Long-run growth rate in a random multiplicative model

TL;DR

This paper analyzes the long-term growth rate of a random multiplicative process with Markovian dependence, revealing a phase transition analogous to a lattice gas in thermodynamics and providing an exact solution for the growth rate.

Contribution

It establishes a novel connection between the growth rate of a stochastic process and a lattice gas model, deriving an exact thermodynamic description.

Findings

Lyapunov exponent characterized by the lattice gas equation of state.

Discontinuous derivatives indicating a phase transition.

Exact solution for the growth rate in the thermodynamic limit.

Abstract

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 $n\to \infty$.