Long-run growth rate in a random multiplicative model

Abstract

We consider the long-run growth rate of the average value of a random multiplicative process xi + 1 = aixi where the multipliers \documentclass[12pt]{minimal}\begin{document}$a_i=1+\rho \exp (\sigma W_i - \frac{1}{2}\sigma ^2 t_i)$\end{document}ai=1+ρexp(σWi−12σ2ti) have Markovian dependence given by the exponential of a standard Brownian motion Wi. The average value ⟨xn⟩ is given by the grand partition function of a one-dimensional lattice gas with two-body linear attractive interactions placed in a uniform field. We study the Lyapunov exponent \documentclass[12pt]{minimal}\begin{document}$\lambda =\lim _{n\rightarrow \infty } \frac{1}{n}\log \langle x_n\rangle$\end{document}λ=limn→∞1nlog⟨xn⟩, at fixed \documentclass[12pt]{minimal}\begin{document}$\beta = \frac{1}{2} \sigma^2 t_n n$\end{document}β=12σ2tnn, and show that it is given by the equation of state of the lattice gas in thermodynamical equilibrium. The Lyapunov exponent has discontinuous partial derivatives along a curve in the (ρ, β) plane ending at a critical point (ρC, βC) which is related to a phase transition in the equivalent lattice gas. Using the equivalence of the lattice gas with a bosonic system, we obtain the exact solution for the equation of state in the thermodynamical limit n → ∞.