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Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks

Ali Rayat, Gia-Wei Chern

arXiv:2610.18742v1cond-mat.stat-mechcond-mat.str-elcs.LGhep-lat

TL;DR

Gauge-equivariant message passing generates long, ordered gauge-link paths, but receptive-field size alone does not describe their geometry. The paper analyzes the recursively generated loop ensemble independently of training and finds that contour length grows much faster than spatial extent, with extra steps largely revisiting nearby regions.

  • Problem

    The receptive field bounds spatial reach but does not describe the complexity of paths composed within that region, motivating a statistical characterization of the generated loop ensemble.

    Source: §I · p. 1

  • Method

    The paper analyzes closed walks generated by recursive node- and edge-feature operations independently of network training, using length and spatial-geometry measures.

    Source: §I · p. 2, §V · p. 10

  • Results

    Over the depths studied, contour length grows much faster than spatial extent, with increasingly long paths accumulating repeated traversals in relatively compact regions.

    Source: §V · p. 10

  • Takeaways & Limitations

    The hierarchy distinguishes spatial reach from representational complexity within that reach, while loop expansions connect it to high-order dependence on nearby gauge fields.

    Source: §V · p. 10

    The locality assumption concerns dependence on the surrounding gauge configuration; assigning an observable to a site or link alone does not establish that locality.

    Source: §V · p. 10

Abstract

from arXiv · show

Gauge-equivariant graph neural networks generate ordered products of gauge links through local message passing, but the geometry of these internal structures is not captured by the receptive field alone. We investigate the statistical ensemble of closed lattice walks generated by recursive node- and edge-feature operations, independently of network training. Analytical and numerical results reveal a separation between rapid growth of contour length and much slower spatial expansion: deeper compositions generate increasingly long paths through repeated traversal of relatively compact regions. Loop expansions of fermionic observables provide a physical interpretation of this hierarchy, showing how spatial locality remains compatible with high-order dependence on nearby gauge fields. These findings identify complementary roles of network depth in extending spatial reach and enriching the representations constructed within that reach, providing a geometrical perspective on learning lattice gauge observables.

I. INTRODUCTION

Gauge-equivariant learning preserves local gauge structure while message passing composes gauge-link products into paths whose complexity is not captured by receptive-field size alone. This work characterizes those loop structures independently of training, finding that contour length grows much faster than spatial extent.

I. INTRODUCTION

  • Gauge-equivariant message passing composes gauge-link products into paths and loops, whose contour complexity is not described by the receptive field alone.The receptive field bounds spatial reach, while paths may retrace links, revisit sites, and include nested excursions.Source: §I · p. 1
  • The study characterizes the loop ensemble generated by recursive feature operations independently of network training.It examines length distributions, spatial extent, shape, site exploration, and length diversity.Source: §I · p. 2
  • Contour length grows exponentially with depth while spatial extent grows much more slowly over the studied depths.Longer paths accumulate traversals within relatively compact regions, while retaining substantial geometrical diversity.Source: §I · p. 2
  • Loop expansions of fermionic observables connect the hierarchy to the distinction between hopping order and physical locality.This provides a way to interpret high-order dependence on nearby gauge fields within local message passing.Source: §I · p. 2

II. GAUGE-EQUIVARIANT MESSAGE PASSING

Gauge-equivariant message passing builds path-valued node and edge features by composing local gauge links and plaquette holonomies. Recursive products can generate closed walks with repeated traversals and varied geometries, which the paper samples to characterize their lengths and spatial organization.

II. GAUGE-EQUIVARIANT MESSAGE PASSING

  • Node features begin with plaquette holonomies and edge features with gauge links; both retain their matrix structure for later composition.The untraced plaquettes can be transported and multiplied before gauge-invariant quantities are formed.Source: §II · p. 2
  • Node updates carry features forward, multiply local features, or transport neighboring-node features into the reference site's gauge frame.Transport uses edge features so messages from neighboring sites can be combined without violating gauge covariance.Source: §II · pp. 2–3
  • Edge updates dress open paths with excursions through neighboring loops while preserving the edge's endpoint transformation law.The resulting longer edge features can then contribute to new closed walks in later node updates.Source: §II · pp. 3–4
  • Node features represent closed walks based at a site, while edge features represent open walks connecting neighboring sites.Products concatenate paths when their endpoints match, and the feature transformation laws are preserved through updates.Source: §II · pp. 2–3
  • Recursive compositions generate walks with retraced links, revisited sites, and nested excursions, so contour length need not track spatial extent.The same local rules can add traversals without reaching new sites or introduce excursions into already explored regions.Source: §II · p. 4
  • Some shared-link traversals cancel in the short-loop schematics, but the statistical analysis counts every traversal in the original generated path.Multiplying transported and elementary loops can produce either cancellation or a retained retraced segment depending on their placement and ordering.Source: §II · pp. 4–5
  • The sampling procedure chooses message operations uniformly and recursively generates every factor until reaching the input features.Each realization is an AND/OR derivation subtree; path shapes are not prescribed and learned weights are not used.Source: §II · pp. 5–6
  • At the same recursion depth, sampled paths vary substantially in length and geometry, often concentrating many steps within a small region.Carry-forward choices retain existing structures, while product choices combine recursively generated factors; all traversals are retained.Source: Figure 3 · p. 5, §II · p. 6

