Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes

I. INTRODUCTION

• Simplicial and cell complexes generalize networks by incorporating higher-dimensional simplexes and polytopes that encode geometry and many-body interactions.They motivate studies of percolation and other dynamical processes on structures beyond node-link skeletons.Source: §I · p. 1
• The work extends real-space renormalization-group analysis of link percolation to random pseudo-fractal simplicial and cell complexes.The model iteratively glues m-polygons to links, with polygon sizes sampled from a distribution q_m.Source: §I · p. 1, §II · p. 2
• For deterministic pseudo-fractal simplicial complexes formed by m-polygons, the percolation transition occurs at p_c = 0.The paper distinguishes this behavior from deterministic cell complexes and random pseudo-fractal simplicial complexes.Source: §I · p. 1
• The order parameter is exponentially suppressed as p exp(-α/p^(m-2)), with the exponent reducing to 1/p for m = 3.The derivation also captures the prefactor p, which was not discussed in earlier work cited by the authors.Source: §I · p. 2

II. RANDOM PSEUDO-FRACTAL SIMPLICIAL AND CELL COMPLEXES

• Random pseudo-fractal complexes are constructed by drawing each newly attached polygon size m ≥ 3 from q_m, while delta distributions recover deterministic cases.The cases q_m = δ_m,3 and q_m′ = δ_m′,m produce the simplicial and single-polygon-size deterministic complexes, respectively.Source: §II · p. 2

A. Link probability

• The RG recursion defines T_n as the probability that the two initial nodes are connected at iteration n, starting from T_0 = p.For random complexes, the recursion averages over the polygon-size distribution through Q(T).Source: A. Link probability · p. 2
• For any p > 0, the recursive equations flow to the fixed point T* = 1, whereas at p = 0 the steady state is T* = 0.Thus every positive link-retention probability yields an infinite-network percolation probability of one.Source: A. Link probability · pp. 2–3
• Figure 1 contrasts deterministic m = 3 and m = 4 constructions with a random complex having q_3 = q_4 = 1/2 at iteration n = 3.Colors identify structures introduced at iterations n = 0, 1, 2, and 3.Source: Figure 1 · p. 3
• The resulting upper percolation threshold is p_c = 0.The threshold follows from the distinct fixed-point behavior at positive p and at p = 0.Source: A. Link probability · pp. 2–3

• The generating function T̂_n(x) tracks component sizes connected to both initial nodes, while Ŝ_n(x,y) tracks the two components connected exclusively to one initial node each.Source: B. Generating function · p. 3
• The diagrams distinguish configurations that guarantee connectivity from those that leave the initial nodes disconnected, determining which terms enter T̂_n(x) and Ŝ_n(x,y).Only the connectivity-guaranteeing polygon diagram contributes exclusively to T̂_n(x); the remaining diagrams can contribute to both functions.Source: B. Generating function · p. 4
• Derivatives of T̂_n(x) at x=1 provide the expected size M_n of the giant component connected to the two initial nodes.Source: B. Generating function · pp. 5, 7
• The recursive equations for T̂_n(x) and Ŝ_n(x,y) depend on the full prior RG flow rather than only the immediately preceding iteration.Source: B. Generating function · p. 6
• Introducing the auxiliary function K_n(x) converts the full-flow relations into iterative equations between variables at successive generations.The transformed system uses initial conditions T̂_0(x)=p, Σ̂_0(x)=Ŝ_0(x)=1−p, and K̂_0(x)=1.Source: B. Generating function · p. 6

• The order parameter P∞ is the thermodynamic-limit fraction of nodes belonging to the giant component.Source: C. Order parameter · p. 7
• The expected giant-component size M_n is obtained by differentiating T̂_n(x), with the generating-function vector recast into a recursive derivative system.Source: C. Order parameter · p. 7
• The derivative recursion is governed by the Jacobian J_n evaluated along the RG flow, with initial condition V̇_0=0 because the initial nodes are excluded from the count.Source: C. Order parameter · p. 7
• For n>0 and T_n<1, the homogeneous derivative term dominates the non-homogeneous term, while the latter must be retained at n=0 because the homogeneous term vanishes.Source: C. Order parameter · p. 8
• The dominant eigenvalue λ_n and its left and right eigenvectors determine the recursive expression for M_{n+1} and support the subsequent critical analysis.Source: C. Order parameter · pp. 8–9

