Open Access
February 2019 Random polymers on the complete graph
Francis Comets, Gregorio Moreno, Alejandro F. Ramí rez
Bernoulli 25(1): 683-711 (February 2019). DOI: 10.3150/17-BEJ1002


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. $N\times N$ 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 $\alpha\in(0,1)$. 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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Francis Comets. Gregorio Moreno. Alejandro F. Ramí rez. "Random polymers on the complete graph." Bernoulli 25 (1) 683 - 711, February 2019.


Received: 1 July 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

MathSciNet: MR3892333
Digital Object Identifier: 10.3150/17-BEJ1002

Keywords: Directed polymers , exactly solvable model , Product of random matrices , random medium , Stable laws

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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