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Fast computation and convergence analysis of the infinite-product representation of the Schottky--Klein prime function

Shuntaro Yamamoto, Hiroyuki Miyoshi

arXiv:2609.00785v1math.NA

TL;DR

The paper addresses the cost and poor geometric ordering of word-length truncation for the Schottky–Klein prime function. It uses a cross-ratio potential with uniform increment bounds to enumerate retained factors efficiently, obtaining exponential error decay and improved performance in challenging configurations.

  • Problem

    Word-length truncation has exponential cost and can converge slowly in closely packed geometries, while convergence of the infinite product is not known for every multiply connected circular domain.

  • Method

    The paper ranks product factors with the cross-ratio potential Uζ,α(θ) and uses uniform increment bounds to prune a depth-first enumeration of factors below a cutoff.

  • Results

    The relative error decays exponentially in Umax at rate 1 −δ, and experiments show superior efficiency and global accuracy over level truncation, especially for closely packed configurations.

  • Takeaways & Limitations

    Potential truncation provides rigorous a priori error control whose accuracy can be systematically improved by increasing the cutoff.

  • Takeaways & Limitations

    The no-negative-cycle condition is sufficient for convergence but does not settle convergence of the infinite product in full generality.

Abstract

from arXiv · show

The Schottky--Klein prime function is a standard tool for boundary-value problems on multiply connected circular domains. Because this function is represented as an infinite product over a Schottky group, numerical evaluation requires truncation to finitely many factors. The standard word-length truncation grows exponentially in cost and becomes inefficient when the boundary circles nearly touch one another or the unit circle. To address this difficulty, we assign to each group element a cross-ratio potential measuring the size of its contribution, and retain only terms below a prescribed threshold. We establish uniform closed-form bounds on the change in this potential when prepending Schottky-group generators, and from these bounds we derive an efficient enumeration algorithm. The resulting relative error decays exponentially with the threshold at a rate determined by the Hausdorff dimension of the limit set of the Schottky group. Numerical experiments demonstrate that the proposed formulation achieves substantial computational speedups over word-length truncation in challenging geometric configurations.

1. Introduction.

The paper targets the exponential cost and poor geometric ordering of classical word-length truncation for the Schottky–Klein prime function. It introduces potential-based ordering and efficient enumeration, with exponential error control and improved performance in difficult configurations.

  • The Schottky–Klein prime function represents solutions for boundary-value problems on multiply connected circular domains.
  • The infinite product is not known to converge for every multiply connected circular domain, and general convergence theory remains open.Classical treatments impose geometric restrictions, while separated-hole convergence is described as heuristic without a quantitative rate.
  • Level truncation costs exponentially because the number of level-ℓ elements is M(2M −1)^(ℓ−1).Its word-length ordering also performs poorly when holes approach one another or the unit circle because factor decay need not track word length.
  • The proposed potential truncation ranks factors by cross-ratio size rather than word length and retains factors with potential at most Umax.Selection depends on the reference points and hole configuration, so it can cut across tree levels.
  • Uniform potential-increment bounds enable efficient enumeration and subtree pruning, while the relative error decays exponentially in Umax at rate 1 −δ when δ < 1.Here δ is the Hausdorff dimension of the Schottky-group limit set, and the same counting estimate controls enumeration complexity.
  • Numerical experiments report superior efficiency and global accuracy over level truncation, especially for closely packed holes, with rigorous a priori error control compared with SKPrime.

2. Preliminaries.

The preliminaries define circular multiply connected domains, their Schottky-group tree, and the infinite-product representation of the Schottky–Klein prime function. They also introduce hyperbolic-distance counting results that motivate potential-based ordering.

  • 2.1. Multiply connected domains and the Schottky group.: The domain is the unit disk with M pairwise disjoint circular holes strictly inside it.Each hole Dm has center δm, radius qm, and boundary Cm.
  • 2.1. Multiply connected domains and the Schottky group.: Basic Möbius transformations are formed by composing reflection in the unit circle with reflection in each hole circle.These transformations map complementary disks according to the circular-domain construction.
  • 2.1. Multiply connected domains and the Schottky group.: The Schottky group is the free group generated by θ1,…,θM, with reduced words formed by excluding adjacent inverse indices.A reduced word’s level is its word length, and the rightmost factor acts first.
  • 2.2. The Schottky–Klein prime function.: The reduced-word tree has the identity at its root, and each nonroot vertex has 2M −1 children obtained by admissible left composition.The quotient set Θ′′ omits the identity and one representative from each inverse pair.
  • 2.2. The Schottky–Klein prime function.: The Schottky–Klein prime function is represented by an infinite product over Θ′′ whose factors are built from the inverse cross-ratio Rζ,α(θ).The product is evaluated numerically by replacing it with a finite product over selected group words.
  • 2.2. The Schottky–Klein prime function.: Level truncation retains all group elements through Lmax, requiring exponentially many factors as Lmax grows.
  • 2.3. Hyperbolic geometry.: Pollicott’s counting theorem gives Card{θ ∈Θ : |Rζ,α(θ) −1| ≥1/T} ∼ CT^δ, where δ is the Hausdorff dimension of the limit set.The associated factor-size asymptotic motivates ordering by hyperbolic distance rather than word length.
  • 2.4. Potential-based ordering.: The proposed scheme ranks factors using a scalar cross-ratio potential, with large potential corresponding to negligible product factors.