III. STATISTICAL GEOMETRY OF GENERATED LOOPS

The generated loops become much longer with depth while expanding spatially much more slowly, with revisitation and dimensionality shaping how they explore the lattice. Length diversity also broadens, while the measured geometry describes the specified recursion rather than trained-network weights or an independent operator basis.

Contour-length statistics

  • Mean contour length grows asymptotically as (14/5)2^n, and its distribution is dimension-independent under the stated recursion.The permitted maximum grows as 3n, faster than the mean; the observed maximum in finite samples can lie well below that bound.Source: §III · pp. 6–7
  • Because each finite-depth length distribution has bounded support, its observed tail shape does not establish an asymptotic tail law.The generating-function recursion permits sampling-free evaluation of Pn(ℓ).Source: §III · p. 7

Spatial extent

  • At n=6, ⟨Rg⟩ is about 1.25 lattice spacings after rising from about 0.9 at n=2, while mean contour length grows by approximately a factor of 16.The curves for d=2, 3, and 4 nearly coincide; maximum excursion reaches about 2.4–2.6 lattice spacings at n=6.Source: §III · p. 7, Figure 5 · p. 7

Shape

  • Effective dimension remains near 1.8 in d=2 but increases with depth in d=3 and 4, indicating more balanced directional spread without comparable radius growth.Deff measures shape anisotropy, not fractal dimension or spatial filling.Source: §III · p. 7, Figure 5 · p. 7

Site exploration and revisitation

  • At n=6, loops visit roughly 17, 28, and 37 distinct sites in d=2, 3, and 4, respectively, while revisitation reaches about 0.74 even in d=4.Node products concatenate loops at a common site, and transported node messages follow an edge path and its exact reverse, causing revisitation.Source: §III · p. 8

Length diversity

  • Length-distribution entropy grows approximately linearly with depth, while the effective number of populated length sectors grows approximately exponentially and does not count distinct loop shapes.The entropy curves nearly coincide across dimensions, consistent with the dimension-independent length distribution.Source: §III · p. 8

Scope and interpretation

  • The sampled-depth separation between exponential mean contour growth and slower spatial expansion does not establish growth of an independent operator basis or trained-network weights.Backtracking cancellations and trace identities can relate paths of different lengths, and correlations reveal geometric variation among loops of comparable length.Source: §III · p. 8

IV. IMPLICATIONS FOR LEARNING LATTICE GAUGE OBSERVABLES

Fermionic loop expansions illustrate how long contour lengths can coexist with spatial locality: repeated propagation may remain within a finite neighborhood. The physical importance of those paths depends on the observable’s weights and regime, not geometry alone.

Loop expansions of observables

  • At high temperature, higher hopping orders are suppressed; length-two walks are gauge independent, and the first gauge-dependent hypercubic-lattice contributions arise from length-four plaquettes.The series converges when β∥h∥ < π.Source: §IV · p. 9
  • At zero temperature and exact half filling, a Chebyshev representation groups even-length closed-walk sums, whose coefficients decay as m^-2 and whose grouping is essential for convergence.The coefficient decay reflects the cusp in the energy function; the representation organizes polynomial combinations of loop lengths rather than individual length sectors.Source: §IV · p. 9

Scope and limitations

  • A site decomposition alone does not establish local gauge-field dependence; at zero temperature, gapless metals have algebraic density-matrix tails, and fixed-filling chemical-potential dependence adds a global dependence.The site energy expectation generally depends on the entire gauge configuration.Source: §IV · pp. 9–10

Locality

  • For bounded finite-range hopping at finite temperature, local expectation values admit exponentially accurate approximations within a graph-distance-R neighborhood at fixed chemical potential and temperature.Leaving that neighborhood and returning requires at least 2R + 2 steps, so local moments through degree 2R+1 are unchanged, while arbitrarily long walks can remain inside it.Source: §IV · p. 9

Implications for locality

  • Spatial locality and high-order gauge-field dependence are compatible, but loop weights determine which available paths contribute appreciably.Measured geometry describes possible propagation structures, not their physical importance; compactness alone does not enhance a walk’s contribution.Source: §IV · pp. 9–10

V. CONCLUSION AND OUTLOOK

Recursive gauge-equivariant message passing produces a hierarchy in which contour length grows much faster than typical spatial extent. This separates the depth needed for spatial coverage from the depth that enriches combinations of nearby gauge fields, with usefulness depending on the observable and regime.

