论文标题
随机离散凹入在与周期性黑姐妹的等边格上的功能
Random discrete concave functions on an equilateral lattice with periodic Hessians
论文作者
论文摘要
由与随机矩阵，利特伍德 - 富丽堂子系数和瓷砖的连接动机，我们研究了在等边晶格上的随机离散凹面功能。我们表明，这种功能具有固定的平均值$ -S $浓度的周期性Hessian，围绕二次函数。我们考虑等边晶格$ \ mathbb l $上的所有凹功能$ g $的集合，该元素被$ n \ mathbb l $移动时，具有定期的离散性hessian，带有$ n \ nathbb l $。我们添加了Hessian $ s $的凸二次；然后将总和与周期$ n \ mathbb l $进行周期性，并将其视为torus $ \ mathbb {t} t} _n：= \ frac {\ mathbb {Z} {Z} {n \ z n \ z n \ z { \ frac {\ mathbb {z}}} {n \ mathbb {z}} $，其Hessian由$ S $主导。由此产生的Semiconcave函数组形成了凸polytope $ p_n（s）$。 $ \ ell_ \ infty $ $ p_n（s）$的直径在下面由$ c（s）n^2 $限制，其中$ c（s）$仅取决于$ s $。我们的主要结果是，在某些条件下，例如在$ s_0 = s_1 \ leq s_2 $的情况下，对于任何$ε> 0，$，我们有$ \ lim_ {n \ rightarrow 0} \ mathbb {p} \ weft [\ | 0 $$如果$ g $是从$ p_n（s）$的统一度量中取样的。 p_n（s）$中的每个$ g \对应于一种蜂窝。我们也获得了这些结果的浓度结果。
Motivated by connections to random matrices, Littlewood-Richardson coefficients and tilings, we study random discrete concave functions on an equilateral lattice. We show that such functions having a periodic Hessian of a fixed average value $- s$ concentrate around a quadratic function. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$ have a periodic discrete Hessian, with period $n \mathbb L$. We add a convex quadratic of Hessian $s$; the sum is then periodic with period $n \mathbb L$, and view this as a mean zero function $g$ on the set of vertices $V(\mathbb{T}_n)$ of a torus $\mathbb{T}_n := \frac{\mathbb{Z}}{n\mathbb{Z}}\times \frac{\mathbb{Z}}{n\mathbb{Z}}$ whose Hessian is dominated by $s$. The resulting set of semiconcave functions forms a convex polytope $P_n(s)$. The $\ell_\infty$ diameter of $P_n(s)$ is bounded below by $c(s) n^2$, where $c(s)$ is a positive constant depending only on $s$. Our main result is that under certain conditions, that are met for example when $s_0 = s_1 \leq s_2$, for any $ε> 0,$ we have $$\lim_{n \rightarrow 0} \mathbb{P}\left[\|g\|_\infty > n^{\frac{7}{4} + ε}\right] = 0$$ if $g$ is sampled from the uniform measure on $P_n(s)$. Each $g \in P_n(s)$ corresponds to a kind of honeycomb. We obtain concentration results for these as well.
