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.
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https://hal.archives-ouvertes.fr/hal-01561548
Contributor : Francis Comets <>
Submitted on : Thursday, November 30, 2017 - 10:11:50 AM
Last modification on : Thursday, March 21, 2019 - 1:09:08 PM

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  • HAL Id : hal-01561548, version 2
  • ARXIV : 1707.01588

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Francis Comets, Gregorio Moreno, Alejandro F. Ramirez. Random polymers on the complete graph. 2017. ⟨hal-01561548v2⟩

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