离散环面上对称排斥过程的典范局部平衡与截断轮廓
Canonical Local Equilibrium and Cutoff Profiles for the Symmetric Exclusion Process on Discrete Tori
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中文总结 AI 辅助
本文证明了离散环面上对称简单排斥过程的典范局部平衡定理，建立了全变差距离的高斯截断轮廓，其坐标由单粒子最小非零特征值决定，并给出了严格的数学证明。
中文摘要 AI 辅助
我们证明了离散环面 $\mathbb T_N^D$（$D\ge2$）上对称简单排斥过程在粒子密度远离 $0$ 和 $1$ 时的典范（或固定粒子数）局部平衡定理，该定理对具有给定粒子数的所有确定性初始构型一致成立。在时间 \\[ t_N(s)=\frac{\log(N^D)+s}{2\gamma_N}, \qquad \gamma_N=2-2\cos\left(\frac{2\pi}{N}\right), \\] 过程相对于平衡态的 Radon--Nikodym 密度在 $L^2$ 中收敛到一个由唯一的小均值零校准场生成的典范指数倾斜，该校准场的单点边际与演化的单粒子热轮廓相匹配。因此，每当轮廓坐标收敛时，相应的全变差轮廓就是高斯平移。更精确地说，对于确定性初始序列 $(S_N)$，如果在固定 $s$ 处协方差归一化的平方振幅 $\mathfrak q_N^{S_N}(s)$ 收敛到标量 $\mathfrak q$，则到平稳性的距离收敛到 $ 2\Phi\\!\left({\sqrt{\mathfrak q}}/2\right)-1 $，其中 $\Phi$ 是标准正态分布函数。轮廓坐标渐近地由对应于最小非零特征值的单粒子特征空间决定。证明结合了校准的典范比较、独立动力学与排斥动力学之间的固定度比较，以及通过配对能量估计和 Strong--Rayleigh 性质的保持获得的全度控制。
英文摘要
We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus $\mathbb T_N^D$, $D\ge2$, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times \[ t_N(s)=\frac{\log(N^D)+s}{2γ_N}, \qquad γ_N=2-2\cos\left(\frac{2π}{N}\right), \] the Radon--Nikodym density of the process relative to equilibrium converges in $L^2$ to a canonical exponential tilt generated by the unique small mean-zero calibration field whose one-site marginals match the evolving one-particle heat profile. Consequently, whenever the profile coordinate converges, the corresponding total variation profile is a Gaussian shift. More precisely, for a deterministic initial sequence $(S_N)$, if the covariance-normalized squared amplitude $\mathfrak q_N^{S_N}(s)$ converges to a scalar $\mathfrak q$ at a fixed $s$, then the distance to stationarity converges to $ 2Φ\!\left({\sqrt{\mathfrak q}}/2\right)-1 $, where $Φ$ is the standard normal distribution function. The profile coordinate is asymptotically determined by the one-particle eigenspace corresponding to the smallest nonzero eigenvalue. The proof combines a calibrated canonical comparison, a fixed-degree comparison between independent and exclusion dynamics, and all-degree control obtained from pair energy estimates and preservation of the Strong--Rayleigh property.
发表机构
- Colgate University（科尔盖特大学）
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