A Spectral Local-to-Global Principle for Spin Systems on Graphs with Girth At Least Five

Xiaoyu Chen, Kuikui Liu

Mathematics

Theoretical Computer Science

For each δ∈(0,1), sufficiently large maximum degree Δ and finite simple n-vertex graphs of girth at least 5, Theorem 1 gives irreducible proper-q-coloring Glauber dynamics with spectral gap Ωδ(1/n) and mixing time Oδ(n² log q+n log(1/ε)) when q≥(1+δ)Δ. Theorem 3 extends the method to antiferromagnetic Potts models under its stated β and q conditions; Theorem 5 gives a local spectral-contraction criterion for general multi-spin systems. The result is rapid mixing, not the conjectured exact q≥Δ+2 threshold or uniformly optimal O(n log n) bound.

What was already known

For girth 5, earlier Glauber results reached q>(α*+ε)Δ with α*≈1.763. The near-Δ threshold was available at girth at least 11 for a Metropolis chain, including a 2026 constant-degree result; fixed-degree q≥Δ+3 Glauber mixing needed girth growing with Δ.

AI involvement AI-centered

Chen and Liu brought the girth-five coloring question to GPT-5.6 Sol Ultra, which proposed the local-to-global strategy. Xiaoyu Chen and Kuikui Liu checked that argument, developed the multi-spin extension and explain the final proofs.

The model supplied the central new proof strategy for a human-chosen problem; the human authors verified and generalized it.

Importance Notable

The result lowers the girth requirement for near-Δ rapid mixing from 11 to 5 and supplies a broader multi-spin method.

Limitations

The multi-spin extension was developed with human input. No independent external check of the complete extension is documented.

Technical verification Formal (human)

The authors explicitly report verifying the model-proposed coloring argument before generalizing it; this is a reported human check of the AI-origin core, not independent reproduction.

Human understanding Full

The authors explain the local-to-global Bochner reduction and Fourier analysis of the star dynamics.

Correction and publication history

2026-10-18 · publication