Random polymers on the complete graph

Abstract : Consider directed polymers in a random environment on the complete graph of size N. This model can be formulated as a product of i.i.d. N×N random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path. Next, we observe that the model becomes exactly solvable when the disorder variables are located on edges of the complete graph and follow a totally asymmetric stable law of index α∈(0,1). Then, a certain notion of mean height of the polymer behaves like a random walk and we show that the height function is distributed around this mean according to an explicit law. Large N asymptotics can be taken in this setting, for instance, for the free energy of the system and for the invariant law of the polymer height with a shift. Moreover, we give some perturbative results for environments which are close to the totally asymmetric stable laws.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01561548
Contributeur : Francis Comets <>
Soumis le : jeudi 13 juillet 2017 - 08:17:27
Dernière modification le : jeudi 20 juillet 2017 - 01:11:03

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cmr-170628.pdf
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  • HAL Id : hal-01561548, version 1
  • ARXIV : 1707.01588

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INSMI | PMA | UPMC

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Francis Comets, Gregorio Moreno, Alejandro F. Ramirez. Random polymers on the complete graph. 2017. <hal-01561548>

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