  We map the Markov Switching Multi-fractal model (MSM) onto the Random Energy
Model (REM). The MSM is, like the REM, an exactly solvable model in 1-d space
with non-trivial correlation functions. According to our results, four
different statistical physics phases are possible in random walks with
multi-fractal behavior. We also introduce the continuous branching version of
the model, calculate the moments and prove multiscaling behavior. Different
phases have different multi-scaling properties.
