Summary: We consider the long-run growth rate of the average value of a random multiplicative process \(x_{i + 1} = a_{i}x_{i}\) where the multipliers \(a_i=1+\rho \exp (\sigma W_i - \frac{1}{2}\sigma ^2 t_i)\) have Markovian dependence given by the exponential of a standard Brownian motion \(W_{i}\). The average value \(\langle x_{n}\rangle\) 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 \(\lambda =\lim _{n\to \infty } \frac{1}{n}\log \langle x_n\rangle\), at fixed \(\beta = \frac{1}{2} \sigma^2 t_n n\), 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 ({\(\rho\)}, {\(\beta\)}) plane ending at a critical point ({\(\rho_{C}\)}, {\(\beta_{C}\)}) 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 \(n \to \infty\).{
©2014 American Institute of Physics}