3. Potential and proposed truncation.

The paper defines a cross-ratio potential that measures factor size and uses it to select a sparse truncation set rather than a complete word-length level. Uniform increment bounds support enumeration and error analysis.

  • 3.1. Potential associated with a group element.: The potential is Uζ,α(θ) := −log|Rζ,α(θ) −1|, where Rζ,α(θ) is the inverse cross-ratio of ζ, α, θ(ζ), and θ(α).
  • 3.1. Potential associated with a group element.: The exact factor-size relation is |Rζ,α(θ) −1| = e^(−Uζ,α(θ)).Thus larger potential means a smaller multiplicative perturbation.
  • 3.2. Potential truncation.: Potential truncation retains group elements satisfying Uζ,α(θ) ≤ Umax rather than retaining every word through a fixed level.The retained vertices form a sparse selection that can cross multiple tree levels.
  • 3.2. Potential truncation.: Potential-based selection depends on the reference pair and hole configuration, so words of any level may be included or excluded.The resulting enumeration problem is to find all retained elements without exploring every deep tree vertex and to quantify error and cost versus Umax.

4. Efficient enumeration algorithm.

The algorithm enumerates potential-truncated Schottky-group factors with a depth-first search, using finite-graph lower bounds to prune subtrees whose descendants cannot meet the cutoff. It precomputes these bounds, assembles the partial product, and aborts when a negative cycle makes pruning ineffective.

  • Enumeration strategy: A depth-first search enumerates the truncation set and partial product while avoiding an unbounded direct scan of the infinite group tree.At each vertex, the algorithm decides whether its subtree may contain elements below the cutoff and discards it otherwise.
  • Finite-graph bounds: One-step lower bounds on potential increments depend only on consecutive generators, enabling path bounds on the tree and walk weights on a finite directed graph.The graph has 2M vertices and 2M(2M −1) edges, excluding only immediate generator cancellation.
  • Pruning rule: The minimum walk weight Wmin supplies a uniform descendant bound, U(ψℓ′) ≥ U(ψℓ) + Wmin, used to certify subtree pruning.If U(ψℓ) > Umax −Wmin, every descendant exceeds Umax and the subtree can be discarded.
  • Limitation: A negative cycle makes Wmin = −∞, so the pruning rule becomes vacuous and the algorithm aborts rather than safely discarding subtrees.This obstruction does not by itself certify divergence of the full infinite product.
  • Implementation: The implementation precomputes edge weights and Wmin, then traverses the group tree depth-first while evaluating exact potentials only for words surviving pruning.The partial product is updated for retained factors, with inverse representatives handled to match the quotient truncation set.

5. A uniform bound on the potential increment.

The paper derives uniform closed-form bounds for potential increments by reducing their dependence on a word's tail to boundary optimizations. These bounds depend only on adjacent generators and reference points, so they can be precomputed as graph edge weights.

  • Increment reduction: The potential increment under prepending a generator is rewritten using image points and cross-ratio invariance, leaving dependence on the child word through those image points.After fixing the reference points and prepended generator, the transformed expression depends only on the image pair.
  • Domain restriction: Reduced-word geometry confines the image points to the generator's associated domain, independently of the word's tail.Thus, bounding increments for all words with the same leftmost generator reduces to optimization over that domain.
  • Boundary optimization: Harmonicity and the maximum–minimum principles reduce the four-dimensional optimization to two one-dimensional optimizations over a boundary circle.The resulting extrema depend only on θj, θk, ζ, and α.
  • Uniform bound: Theorem 5.1 gives lower and upper increment bounds for every reduced word, with constants determined entirely by the adjacent generators and reference points.Telescoping these bounds along prefix paths produces the graph edge weights used by the enumeration algorithm.
  • Precomputation: All 2M(2M −1) edge-weight constants can be precomputed in O(M 2) time and memory before enumeration begins.The bounds depend on hole geometry and reference points, not on Umax, the word, or other runtime quantities.

6. Error bound and algorithmic complexity.

The potential-counting asymptotic yields exponential relative-error decay, while the pruning algorithm enumerates all retained factors exactly once with controlled complexity. The analysis links the decay rate to the Hausdorff dimension and tests the approach on increasingly difficult circular-hole configurations.