V. CONCLUSION AND OUTLOOK

  • The mean contour length grows exponentially as ⟨ℓ⟩∼(14/5)2^n, while spatial extent grows much more slowly over the depths studied.The specified sampling rules make the length distribution independent of spatial dimension, although path geometry depends on dimension.Source: §V · p. 10
  • Longer paths accumulate increasing fractions of steps through revisits, while loops can distribute spatial variance more evenly without becoming substantially larger.Loops of comparable length also vary considerably in spatial extent and shape, so hopping order alone does not determine geometry.Source: §V · p. 10
  • For GNN design, maximum receptive-field reach and typical loop radius are distinct: distant-link observables still require sufficient spatial coverage.The typical loop radius describes visited-position distributions, not the full receptive field or the trained model's spatial sensitivity.Source: §V · p. 10
  • For local observables, additional depth can build high-order combinations of nearby links after the necessary spatial coverage is reached.Fermionic quantities can involve repeated propagation through the surrounding gauge field, unlike the plaquette dependence of the Wilson gauge-action density.Source: §V · p. 10
  • The benefit of added composition depends on the observable and physical regime: short loops may suffice in the high-temperature example, while many relevant hopping orders may favor deeper layers.Carry-forward messages retain access to shorter structures, so learned combinations need not emphasize the longest paths.Source: §V · p. 11
  • The recursion reaches mean formal hopping orders of size L at depth n = log2 L + O(1), but formal contour growth does not establish better prediction accuracy.The authors identify how training selects and combines structures, and whether depth helps through distant-field access or more elaborate combinations within existing support, as open questions.Source: §V · p. 11

Appendix A: Recursive generation of gauge loops

The appendix defines a stochastic construction that follows the message-passing products to generate closed lattice paths without trained weights. Recursive operation choices and required factors determine each path; generating functions describe the resulting contour-length distributions.

Appendix A: Recursive generation of gauge loops

  • The ensemble is generated by recursively choosing contributions to node and edge messages until reaching input links and plaquettes, without using learned channel weights.Each realization specifies an ordered product contributing to an output node feature, whose trace is a gauge-invariant loop observable.Source: Appendix A · p. 11
  • The construction is an AND/OR derivation tree: each operation selects an alternative, and its product requires all constituent factors to specify the complete loop.A single root-to-input branch traces only one factor's ancestry, not the complete loop.Source: Appendix A · p. 11, Figure 7 · p. 12
  • Node operations are chosen with equal probability among three options, while edge objects are chosen uniformly among four products; X3 uses the same sampled transporter in both directions.The X3 neighbor is selected uniformly from the 2d nearest neighbors.Source: Appendix A · p. 11
  • Input node objects are closed plaquettes and depth-zero edge objects are oriented links; recursive operations preserve closed node paths and specified edge endpoints.Generated paths may self-intersect, revisit sites, and retrace links, which remain in the ordered path rather than being canceled.Source: Appendix A · pp. 11–12
  • Paths are stored as ordered unit displacements, with feature products concatenating paths in matrix-multiplication order and transported node paths returning along the inverse route.For an individual edge holonomy, reversal gives its inverse, but that need not hold for a general learned edge feature.Source: Appendix A · pp. 11–12
  • Generating functions give a sampling-free contour-length distribution, whose mean is the first moment and whose highest nonzero power gives the maximum permitted length.The length distribution is dimension-independent under these rules, although generated path geometry depends on spatial dimension.Source: Appendix A · p. 12

Appendix B: Correlations between loop length and geometry

Individual-loop correlations show that contour length relates to spatial geometry only imperfectly. Longer loops generally explore more sites, but their typical radius grows modestly and both radius and site count vary with dimension.

Appendix B: Correlations between loop length and geometry

  • Longer loops generally have larger Rg, but the increase is modest and many paths with hundreds of traversals have radii of only a few lattice spacings or less.At fixed length, Rg varies appreciably; the scatter does not establish a particular asymptotic scaling law.Source: Appendix B · p. 13
  • Longer loops generally visit more distinct sites, but distinct-site counts rise more slowly than total traversals, reflecting both exploration and revisitation.At comparable contour lengths, higher-dimensional loops typically visit more distinct sites despite similar characteristic radii.Source: Appendix B · p. 13

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