Large Fluctuations in Amplifying Graphs

TL;DR

This paper investigates chaotic diffusion with amplification on graphs, identifying conditions for fat-tailed invariant measures and linking non-Gaussian statistics to generalized Lyapunov exponents, with insights into large graph behavior.

Contribution

It introduces a model for chaotic diffusion with amplification on graphs, analyzing invariant measures and their fat tails, and draws an analogy with directed polymers for physical interpretation.

Findings

Conditions for fat-tailed invariant measures are established.

Connection between non-Gaussian statistics and generalized Lyapunov exponents is demonstrated.

Results on large graphs are reported.

Abstract

We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval [S. Lepri, Chaos Solitons & Fractals, 139,110003 (2020)]. We determine the conditions for having fat-tailed invariant measures by considering approximate solution of the Perron-Frobenius equation for generic graphs. An analogy with the statistical mechanics of a directed polymer is presented that allows for a physically appealing interpretation of the statistical regimes. The connection between non-Gaussian statistics and the generalized Lyapunov exponents $L(q)$ is illustrated. Finally, some results concerning large graphs are reported.