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Directed polymers on hierarchical lattices with site disorder

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  • Lacoin, Hubert
  • Moreno, Gregorio

Abstract

We study a polymer model on hierarchical lattices very close to the one introduced and studied in Derrida and Griffith (1989)Â [19] and Cook and Derrida (1989)Â [16]. For this model, we prove the existence of free energy and derive the necessary and sufficient condition for which very strong disorder holds for all [beta], and give some accurate results on the behavior of the free energy at high temperature. We obtain these results by using a combination of fractional moment method and change of measure over the environment to obtain an upper bound, and a second moment method to get a lower bound. We also get lower bounds on the fluctuation exponent of logZn, and study the infinite polymer measure in the weak disorder phase.

Suggested Citation

  • Lacoin, Hubert & Moreno, Gregorio, 2010. "Directed polymers on hierarchical lattices with site disorder," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 467-493, April.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:4:p:467-493
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    References listed on IDEAS

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    1. Lesigne, Emmanuel & Volný, Dalibor, 2001. "Large deviations for martingales," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 143-159, November.
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    Cited by:

    1. Alberts, Tom & Clark, Jeremy & Kocić, Saša, 2017. "The intermediate disorder regime for a directed polymer model on a hierarchical lattice," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3291-3330.
    2. Clark, Jeremy Thane, 2020. "Continuum directed random polymers on disordered hierarchical diamond lattices," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1643-1668.
    3. Clark, Jeremy & Lochridge, Casey, 2023. "Weak-disorder limit for directed polymers on critical hierarchical graphs with vertex disorder," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 75-102.
    4. Tom Alberts & Jeremy Clark, 2019. "Nested Critical Points for a Directed Polymer on a Disordered Diamond Lattice," Journal of Theoretical Probability, Springer, vol. 32(1), pages 64-89, March.

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