具有周期Hessians的等边格上的随机凹函数I：熵和Laplacian
Random concave functions on an equilateral lattice with periodic Hessians I: entropy and Laplacians
摘要：
证明了在等边格上具有周期Hessian的随机凹函数，如果其平均Hessian满足一定的条件，则具有二次标度极限。我们考虑等边格点$\mathbb L$上的所有凹函数$g$的集合，当被$n \mathbb L$的一个元素移位时，引起线性函数的加法（这个条件等价于$g$的hessian的周期性）。我们确定这一套，除了由一个常数，凸多面体$P_n(s)$，其中$s$对应于平均海森。我们发现，$P_n(s)$的$\ell_\infty$直径是有界的下面由$c(s) n^2$，其中$c(s)$是一个正常数，只依赖于$s$。我们的主要结果是，对于任何$ε_0 > 0$，$P_n(s)$中所有不包含在边长为$2 ε_0 n^2$的$n^2$维立方体$Q$中的点的归一化勒贝格测度，以唯一的（直到添加线性项）具有Hessian $s$的二次多项式为中心，趋于$0$，因为$n$趋于$\infty$。
英文摘要：
We show that a random concave function having a periodic hessian on an equilateral lattice has a quadratic scaling limit, if the average hessian of the function satisfies certain conditions. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$, incur addition by a linear function (this condition is equivalent to the periodicity of the hessian of $g$). We identify this set, up to addition by a constant, with a convex polytope $P_n(s)$, where $s$ corresponds to the average hessian. We show that the $\ell_\infty$ diameter of $P_n(s)$ is bounded below by $c(s) n^2$, where $c(s)$ is a positive constant depending only on $s$. Our main result is that, for any $ε_0 > 0$, the normalized Lebesgue measure of all points in $P_n(s)$ that are not contained in a $n^2$ dimensional cube $Q$ of sidelength $2 ε_0 n^2$, centered at the unique (up to addition of a linear term) quadratic polynomial with hessian $s$, tends to $0$ as $n$ tends to $\infty$.
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