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  "slug": "stable-twisted-kuramoto-equilibria",
  "title": "Stable twisted equilibria on random geometric tori in every fixed dimension",
  "shortTitle": "Random points, stable twist: twisted Kuramoto equilibria on random geometric tori",
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  "oneLine": "Identical Kuramoto oscillators scattered at random on a flat torus of any dimension have stable twisted equilibria near every integer winding up to a growing limit, proving the Cirelli–Groisman–Huang–Vivas conjecture for every kernel in their class with |z|K(z) bounded.",
  "abstract": "Place n identical Kuramoto oscillators at independent uniform points of the flat torus T^d, d ≥ 1, and couple points closer than ε_n with weight K(z/ε_n). If ε_n → 0 and nε_n^(d+2)/log n → ∞, then with high probability, for every integer twist k ∈ Z^d with |k| ≤ c(ε_n+δ_n)^(−1), where δ_n = (log n/(nε_n^(d+2)))^(1/2), there is an asymptotically stable equilibrium whose phase differs from k·x by at most C|k|δ_n at every vertex. The kernel need only be even, supported in the unit ball, positive almost everywhere near the origin and satisfy sup|z|K(z) < ∞; this covers the hard-radius graph, every bounded kernel and |z|^(−α), 0 < α ≤ 1. The result proves the twisted-equilibrium conjecture of Cirelli, Groisman, Huang and Vivas (SIAM J. Appl. Math. 2025) for every kernel in their class with |z|K bounded, and extends it from coordinate twists to all k ∈ Z^d. Further results: at least c(ε_n+δ_n)^(−d) distinct stable constructed states; a basin of fixed size; for bounded radial kernels, a heat-equation limit uniform over all times; and stable weighted harmonic lifts under smooth nonuniform sampling. The key tool is a sup-norm bound for the inverse graph Laplacian, uniform over all conductances above a fixed bounded backbone. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-04",
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  "keywords": [
    "Kuramoto model",
    "random geometric graphs",
    "twisted states",
    "synchronisation",
    "graph Laplacian",
    "graph Poisson equation",
    "Nash inequality",
    "singular kernels"
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  "keyResults": [
    "Stable twisted equilibria (Theorem 2.1, proof): if ε_n → 0 and nε_n^(d+2)/log n → ∞, then with high probability and almost surely eventually, every k ∈ Z^d with |k| ≤ c/(ε_n+δ_n) has an asymptotically stable equilibrium θ = k·X + u with mean-zero correction ‖u‖_∞ ≤ C|k|δ_n, δ_n = (log n/(nε_n^(d+2)))^(1/2).",
    "Kernel class and the conjecture (Theorem 2.1 and Remark 2.2): even kernels supported in the unit ball, positive almost everywhere near 0, with sup|z|K(z) < ∞ (hard radius, bounded kernels, |z|^(−α) for 0 < α ≤ 1); this proves the Cirelli–Groisman–Huang–Vivas conjecture for that part of their class and for all twists k ∈ Z^d, not only coordinate twists.",
    "Many coexisting states (Corollary 2.3, proof): at least c(ε_n+δ_n)^(−d) distinct stable constructed rotation orbits, a lower bound rather than a count of all equilibria; balancing ε and δ gives order (n/log n)^(d/(d+4)).",
    "Basin and long times (Proposition 5.1 and Theorem 5.2, proofs): a basin of fixed size with uniform weighted L² decay; for bounded radial kernels the heat-equation approximation of Cirelli et al. holds uniformly over all times for each fixed smooth winding lift.",
    "Nonuniform sampling (Theorem 6.1, proof): for smooth positive sampling densities ρ and bounded kernels, stable equilibria near weighted harmonic lifts solving div(ρ²M_K∇φ_k) = 0.",
    "Key estimate (Proposition 3.2, proof): a sup-norm bound for the mean-zero inverse graph Laplacian that holds uniformly over all conductance fields dominating a bounded backbone, with no upper bound on the conductances."
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  "relatedWorks": [
    {
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      "url": "https://doi.org/10.1137/24M1641518",
      "doi": "10.1137/24M1641518"
    },
    {
      "citation": "De Vita, Fernández Bonder and Groisman (2025). The Energy Landscape of the Kuramoto Model in Random Geometric Graphs in a Circle. SIAM Journal on Applied Dynamical Systems. Twisted states on the circle in a different scaling regime.",
      "url": "https://doi.org/10.1137/24M1642111",
      "doi": "10.1137/24M1642111"
    },
    {
      "citation": "Diaconis and Saloff-Coste (1996). Nash inequalities for finite Markov chains. Journal of Theoretical Probability. The classical Nash-inequality route to mixing used in the inverse estimate.",
      "url": "https://doi.org/10.1007/BF02214660",
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    },
    {
      "citation": "Calder, García Trillos and Lewicka (2022). Lipschitz Regularity of Graph Laplacians on Random Data Clouds. SIAM Journal on Mathematical Analysis. Graph Poisson regularity on random point clouds.",
      "url": "https://doi.org/10.1137/20M1356610",
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    },
    {
      "citation": "Bramburger, Holzer and Williams (2025). Persistence of Steady States for Dynamical Systems on Large Networks. Journal of Nonlinear Science. Persistence near a fixed graphon limit.",
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    },
    {
      "citation": "Medvedev and Mizuhara (2026). The Kuramoto Model on the Sierpinski Gasket II: Twisted States. Journal of Nonlinear Science. Twisted states on graph approximations of a fractal.",
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    },
    {
      "citation": "Zhai, Xiao and Zheng (2026). Twisted States in Coupled Phase Oscillator Systems with Different Coupling Weight Schemes. Chinese Journal of Physics. Recent study cited from its bibliographic record only.",
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    },
    {
      "citation": "De Vita, Groisman and Huang (2025). Phase synchronization in random geometric graphs on the 2D sphere. arXiv:2504.01151. Synchronisation on the sphere; the torus is the basic case where it fails.",
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      "doi": "10.48550/arXiv.2504.01151"
    },
    {
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      "url": "https://arxiv.org/abs/2609.02618",
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    },
    {
      "citation": "Lee and Maggs (2026). The geometry of cochains on sampled Vietoris–Rips complexes. arXiv:2608.11137 version 2. Harmonic representatives on samples.",
      "url": "https://arxiv.org/abs/2608.11137v2",
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    },
    {
      "citation": "Grande and Meila (2026). Selecting interpretable circular coordinates from data. arXiv:2607.08230. Density-corrected inner products for one-forms.",
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      "doi": "10.48550/arXiv.2607.08230"
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