Soliton Turbulence as a Thermodynamic Limit of Stochastic Soliton Lattices

We use recently introduced notion of stochastic soliton lattice for quantitative description of soliton turbulence. We consider the stochastic soliton lattice on a special band-gap scaling of the spectral surface of genus $N$ so that the integrated density of states remains finite as $N \to \infty$ (thermodynamic type limit). We prove existence of the limiting stationary ergodic process and associate it with the soliton turbulence. The phase space of the soliton turbulence is a one-dimensional space with the random Poisson measure. The zero density limit of the soliton turbulence coincides with the Frish - Lloyd potential of the quantum theory of disordered systems.