IV. RG FLOW

• The RG flow is reformulated with y_n=-ln(1-T_n), initial condition y_0=-ln(1-p), and an expansion near p_c=0.For small p, the leading evolution is obtained by expanding y_{n+1}-y_n near T_c=y_c=0.Source: §IV · p. 9
• y diverges at a finite iteration n=n_c, marking a discontinuous jump of the linking probability T_n to 1.The divergence prediction follows from integrating the continuous RG equation and is supported by the reported flow behavior.Source: §IV · p. 9
• For n≪n_c, numerical integration agrees very well with the asymptotic expression for the RG flow.The asymptotic form is derived for y≪1 and p≪1, while the numerical flow retains its structure beyond that limited expansion regime.Source: §IV · p. 9
• Figure 5 compares m=3 at p=10^-3 with m=4 at p=6×10^-3 and displays divergence of y_n near n_c.The plotted quantity is y_n=-ln(1-T_n), with T_n the percolation probability.Source: Figure 5 · p. 10

A. Critical region

• The analysis studies the order parameter near p_c=0 by first characterizing the expected number of giant-component nodes M_n.M_n is approximated using the derivation from the RG equations and the linking probability.Source: A. Critical region · p. 10
• The resulting giant-component estimate is organized through A_n and D_n, whose definitions determine the n-dependent scaling of M_n.The section introduces these factors before applying the near-critical approximation.Source: A. Critical region · p. 10
• Near p_c, D_n is approximately independent of n and close to one because the right and left eigenvectors change slowly.This approximation allows the expression for M_n to be rewritten in a simpler asymptotic form.Source: A. Critical region · p. 10

B. Critical expansions

• For y_n≪1, the leading Jacobian eigenvalue is expanded through H(T_n) and Q′(T_n), with the smallest polygon size satisfying q_m>0 setting the leading term.The expansion is performed after writing T_n=1-e^-y_n.Source: B. Critical expansions · p. 10
• For yn ≪ 1, the inhomogeneous term ∂Fn/∂x can be expanded; at n = 0, the homogeneous term vanishes due to the trivial initial condition V̇0 = 0, making the inhomogeneous term leading.Source: B. Critical expansions · p. 10
• For p ≪ 1, ⟨v1|V̇1⟩ scales linearly with p, and the leading term of v1 is given by an explicit expansion in p.Source: B. Critical expansions · p. 10
• The large-n giant-component fraction M_n is obtained by combining the scaling relations for the RG eigenvalue and eigenvector factors.These relations are substituted into the expression for the expected number of nodes in the giant component.Source: B. Critical expansions · p. 11

Asymptotic order parameter

• Near p_c = 0, the order parameter P_∞ is obtained by approximating the expected giant-component size M_n in the critical region 0 < p ≪ 1.The derivation uses the asymptotic behavior of M_n and the RG scaling relations.Source: C. Critical scaling of the order… · pp. 11–12

RG flow

• The RG analysis identifies a discontinuity in 1 − ψ_n at the crossover n_c, where T_n reaches 1 and λ_n reaches ⟨m⟩.The scaling function y(n) diverges at n = n_c, producing the stated jump in the RG-flow quantities.Source: C. Critical scaling of the order… · p. 11

Numerical validation

• The scaling prediction is validated by numerically integrating the RG equation and applying finite-size scaling to P_n.P_n is defined as the fraction of nodes in the giant component of a complex evolved to iteration n.Source: C. Critical scaling of the order… · p. 12

Finite-size scaling

• For n > n_c, P_n follows the asymptotic scaling predicted for the infinite pseudo-fractal complex, whereas for n < n_c it saturates to a constant.The crossover n_c depends on p, and the finite-size behavior is shown numerically in Fig. 6.Source: Figure 6 · p. 11, C. Critical scaling of the order… · p. 12