任意小$n$时环路$O(n)$模型中的移动临界阈值
Shifted critical threshold in the loop $O(n)$ model at arbitrary small $n$
摘要：
在循环$O(n)$模型中，在格的有限域上随机绘制相互不相交的自避免循环的集合，其概率与$${λ^{\# \mbox{edges}} n^{\# \mbox{loops}},}$$成比例，其中$λ, n \in [0, \infty)$。令$μ$是格的连接常数，对于任何$n \in [0, \infty)$，令$λ_c(n)$是$λ$的最大值，使得循环长度允许一致有界指数矩。这是不难证明，$λ_c(n) =1/μ$时，$n=0$（在这种情况下，该模型对应于自回避行走），并为任何$n \geq 0$，$λ_c(n) \geq 1/μ$。在本说明中，我们证明，\begin{align*} λ_c(n) & > 1/μ\, \, \, \, \, \, \, \, \, \, \, \mbox{whenever $n >0$}, \\ λ_c(n) & \geq 1/μ\, + \, c_0 \, n \, + \, O(n^2), \end{align*}上的$\mathbb{Z}^d$，与$d \geq 2$，和六方晶格，其中$c_0>0$。这意味着，当$n$为正（甚至任意小）时，由于环之间的相互排斥，相变只能发生在比自避免行走严格更大的临界阈值处。
英文摘要：
In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to $${λ^{\# \mbox{edges}} n^{\# \mbox{loops}},}$$ where $λ, n \in [0, \infty)$. Let $μ$ be the connective constant of the lattice and, for any $n \in [0, \infty)$, let $λ_c(n)$ be the largest value of $λ$ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $λ_c(n) =1/μ$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and that for any $n \geq 0$, $λ_c(n) \geq 1/μ$. In this note we prove that, \begin{align*} λ_c(n) & > 1/μ\, \, \, \, \, \, \, \, \, \, \, \mbox{whenever $n >0$}, \\ λ_c(n) & \geq 1/μ\, + \, c_0 \, n \, + \, O(n^2), \end{align*} on $\mathbb{Z}^d$, with $d \geq 2$, and on the hexagonal lattice, where $c_0>0$. This means that, when $n$ is positive (even arbitrarily small), as a consequence of the mutual repulsion between the loops, a phase transition can only occur at a strictly larger critical threshold than in the self-avoiding walk.
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