Random polymers on the complete graph

TL;DR

This paper analyzes directed polymers on a complete graph, linking their behavior to random matrix products, and finds exact solutions when disorder follows a stable law, providing insights into their asymptotics and invariant distributions.

Contribution

It establishes a detailed correspondence between polymer models and random matrix theory, and derives exact solutions for models with stable law disorder.

Findings

Long-time asymptotics characterized by Lyapunov exponents.

Exact solvability when disorder follows a stable law.

Explicit distribution laws for polymer height and free energy.

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. $N\times N$ 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.