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The problem of Sobolev norm growth—namely, the search for mechanisms of energy transfer from low to arbitrarily high modes—has Ìýbeen intensively studied in Hamiltonian systems with an organized, Ìýdeterministic structure of mode interactions, such as the cubic Ìýnonlinear Schrödinger equation. In this talk, I will present an Ìýextension of this problem to Hamiltonian systems dominated by random nonlinear interactions. I will introduce analytic solutions describing three types of energy cascades that lead either to unbounded growth or to finite-time blow-up of Sobolev norms.I will then present numerical simulations demonstrating the emergence Ìýof these dynamics from incoherent initial conditions. Taken together, these results demonstrate that coherent (phase-organized) energy Ìýcascades can be robust mechanisms of energy transfer even in systems with a random structure.