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1D self-similar fractals with centro-symmetric Jacobians: asymptotics and modular data

Radhakrishnan Balu

Abstract

We establish asymptotics of growing one dimensional self-similar fractal graphs, they are networks that allow multiple weighted edges between nodes, in terms of quantum central limit theorems for algebraic probability spaces in pure state. An additional structure is endowed with 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 centro-symmetric Jacobians formulated as orthogonal polynomials that satisfy three term recurrence relations support such limits. The construction proceeds with the interacting fock spaces, T-algebras endowed with a quantum probability space, corresponding to the Jacobi coefficients of the recurrence relations and when some elements of the centro-symmetric matrix are constrained in a specific way we obtain, as the same Jacobian structure is repeated, the central limits. The generic formulation of Leonard pairs that form bases of conformal blocks and probablistic laplacians used in physics provide choice of centro-symmetric 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 anyon systems with modular invariance.

1D self-similar fractals with centro-symmetric Jacobians: asymptotics and modular data

Abstract

We establish asymptotics of growing one dimensional self-similar fractal graphs, they are networks that allow multiple weighted edges between nodes, in terms of quantum central limit theorems for algebraic probability spaces in pure state. An additional structure is endowed with 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 centro-symmetric Jacobians formulated as orthogonal polynomials that satisfy three term recurrence relations support such limits. The construction proceeds with the interacting fock spaces, T-algebras endowed with a quantum probability space, corresponding to the Jacobi coefficients of the recurrence relations and when some elements of the centro-symmetric matrix are constrained in a specific way we obtain, as the same Jacobian structure is repeated, the central limits. The generic formulation of Leonard pairs that form bases of conformal blocks and probablistic laplacians used in physics provide choice of centro-symmetric 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 anyon systems with modular invariance.
Paper Structure (8 sections, 2 theorems, 29 equations, 3 figures)

This paper contains 8 sections, 2 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Association Schemes and T-algebras
  3. Interacting Fock spaces
  4. Self-similar $p$ Laplacians on the half-integer lattice
  5. Modular Data of Rational Conformal Field Theory
  6. Summary and Conclusions
  7. Acknowledgement
  8. Declarations

Key Result

Theorem 1

Let $\mathcal{G}^\nu = (V^\nu, E^\nu)$ be a growing distance-regular graph with an adjacency matrix $A_\nu$. Let us denote the degree as $\kappa(\nu)$ and assume the following conditions in terms of intersection numbers hold: Let $\Gamma_{\omega_n} = (\mathscr{G}, \{\Phi_n\}, B^+, B^-)$ be an interacting Fock space associated with $\{\omega_n\}$ and $B^o = \alpha_{N + 1}$ be the diagonal operator

Figures (3)

  • Figure 1: Spiderweb Diagram - an example of a stratified graph on which an IFS can be defined.
  • Figure 2: Cantor Set that is a self-similar graph
  • Figure 3: Construction of self-similar graph from repeating units. (Top) A copy of the basic building block. The deleted edges correspond to the edges that are replaced when applying the substitution rule. (Bottom) The fractal graph is constructed by inserting the three copies of the building block in outer graph which is the 1D lattice While the vertices are labeled by the sequentially, the labeling of the edges represents the transition probabilities (off-diagonal entries in the self-similar Laplacian).

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3
  • Example 2
  • Example 3
  • Example 4
  • Theorem 1
  • proof
  • Example 5
  • ...and 3 more