AI 中文总结
研究有限体积理想玻色气体宏观费曼循环的极限定理，核心方法是去除确定性背景密度，主要贡献是证明标记宏观循环过程收敛到特定泊松 - 金曼桥，不同条件下有不同表现。
AI 中文摘要
我们证明了有限体积理想玻色气体宏观费曼循环的一个规范极限定理。循环带有一般波兰空间\(\mathsf{M}\)中的标记，编码空间、几何、光谱或内部数据。去除确定性背景密度\(\rho_{\mathrm{bg}}\)后，标记宏观循环过程在正则系综中收敛到总质量为\(\rho - \rho_{\mathrm{bg}}\)的标记泊松 - 金曼桥。桥由强度为\(x^{-1}\eta_x(dm)\,dx\)的标记泊松点过程构建，条件是总质量为\(\rho - \rho_{\mathrm{bg}}\)，其中核\(x \mapsto \eta_x\)及其总质量分布\(\phi(x) = \eta_x(\mathsf{M})\)由尺度\(j \sim V_L\)上可见的低能谱数据确定。当\(\phi\)为常数时，桥简化为伽马桥且排序后的循环长度遵循泊松 - 狄利克雷定律。我们在\(d > 2\)维的理想玻色气体在周期、狄利克雷和诺伊曼边界条件下验证了这一点：在所有三种情况下\(\phi \equiv 1\)且排序后的长度收敛到\(\mathrm{PD}(0,1)\)，而标记核通过其缠绕、被杀桥和反射桥几何区分这三种模型。当\(\phi\)不是常数时，桥不再是伽马桥且排序后的长度不是泊松 - 狄利克雷分布。作为一个具体例子，一个隧穿分裂满足\(V_L \Delta_L \to \gamma\)的临界双阱势给出\(\phi_\gamma(x) = 1 + e^{-\beta\gamma x}\)；更一般地，具有\(R\)个分量的有限型可见光谱给出\(\phi(x) = \sum_{r=1}^{R} \theta_r e^{-\beta\lambda_r x}\)。这些结果将泊松 - 金曼桥确定为标记宏观玻色循环的规范普遍性类，可见的低能谱选择特定的桥。
英文摘要
We prove a canonical limit theorem for the macroscopic Feynman cycles of finite-volume ideal Bose gases. Cycles carry marks in a general Polish space $\mathsf{M}$, encoding spatial, geometric, spectral, or internal data. After removing a deterministic background density $ρ_{\mathrm{bg}}$, the marked macroscopic cycle process converges in the canonical ensemble to a marked Poisson--Kingman bridge of total mass $ρ- ρ_{\mathrm{bg}}$. The bridge is constructed from a marked Poisson point process with intensity $x^{-1}η_x(dm)\,dx$, conditioned on total mass~$ρ- ρ_{\mathrm{bg}}$, where the kernel $x \mapsto η_x$ and its total-mass profile $ϕ(x) = η_x(\mathsf{M})$ are determined by the low-energy spectral data visible on the scale $j \sim V_L$. When $ϕ$ is constant, the bridge reduces to a Gamma bridge and the ranked cycle lengths follow the Poisson--Dirichlet law. We verify this for the ideal Bose gas in dimension $d > 2$ under periodic, Dirichlet, and Neumann boundary conditions: in all three cases $ϕ\equiv 1$ and the ranked lengths converge to $\mathrm{PD}(0,1)$, while the mark kernels distinguish the three models through their winding, killed-bridge, and reflected-bridge geometry. When $ϕ$ is not constant, the bridge is no longer Gamma and the ranked lengths are not Poisson--Dirichlet. As a concrete example, a critical double-well potential whose tunnelling splitting satisfies $V_L Δ_L \to γ$ gives $ϕ_γ(x) = 1 + e^{-βγx}$; more generally, a finite-type visible spectrum with $Q$ components yields $ϕ(x) = \sum_{r=1}^{Q} θ_r e^{-βλ_r x}$. These results identify Poisson--Kingman bridges as the canonical universality class for marked macroscopic Bose cycles, with the visible low-energy spectrum selecting the particular bridge.
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