Combinatorics of paths on super-trees with applications in statistical physics and field theory

In the talk we consider the spectral properties of transfer-matrices on "super-trees" (trees whose vertex degree varies with the distance from the root point). At small "branching rates", the path combinatorics on super-trees coincides with the statistics of one-dimensional Dyck paths weighted with the area under the trajectory. We discuss the relationship of statistics on super-trees with the Kardar-Parisi-Zhang statistics, the one-dimensional Anderson localization, and the random matrix theory in the Edelman-Dumitru representation. Trees with variable branching also encode the structure of some Fock space. Random walk on super-trees is a specific example of statistical model related to CFT, on the one hand, and ODE spectral properties, known as ODE/IS correspondence, on the other hand.