Long-run growth rate in a random multiplicative model

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 $a_i=1+\rho\exp(\sigma W_i - \frac12 \sigma^2 t_i)$ 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 $\lambda(\rho,\beta) = \lim_{n\to \infty} \frac{1}{n} \log \langle x_n\rangle$ at fixed $\beta = \frac12 \sigma^2 t_n n$, and show that it is given by the equation of state of the lattice gas in thermodynamical equilibrium. The Lyapunov exponent has discontinuous first 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 $n\to \infty$.