AI 中文总结
该研究针对高温下离散环面的q态铁磁Potts模型Glauber动力学，证明其具有O(1)窗口的截断，通过开发符号影响的信息渗流框架解决了q≥3时直接应用信息渗流的问题，确定了混合时间的加性误差。
AI 中文摘要
我们证明了在离散环面Λₙ=(Z/nZ)ᵈ上的铁磁q态Potts模型的连续时间热浴Glauber动力学，在足够高的温度下具有O(1)窗口的截断。对于每个固定的d≥2和q≥3，存在β₀=β₀(d,q)>0，使得当0<β<β₀时，Λₙ上Potts模型的Glauber动力学在t★=t★⁽ⁿ⁾:=1/(2𝔯) log|Λₙ|附近呈现最优O(1)窗口的截断，其中𝔯∈(0,1)是单格点磁化的指数衰减率。特别地，这确定了混合时间的加性O(1)误差。该混合时间的特征是，从单色初始条件产生的宏观色密度偏差达到平衡涨落尺度的时刻。此外，我们的证明表明，单色初始条件唯一最大化了色偏差。这是首次将信息渗流用于证明非单调自旋系统的截断。与伊辛模型不同，当q≥3时，直接应用信息渗流无法得到Potts动力学匹配的上下界。我们通过开发针对符号影响的信息渗流框架，并将其与符号卷积幂的傅里叶界及历史图的几何控制相结合，克服了这一问题。
英文摘要
We prove cutoff with an $O(1)$ window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus $Λ_n=(\mathbb Z/n\mathbb Z)^d$ at sufficiently high temperature. For every fixed $d\ge2$ and $q\ge3$, there exists $β_0=β_0(d,q)>0$ such that, for $0<β<β_0$, the Glauber dynamics of the Potts model on $Λ_n$ exhibits cutoff with optimal $O(1)$ window around \[ t_\star=t_\star^{(n)}:=\frac{1}{2\mathfrak{r}}\log |Λ_n|, \] where $\mathfrak{r}\in(0,1)$ is the exponential decay rate of the one-site magnetization. In particular, this determines the mixing time up to an additive $O(1)$. It is characterized by the point at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. In contrast with the Ising model, a direct implementation of information percolation does not yield matching upper and lower bounds for the Potts dynamics when $q\ge3$. We overcome this by developing an information-percolation framework for signed influences and combining it with Fourier bounds on signed convolution powers and geometric control of history diagrams.