  • 6.1. Counting estimate: N(u) ∼ Ce^δu as u →∞, where δ is the Hausdorff dimension of the Schottky-group limit set.This counting estimate is the sole counting input for both the error and complexity analyses.
  • 6.2. Asymptotic relative-error bound: Theorem 6.2 derives the relative-error bound by converting the omitted tail into a Stieltjes integral against the counting function N.A product-bound inequality reduces the analysis to estimating the tail sum, followed by integration by parts.
  • 6.2. Asymptotic relative-error bound: The relative error decays exponentially in cutoff Umax at rate 1 −δ.Larger δ corresponds to more crowded hole configurations and therefore slower guaranteed decay.
  • 6.3. Complexity of the enumeration algorithm: When G has no negative cycle, Algorithm 4.1 returns every factor with potential at most Umax exactly once.The correctness proof uses the increment bound to show that no required ancestor is pruned.
  • 6.3. Complexity of the enumeration algorithm: The algorithm visits only words with potential at most Umax − Wmin, with nonnegative edge weights giving Wmin = 0.For negative weights, the additional factor e^−δWmin depends on G but not on Umax.
  • Numerical validation: The numerical study tests the bounds and enumeration algorithm on three circular-hole configurations of increasing geometric difficulty.The cases compare potential truncation with classical level truncation, including closely packed configurations.

7. Numerical experiments.

Numerical experiments show that potential truncation is correct, globally accurate, and especially efficient in closely packed configurations, while its error follows the predicted Hausdorff-dimension-dependent decay.

  • Experimental setup: Cases 1–3 cover well-separated, closely packed, and three-hole closely packed configurations used to compare the truncation schemes.The experiments record relative error and wall-clock time while varying Lmax or Umax.
  • Uniform increment bounds: Theorem 5.1 bounds potential increments using only the leftmost generator pair, and Case 2 samples lie within those bounds.The tested pairs include narrow positive, wider positive, and ranges containing negative increments.
  • Pruning graph: Case 1 has nonnegative pruning weights, whereas Cases 2 and 3 contain negative edges but finite Wmin and no negative cycles.These conditions satisfy the hypothesis used to establish Algorithm 4.1’s correctness.
  • Runtime comparison: Cases 2 and 3 reach the same relative error substantially faster with potential truncation, while the methods are nearly comparable in Case 1.The gain is clearest when word level poorly tracks actual term size; the visited set may also include factors beyond Umax, improving accuracy at fixed cutoff.
  • Error convergence: The predicted decay rates are 1 −δ ≈0.69, 0.54, and 0.23 for Cases 1–3, respectively, making Case 3 the slowest-converging case.Observed errors track the asymptotic bound in Cases 1 and 2, while the bound is conservative for Case 3 at moderate Umax.
  • Comparison with SKPrime: Potential truncation continues below SKPrime’s ∼10−4 error in a near-unit-circle configuration, whereas SKPrime is already ∼10−12 in Case 1.The potential method approaches the Case 1 regime as Umax grows and does so with explicit wall-clock measurements.
  • Spatial accuracy: Across a 100 × 100 grid, potential truncation is more accurate throughout Dζ, with the largest errors near hole boundaries and the unit circle.This global comparison uses a distinct two-hole geometry and fixed α.

8. Conclusions and future work.

The paper concludes that potential truncation combines rigorous error control with efficient enumeration and improved accuracy, especially for closely packed configurations, while identifying sharper bounds and broader convergence theory as future work.

  • Conclusions: Experiments confirm no negative cycles, tracking of the asymptotic error bound, and superior efficiency and global accuracy over level truncation in closely packed configurations.Comparison with SKPrime adds rigorous a priori error control through increasing Umax.
  • Future work: Sharper gaps ∆ub j,k would tighten graph edge weights and improve pruning efficiency.The paper also identifies reliable handling of negative cycles as a practical algorithmic challenge.
  • Future work: A complete convergence theory beyond classical geometric restrictions and the paper’s sufficient condition remains an open theoretical goal.Further directions include understanding δ’s dependence on hole geometry and extending the framework to secondary prime functions and other mapping problems.

SUPPLEMENTARY MATERIALS: FAST COMPUTATION AND CONVERGENCE ANALYSIS OF THE INFINITE-PRODUCT REPRESENTATION OF THE SCHOTTKY–KLEIN PRIME FUNCTION∗

The supplementary materials provide detailed derivations for the cross-ratio decomposition, Möbius-image geometry, asymptotic error analysis, and Bellman–Ford computation of pruning weights.

  • Cross-ratio decomposition: The supplement derives the cross-ratio decomposition used to separate ζ- and α-dependent terms in the potential analysis.The derivation applies Möbius identities and fixed-point separation.
  • Pruning computation: The Bellman–Ford supplement computes the minimum walk weight Wmin and detects negative cycles in the finite pruning graph.Algorithm SM1.1 takes lower-bound edge weights as input and returns Wmin or a NegativeCycle flag.
  • Circle-image geometry: A Möbius-image lemma computes the center and radius of the image of a circle, supporting extrema of the potential increment.Pole inversion and the Apollonius ratio provide the geometric formulas.
  • Error analysis: The asymptotic error proof uses the counting function N(u) and the condition δ < 1 to derive the exponential tail behavior.Integration by parts combines factor counting with the e−u decay of individual corrections.
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