Large Fluctuations in Amplifying Graphs

In this paper, we consider a model for chaotic di®usion with ampli¯cation on graphs associated with piecewise-linear maps of the interval [S. Lepri, Chaotic °uctuations in graphs with ampli¯cation, Chaos, Solitons & Fractals 139 (2020) 110003]. 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 LðqÞ is illustrated. Finally, some results concerning large graphs are reported