The Competition of Roughness and Curvature in Area-Constrained Polymer Models

Abstract

The competition between local Brownian roughness and global parabolic curvature experienced in many random interface models reflects an important aspect of the KPZ universality class. It may be summarized by an exponent triple (1/2, 1/3, 2/3) representing local interface fluctuation, local roughness (or inward deviation), and convex hull facet length. The three effects arise, for example, in droplets in planar Ising models (Alexander in Commun Math Phys 224(3): 733–781, 2001; Hammond in J Stat Phys 142(2):229–276, 2011, Commun Math Phys 310(2):455–509, 2012, Ann Probab 40(3):921–978, 2012). In this article, we offer a new perspective on this phenomenon. We consider the model of directed last passage percolation in the plane, a paradigmatic example in the KPZ universality class, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. The interface suffers a constraint of parabolic curvature, as the Ising droplets do, but now its local interface fluctuation exponent is governed by KPZ relations, and is thus two-thirds rather than one-half. We prove that the facet lengths of the constrained path’s convex hull are governed by an exponent of 3/4, and inward deviation by an exponent of 1/2. That is, the exponent triple is now (2/3, 1/2, 3/4) in place of (1/2, 1/3, 2/3). This phenomenon appears to be shared among various isoperimetrically extremal circuits in local randomness. Indeed, we formulate a conjecture to this effect, concerning such circuits in supercritical percolation, whose Wulff-like first-order behaviour was recently established by Biskup et al. (Comm Pure Appl Math 68:1483–1531, 2015), settling a conjecture of Benjamini.

This is a preview of subscription content, log in to check access.

References

  1. 1

    Aldous D., Diaconis P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103, 199–213 (1995)

    MathSciNet  Article  Google Scholar 

  2. 2

    Alexander K.S.: Cube-root boundary fluctuations for droplets in random cluster models. Commun. Math. Phys. 224(3), 733–781 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3

    Auffinger, A., Damron, M., Hanson, J.: 50 years of first passage percolation (2015). arXiv:1511.03262

  4. 4

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

    MathSciNet  Article  Google Scholar 

  5. 5

    Baik J., Rains E.M.: Algebraic aspects of increasing subsequences. Duke Math. J. 109(1), 1–65 (2001)

    MathSciNet  Article  Google Scholar 

  6. 6

    Baik, J., Rains, E.M.: Symmetrized random permutations. In: Random Matrix Models and Their Applications, Volume 40 of Mathematical Sciences Research Institute Publications, pp. 1–19 (2001)

  7. 7

    Basu, R., Hammond, A.: Localization of near geodesics in Brownian last passage percolation. In preparation

  8. 8

    Basu, R., Sidoravicius, V., Sly, A.: Last passage percolation with a defect line and the solution of the slow bond problem. arXiv:1408.3464

  9. 9

    Biskup, M., Louidor, O., Procaccia, E.B., Rosenthal, R.: Isoperimetry in two-dimensional percolation. Comm. Pure Appl. Math. 68, 1483–1531 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10

    Bollobás B., Brightwell G.: The height of a random partial order: concentration of measure. Ann. Appl. Probab. 2, 1009–1018 (1992)

    MathSciNet  Article  Google Scholar 

  11. 11

    Chatterjee S., Dey P.S.: Central limit theorem for first-passage percolation time across thin cylinders. Probab. Theory Relat. Fields 156(3), 613–663 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12

    Deuschel J.-D., Zeitouni O.: Limiting curves for iid records. Ann. Probab. 23, 852–878 (1995)

    MathSciNet  Article  Google Scholar 

  13. 13

    Deuschel J.-D., Zeitouni O.: On increasing subsequences of iid samples. Comb. Probab. Comput. 8(03), 247–263 (1999)

    Article  Google Scholar 

  14. 14

    Dey, P.S., Peled, R., Joseph, M.: Longest increasing path within the critical strip. arXiv:1808.08407

  15. 15

    Dobrushin R.L., Koteckỳ R., Shlosman S.: Wulff Construction: A Global Shape from Local Interaction. American Mathematical Society, Providence (1992)

    Google Scholar 

  16. 16

    Durrett R.: Probability: Theory and Examples. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  17. 17

    Ferrari P.L., Spohn H.: Constrained brownian motion: fluctuations away from circular and parabolic barriers. Ann. Probab. 33, 1302–1325 (2005)

    MathSciNet  Article  Google Scholar 

  18. 18

    Gold, J.: Isoperimetry in supercritical bond percolation in dimensions three and higher (2016). arXiv:1602.05598

  19. 19

    Hammond A.: Phase separation in random cluster models iii: Circuit regularity. J. Stat. Phys. 142(2), 229–276 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20

    Hammond A.: Phase separation in random cluster models i: uniform upper bounds on local deviation. Commun. Math. Phys. 310(2), 455–509 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  21. 21

    Hammond A.: Phase separation in random cluster models ii: the droplet at equilibrium, and local deviation lower bounds. Ann. Probab. 40(3), 921–978 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22

    Hammond A., Peres Y.: Fluctuation of a planar brownian loop capturing a large area. Trans. Am. Math. Soc. 360(12), 6197–6230 (2008)

    MathSciNet  Article  Google Scholar 

  23. 23

    Ioffe D., Schonmann R.H.: Dobrushin–Koteckỳ–Shlosman theorem up to the critical temperature. Commun. Math. Phys. 199(1), 117–167 (1998)

    ADS  Article  Google Scholar 

  24. 24

    Johansson K.: Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Relat. Fields 116(4), 445–456 (2000)

    MathSciNet  Article  Google Scholar 

  25. 25

    Kardar M., Parisi G., Zhang Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)

    ADS  Article  Google Scholar 

  26. 26

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

    MathSciNet  Article  Google Scholar 

  27. 27

    Löwe M., Merkl F.: Moderate deviations for longest increasing subsequences: the upper tail. Commun. Pure Appl. Math. 54, 1488–1519 (2001)

    MathSciNet  Article  Google Scholar 

  28. 28

    Löwe M., Merkl F., Rolles S.: Moderate deviations for longest increasing subsequences: the lower tail. J. Theor. Probab. 15(4), 1031–1047 (2002)

    MathSciNet  Article  Google Scholar 

  29. 29

    Seppäläinen T.: Large deviations for increasing sequences on the plane. Probab. Theory Relat. Fields 112(2), 221–244 (1998)

    MathSciNet  Article  Google Scholar 

  30. 30

    Vershik, A.M., Kerov, S.V.: Asymptotics of the plancherel measure of the symmetric group and the limiting form of young tables. Soviet Math. Dokl. 18:527–531 (1977). Translation of Dokl. Acad. Nauk. SSSR 233:1024–1027 (1977)

Download references

Acknowledgements

The authors thank Marek Biskup, Craig Evans and Ofer Zeitouni for useful discussions, and an anonymous referee for comments on the manuscript. R.B. was partially supported by an AMS-Simons Travel Grant during the completion of this work. S.G.’s research was supported by a Miller Research Fellowship at UC Berkeley. A.H. was supported by NSF Grant DMS-1512908.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shirshendu Ganguly.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Basu, R., Ganguly, S. & Hammond, A. The Competition of Roughness and Curvature in Area-Constrained Polymer Models. Commun. Math. Phys. 364, 1121–1161 (2018). https://doi.org/10.1007/s00220-018-3282-x

Download citation