Weighted infinite tori as quantitative phase spaces

Definitions and mathematical statement

The source asks for an application in which an infinite-dimensional Alexandrov space is a limit of manifolds whose dimensions increase. Here "Alexandrov" must mean a complete geodesic space satisfying the lower-curvature comparison condition CBB$(0)$ without building finite dimension into the definition. The supplied example is a countable phase configuration space.

Let $(r_j)_{j\geq1}$ be positive real numbers with $\sum_jr_j^2<\infty$. Write each circle of radius $r_j$ as angles $\theta_j\in{\mathbb{R}}/(2\pi{\mathbb{Z}})$ and define \[ X=\prod_{j\geq1}S^1(r_j),\qquad d(\theta,\varphi)^2=\sum_{j\geq1}r_j^2\operatorname{dist}_{{\mathbb{R}}/(2\pi{\mathbb{Z}})}(\theta_j,\varphi_j)^2. \] Let $X_N=\prod_{j=1}^NS^1(r_j)$, embedded by fixing all tail angles at zero, and let $p_N:X\to X_N$ be coordinate projection. The aid asserts that $X$ is compact, complete, geodesic, and CBB$(0)$, with infinite dimension since it contains closed tori of every finite dimension. Each $X_N$ is a smooth closed flat $N$-manifold. With \[ \delta_N=\pi\left(\sum_{j>N}r_j^2\right)^{1/2}, \] one has $d_H^X(X_N,X)=\delta_N$ exactly. The graph of $p_N$ has distortion at most $\delta_N$, so \[ d_{\mathrm{GH}}(X_N,X)\leq\delta_N/2\longrightarrow0. \] The ambient Hausdorff equality is not asserted to be an exact GH-distance formula.

For the countable $U(1)$-phase/rotor interpretation, let $F:X\to{\mathbb{R}}$ be $L$-Lipschitz and put $F_N=F|_{X_N}$. Then \[ |F(x)-F_N(p_Nx)|\leq L\delta_N, \qquad\left|\min_XF-\min_{X_N}F_N\right|\leq L\delta_N. \] Let $\mu$ and $\mu_N$ be normalized product Haar probability measures. The projection coupling gives \[ \left|\int_XF\,{\,\mathrm{d}}\mu-\int_{X_N}F_N\,{\,\mathrm{d}}\mu_N\right|\leq L\delta_N. \] For inverse temperature $\beta>0$, define $Z_\beta=\int_Xe^{-\beta F}\,{\,\mathrm{d}}\mu$ and $Z_{\beta,N}=\int_{X_N}e^{-\beta F_N}\,{\,\mathrm{d}}\mu_N$. Their free energies satisfy \[ \left|-\beta^{-1}\log Z_\beta+\beta^{-1}\log Z_{\beta,N}\right|\leq L\delta_N. \] These are quantitative truncation bounds for fixed Lipschitz observables or energies on this weighted phase space. Normalized Haar measure and a finite uniform Lipschitz constant are essential; arbitrary infinite rotor Hamiltonians or unnormalized product volumes are not covered. The example supplies a mathematical application scheme rather than a claim of a particular physical model or all infinite-dimensional Alexandrov phenomena.

Short English statement

The supplied weighted infinite torus is an infinite-dimensional lower-curvature-comparison space obtained as a compact GH limit of flat finite-dimensional tori. Square-summable radii give explicit tail errors for distances, Lipschitz observables, minima, Haar averages, and finite-temperature free energies. The application requires this weighted metric and normalized probability measures; it does not cover arbitrary infinite rotor energies.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-topology-notes.json, notes[233].
• AIM workshop Manifolds with nonnegative sectional curvature, Problem 21, https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. The appended "3. Group Actions and Submersions" is a following section heading, not part of the mathematical request. No proposer or proposal year is supplied.
• Supplied research attempt 1, registry updated 14 August 2026: partial result, low novelty confidence, machine-generated and not peer reviewed. It supplies the weighted-circle construction and quantitative phase-space interpretation. Original remarks and literature field are empty; no complete authoritative external references are supplied.
• Verification reproduces the extraction from input.json but supplies no further definition of infinite-dimensional Alexandrov space or application citation.