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_ix_i where the multipliers a_i=1+ρ exp (σ W_i - 1/2 σ ²t_i) have Markovian dependence given by the exponential of a standard Brownian motion W_i. The average value 〈x_n〈 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 λ = lim _{n →∞} 1/n log 〈 x_n 〉 at fixed β = 1/2 σ 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 (ρ, β) plane ending at a critical point (ρ_C, β_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 → ∞.