高维Lipschitz函数是典型平坦的
High-Dimensional Lipschitz Functions are Typically Flat
摘要：
$d$维环面$\mathbb{Z}_n^d$上的同态高度函数是在环面的顶点上取整数值的函数，其中连续整数被分配给相邻顶点。Lipschitz高度函数的定义类似，但也可以在相邻顶点上取相等的值。在每个模型中，我们考虑在这样的函数的均匀分布，受边界条件。我们证明，在高维，零边界值，一个典型的功能是非常平坦的，有界方差在任何固定的顶点，并采取最多$C(\log n)^{1/d}$值的概率很高。我们的结果推广到任何维度$d\ge 2$，如果$\mathbb{Z}_n^d$被替换为它的一个增强版本，环面$\mathbb{Z}_n^d\times\mathbb{Z}_2^{d_0}$为某些固定的$d_0$。这在$2$维度中建立了一个粗糙化过渡的一侧。建立了一类不等边长环面的全过渡。我们还发现，当$d$被带到无穷大，而$n$保持固定，一个典型的功能，以最多$r$值具有高概率，其中$r=5$的同态模型和$r=4$的Lipschitz模型。当$n$与$d$一起生长时，可得到适当的推广。我们的结果也适用于相关模型的均匀3-着色和建立，对于某些边界条件，均匀采样适当的3-着色的$\mathbb{Z}_n^d$将是几乎恒定的偶数或奇数的子格。我们的证明是基于一个组合变换和仔细分析一类割集，我们任期奇割集的性质。对于Lipschitz模型，我们的结果也依赖于Yadin的双射。推广了Galvin和Kahn的结果，反驳了Benjamini，Yadin和Yehudayoff的一个猜想，回答了Benjamini，Häggström和Mossel的一个问题.
英文摘要：
A homomorphism height function on the $d$-dimensional torus $\mathbb{Z}_n^d$ is a function taking integer values on the vertices of the torus with consecutive integers assigned to adjacent vertices. A Lipschitz height function is defined similarly but may also take equal values on adjacent vertices. In each model, we consider the uniform distribution over such functions, subject to boundary conditions. We prove that in high dimensions, with zero boundary values, a typical function is very flat, having bounded variance at any fixed vertex and taking at most $C(\log n)^{1/d}$ values with high probability. Our results extend to any dimension $d\ge 2$, if $\mathbb{Z}_n^d$ is replaced by an enhanced version of it, the torus $\mathbb{Z}_n^d\times\mathbb{Z}_2^{d_0}$ for some fixed $d_0$. This establishes one side of a conjectured roughening transition in $2$ dimensions. The full transition is established for a class of tori with non-equal side lengths. We also find that when $d$ is taken to infinity while $n$ remains fixed, a typical function takes at most $r$ values with high probability, where $r=5$ for the homomorphism model and $r=4$ for the Lipschitz model. Suitable generalizations are obtained when $n$ grows with $d$. Our results apply also to the related model of uniform 3-coloring and establish, for certain boundary conditions, that a uniformly sampled proper 3-coloring of $\mathbb{Z}_n^d$ will be nearly constant on either the even or odd sub-lattice. Our proofs are based on a combinatorial transformation and on a careful analysis of the properties of a class of cutsets which we term odd cutsets. For the Lipschitz model, our results rely also on a bijection of Yadin. This work generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini, Yadin and Yehudayoff and answers a question of Benjamini, Häggström and Mossel.
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