In this paper, we perform the multifractal analysis of the Lyapunov exponent for random conformal graph directed Markov systems introduced by Roy and Urbański (2011). We also generalize Bowen's formula for the limit set of a random conformal graph directed Markov system established by Roy and Urbański. To do this, we develop several refined properties of random finitely primitive countable Markov shifts.
Y. Arima (Thu,) studied this question.