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Communications in Mathematical Physics

, Volume 370, Issue 3, pp 781–806 | Cite as

Delocalization of Polymers in Lower Tail Large Deviation

  • Riddhipratim BasuEmail author
  • Shirshendu Ganguly
  • Allan Sly
Article
  • 83 Downloads

Abstract

Directed last passage percolation models on the plane, where one studies the weight as well as the geometry of optimizing paths (called polymers) in a field of i.i.d. weights, are paradigm examples of models in KPZ universality class. In this article, we consider the large deviation regime, i.e., when the polymer has a much smaller (lower tail) or larger (upper tail) weight than typical. Precise asymptotics of large deviation probabilities have been obtained in a handful of the so-called exactly solvable scenarios, including the Exponential (Johansson in Commun Math Phys 209(2):437–476, 2000) and Poissonian (Deuschel and Zeitouni in Comb Probab Comput 8(03):247–263, 1999; Seppäläinen in Probab Theory Relat Fields 112(2):221–244, 1998) cases. How the geometry of the optimizing paths change under such a large deviation event was considered in Deuschel and Zeitouni (1999) where it was shown that the paths [from (0, 0) to (nn), say] remain concentrated around the straight line joining the end points in the upper tail large deviation regime, but the corresponding question in the lower tail was left open. We establish a contrasting behaviour in the lower tail large deviation regime, showing that conditioned on the latter, in both the models, the optimizing paths are not concentrated around any deterministic curve. Our argument does not use any ingredient from integrable probability, and hence can be extended to other planar last passage percolation models under fairly mild conditions; and also to other non-integrable settings such as last passage percolation in higher dimensions.

Notes

Acknowledgements

The authors thank Timo Seppäläinen for pointing out some relevant references, and Ofer Zeitouni for useful conversations. We also thank an anonymous referee for a careful reading of the manuscript as well as several insightful comments. Research of RB is partially supported by an ICTS-Simons Junior Faculty Fellowship and a Ramanujan Fellowship (SB/S2/RJN-097/2017) by Govt. of India. SG’s research was supported by a Miller Research Fellowship. AS is supported by NSF Grant DMS-1352013 and a Simons Investigator grant.

References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basu, R., Ganguly, S., Sly, A.: Upper tail large deviations in first passage percolation. Preprint, arXiv:1712.01255
  3. 3.
    Basu, R., Ganguly, S., Sly, A.: Delocalization of polymers in lower tail large deviation. Preprint, arXiv:1710.11623
  4. 4.
    Basu, R., Sarkar, S., Sly, A.: Coalescence of geodesics in exactly solvable models of last passage percolation. Preprint, arXiv:1704.05219
  5. 5.
    Basu, R., Sidoravicius, V., Sly, A.: Last passage percolation with a defect line and the solution of the slow bond problem. Preprint, arXiv:1408.3464
  6. 6.
    Chow, Y., Zhang, Y.: Large deviations in first-passage percolation. Ann. Appl. Probab. 13, 1601–1614 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Theodore Cox, J., Gandolfi, A., Griffin, P.S., Kesten, H.: Greedy lattice animals I: upper bounds. Ann. Appl. Probab. 3(4), 1151–1169 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deuschel, J.-D., Zeitouni, O.: Limiting curves for iid records. Ann. Probab. 23, 852–878 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Deuschel, J.-D., Zeitouni, O.: On increasing subsequences of iid samples. Comb. Probab. Comput. 8(03), 247–263 (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gandolfi, A., Kesten, H.: Greedy lattice animals II: linear growth. Ann. Appl. Probab. 4(1), 76–107 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Johansson, K.: Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Relat. Fields 116(4), 445–456 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kesten, H.: École d’Été de Probabilités de Saint Flour XIV-1984, chapter Aspects of first passage percolation, pp. 125–264 (1986)Google Scholar
  14. 14.
    Kesten, H.: First-passage percolation. In: From Classical to Modern Probability, vol. 54 of Progress in Probability, pp. 93–143. Birkhäuser, Basel (2003)Google Scholar
  15. 15.
    Martin, J.B.: Linear growth for greedy lattice animals. Stoch. Process. Their Appl. 98(1), 43–66 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rost, H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Zeitschrift f. Warsch. Verw. Gebiete 58(1), 41–53 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Seppäläinen, T.: Coupling the totally asymmetric simple exclusion process with a moving interface. Markov Process. Relat. Fields 4(4), 593–628 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Seppäläinen, T.: Large deviations for increasing sequences on the plane. Probab. Theory Relat. Fields 112(2), 221–244 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. l’IHES 81(1), 73–205 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Riddhipratim Basu
    • 1
    Email author
  • Shirshendu Ganguly
    • 2
  • Allan Sly
    • 3
  1. 1.International Centre for Theoretical SciencesTata Institute of Fundamental ResearchBangaloreIndia
  2. 2.Department of StatisticsUC BerkeleyBerkeleyUSA
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA

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