Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices

Technical Summary: Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices

Problem Statement
The paper investigates Bernoulli percolation on hierarchical lattices, specifically addressing the question of whether geometric and dimensional data alone can determine the universality class of critical exponents. While statistical physics on Euclidean lattices often relies on dimension-based universality, previous work by Hauser and Saxena (1986) and Halberstam and Hutchcroft suggested that on hierarchical lattices, the percolation exponent $\nu_p$ and Fisher exponent $\tau$ depend strongly on lattice topology, even when intrinsic dimensions and connectivities are identical. This raises the fundamental problem: can a finite family of "substitution-multiplicative dimensions" (such as Hausdorff dimension) completely classify the critical-exponent universality classes of these systems? Furthermore, does the existence of identical critical exponents imply that the underlying discrete renormalisation scales are commensurate (i.e., related by integer powers)?

Methodology
The paper formalizes the study of these systems using Edge Iterated Graph Systems (EIGS).

1. Framework: They define a "classical EIGS hierarchical lattice" as a deterministic system generated from a single edge by a finite, connected, simple, and terminal-symmetric rule graph $R$. This generalizes the diamond hierarchical lattice (DHL).
2. Random EIGS Representation: Building on the work of Hambly and Kumagai (2010) and extending recent results by Alves et al., the author proves that critical Bernoulli bond percolation, conditioned on terminal connection, induces a random EIGS. This representation allows the critical cluster to be analyzed as a graph-directed random recursive fractal.
3. Spectral Analysis: The core analytical tool is the mass matrix $M_C(p)$, a $3 \times 3$ matrix describing the expected number of "live" child edges of different states (connected, disconnected-but-active, and single-terminal active) under conditional percolation laws. The author analyzes the spectral properties of this matrix at the critical probability $p_c$.
4. Exponent Definition: The study focuses on eight critical exponents ($\beta, \nu, \delta, \eta_{av}, \gamma, \Delta, \alpha, \rho$). The author establishes that four of these ($\beta, \nu, \delta, \eta_{av}$) exist for every classical rule and define the critical-exponent universality class based on the agreement of these four values.
5. Number-Theoretic Tools: To address the commensurability of scales, the paper employs the Six Exponentials Theorem from transcendental number theory, linking the algebraic nature of growth multipliers to the rationality of scale ratios.

Key Contributions and Results

• Characterization of Universality Classes:
  The paper proves that two classical EIGS hierarchical lattices belong to the same critical-exponent universality class if and only if three specific dimensions agree:

  1. Ambient Dimension ($\dim_H(\Xi)$): The Hausdorff dimension of the lattice itself.
  2. Mass Dimension ($\dim_M(C)$): Derived from the spectral radius $\rho(M_C)$ of the critical mass matrix.
  3. Pivotal Dimension ($\dim_P(C)$): Derived from the derivative of the reliability function $\phi'_R(p_c)$.

  The author provides explicit formulas relating the four defining exponents to these dimensions:

  • $\nu = 1 / \dim_P(C)$
  • $\beta = (\dim_H(\Xi) - \dim_M(C)) / \dim_P(C)$
  • $\delta = \dim_M(C) / (\dim_H(\Xi) - \dim_M(C))$
  • $\eta_{av} = 2 + \dim_H(\Xi) - 2\dim_M(C)$

• Failure of Dimension-Based Classification:
  The author provides a rigorous proof that no finite family of substitution-multiplicative dimensions can determine the universality class. They demonstrate that the critical mass dimension $\dim_M(C)$ retains information about the order of substitutions (non-commutative structure) that multiplicative dimensions (which depend only on the set of rules, not their sequence) cannot detect.

  • Example: The paper constructs two distinct lattices, "Tie" and "Gem," which possess different reliability functions, critical probabilities ($p_c$), and mass matrices, yet share identical values for $\dim_H(\Xi)$, $\dim_M(C)$, and $\dim_P(C)$. Conversely, rearranging substitutions can alter $\dim_M(C)$ while preserving multiplicative dimensions, proving that multiplicative dimensions are insufficient for classification.

• Disproving Scale Commensurability:
  The paper addresses the conjecture that identical critical exponents require commensurate discrete renormalisation scales (i.e., $\log \ell_1 / \log \ell_2 \in \mathbb{Q}$).

  • Using the Six Exponentials Theorem, the author shows that if certain irreducibility conditions are met, scales within a universality class must be commensurate.
  • However, they construct a counterexample (Theorem 5.6) showing that a single universality class can contain infinitely many lattices with pairwise incommensurate scales. Specifically, they construct a family of lattices with scale $(19 \cdot 661^k)^{100}$ sharing the same critical dimensions, yet no finite grouping of substitution steps makes their scales compatible.
  • This result disproves the necessity of a common discrete coarse-graining scale for universality.

Significance
The paper provides a rigorous mathematical resolution to questions raised by Hauser and Saxena regarding the limits of dimension-based universality on hierarchical lattices. By establishing that the critical mass dimension depends on the non-commutative structure of the substitution rules, the author demonstrates that geometric data (like Hausdorff dimension) is insufficient to classify percolation universality. Furthermore, the construction of universality classes containing incommensurate scales challenges the physical intuition that universality requires a shared discrete scaling symmetry. The work is supported by formal verification in Lean 4, ensuring the logical soundness of the spectral and number-theoretic arguments.