The Markov Switching Multi-fractal models as a new class of REM-like models in 1-dimensional space

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.