High-Temperature Scaling Limit for Directed Polymers on a Hierarchical Lattice with Bond Disorder

  • Jeremy Thane ClarkEmail author


Diamond “lattices” are sequences of recursively-defined graphs that provide a network of directed pathways between two fixed root nodes, A and B. The construction recipe for diamond graphs depends on a branching number \(b\in {\mathbb {N}}\) and a segmenting number \(s\in {\mathbb {N}}\), for which a larger value of the ratio s / b intuitively corresponds to more opportunities for intersections between two randomly chosen paths. By attaching i.i.d. random variables to the bonds of the graphs, we construct a random Gibbs measure on the set of directed paths by assigning each path an “energy” through summing the random variables along the path. For the case \(b=s\), we propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs, and the partition function (the normalization factor of the Gibbs measure) appears to converge in law. We prove that all of the positive integer moments of the partition function converge in this limiting regime. The motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the \(b<s\) case.


Disordered systems Diamond hierarchical lattice Directed paths Partition function 



  1. 1.
    Alberts, T., Clark, J.: Nested critical points for a directed polymer on a disordered diamond lattice (to appear in Journal of Theoretical Probability) arXiv:1602.06629 (2017)
  2. 2.
    Alberts, T., Clark, J., Kocić, S.: The intermediate disorder regime for a directed polymer model on a hierarchical lattice. Stoch. Process. Appl. 127, 3291–3330 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alberts, T., Khanin, K., Quastel, J.: The intermediate disorder regime for directed polymers in dimension \(1+1\). Ann. Probab. 42(3), 1212–1256 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alberts, T., Khanin, K., Quastel, J.: The continuum directed random polymer. J. Stat. Phys. 154(1–2), 305–326 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bolthausen, E.: A note on the diffusion of directed polymers in a random environment. Commun. Math. Phys. 123, 529–534 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caravenna, F., Sun, R., Zygouras, N.: Polynomial chaos and scaling limits of disordered systems. J. Eur. Math. Soc. 19, 1–65 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caravenna, F., Sun, R., Zygouras, N.: Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27(5), 3050–3112 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Clark, J.T.: Continuum directed random polymers on disordered hierarchical diamond lattices, arXiv:1802.03834
  9. 9.
    Comets, F.: Directed Polymers in Random Environments. Lecture Notes in Mathematics, vol. 2175. Springer, New York (2017)zbMATHGoogle Scholar
  10. 10.
    Comets, F., Yoshida, N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34, 1746–1770 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cook, J., Derrida, B.: Polymers on disordered hierarchical lattices: a nonlinear combination of random variables. J. Stat. Phys. 57, 89–139 (1989)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Derrida, B., Gardner, E.: Renormalisation group study of a disordered model. J. Phys. A 17, 3223–3236 (1984)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Derrida, B., Griffith, R.B.: Directed polymers on disordered hierarchical lattices. Europhys. Lett. 8(2), 111–116 (1989)ADSCrossRefGoogle Scholar
  14. 14.
    Giacomin, G., Lacoin, H., Toninelli, F.L.: Hierarchical pinning models, quadratic maps, and quenched disorder. Probab. Theory Relat. Fields 145, 185–216 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Goldstein, L.: Normal approximation for hierarchical structures. Ann. Appl. Probab. 14(4), 1950–1969 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Griffith, R.B.: Spin systems on hierarchical lattices. Introduction and thermodynamical limit. Phys. Rev. B 26(9), 5022–5032 (1982)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Hambly, B.M., Jordan, J.H.: A random hierarchical lattice: the series-parallel graph and its properties. Adv. Appl. Prob. 36, 824–838 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hambly, B.M., Kumagai, T.: Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice. Adv. Appl. Prob. 36, 824–838 (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kahane, J.P.: Sur le chaos multiplicative. Ann. Sci. Math. Québec 9(2), 105–150 (1985)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lacoin, H., Moreno, G.: Directed polymers on hierarchical lattices with site disorder. Stoch. Proc. Appl. 120(4), 467–493 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lacoin, H.: Hierarchical pinning model with site disorder: disorder is marginally relevant. Probab. Theory Relat. Fields 148(1–2), 159–175 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schlösser, T., Spohn, H.: Sample to sample fluctuations in the conductivity of a disordered medium. J. Stat. Phys. 69, 955–967 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wehr, J., Woo, J.M.: Central limit theorems for nonlinear hierarchical sequences or random variables. J. Stat. Phys. 104, 777–797 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA

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