1D Self-Similar Fractals with Centro-Symmetric Jacobians: Asymptotics and Modular Data

We establish the asymptotics of growing one-dimensional self-similar fractal graphs that are networks with multiple weighted edges between nodes. The asymptotics is described in terms of quantum central limit theorems for algebraic probability spaces in a pure state. We endow an additional structure upon the repeating units of centro-symmetric Jacobians in the adjacency of a linear graph, creating a self-similar fractal. The family of fractals induced by centrosymmetric Jacobians is formulated as orthogonal polynomials that satisfy three-term recurrence relations and support such limits. The construction proceeds with interacting Fock spaces and \(T\)-algebras endowed with a quantum probability space, corresponding to the Jacobi coefficients of the recurrence relations. When some elements of the centrosymmetric matrix are constrained in a specific way, we obtain, as the same Jacobian structure is repeated, quantum central limits. The generic formulation of Leonard pairs that form bases of conformal blocks and probabilistic Laplacians used in physics provide a choice of centrosymmetric Jacobians widening the applicability of the result. We establish that the T-algebras of these 1D fractals, as they form a special class of distance-regular graphs, are thin, and the induced association schemes are self-duals that lead to anyonic systems with modular invariance.