredholm determinant solutions of the painlevé II hierarchy and gap probabilities of determinantal point processes. Int. Math. Res. Not. 2021(4), 2437–2478 (2019). https://doi.org/10.1093/imrn/rnz168. arXiv:1902.05595

  28. Betea, D., Bouttier, J., Walsh, H.: Multicritical random partitions (2020). arXiv:2012.01995 [math.CO]

  29. Jimbo, M., Miwa, T.: Solitons and infinite dimensional lie algebras. Publ. Res. Inst. Math. Sci. Kyoto 19, 943–1001 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  30. Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. 57, 4140–4149 (1998)

    ADS  MathSciNet  Google Scholar 

  31. Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phy. Rev. 58, 7176–7185 (1998)

    ADS  MathSciNet  Google Scholar 

  32. Baik, J., Buckingham, R., DiFranco, J.: Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function. Commun. Math. Phys. 280(2), 463–497 (2008)

    Article  ADS  MATH  Google Scholar 

  33. Dai, D., Xu, S.X., Zhang L.: Asymptotics of Fredholm determinant associated with the Pearcey kernel (2020). https://doi.org/10.1007/s00220-021-03986-3. arXiv:2002.06370

  34. Borodin, A., Okounkov, A., Olshanski, G.: On asymptotics of the Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13, 481–515 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Logan, B., Shepp, L.: A variational problem for random Young tableaux. Adv. Math. 26, 206–222 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vershik, A., Kerov, S.: Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Soviet Math. Dokl. 18, 527–531 (1977)

    MATH  Google Scholar 

  37. Eynard, B., Kimura, T., Ribault, S.: Random matrices (2015). arXiv:1510.04430 [math-ph]

  38. Kimura, T.: Linear Algebra - Theorems and Applications, ch. Gauge Theory Combinatorics, and Matrix Models, pp. 75–98. IntechOpen, London (2012)

    Google Scholar 

  39. Borodin, A., Corwin, I.: Macdonald processes. Prob. Theor. Rel. Fields 158, 225–400 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work has been supported in part by “Investissements d’Avenir” program, Project ISITE-BFC (No. ANR-15-IDEX-0003), EIPHI Graduate School (No. ANR-17-EURE-0002) and the Bourgogne-Franche-Comté Region.

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  1. Institut de Mathématiques de Bourgogne, Université Bourgogne Franche-Comté, Dijon, France

    Taro Kimura & Ali Zahabi

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  1. Taro Kimura
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Correspondence to Taro Kimura.

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Kimura, T., Zahabi, A. Universal edge scaling in random partitions. Lett Math Phys 111, 48 (2021). https://doi.org/10.1007/s11005-021-01389-y

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