FKG不等式的$q$ -模拟及其应用
A $q$-analogue of the FKG inequality and some applications
摘要：
设$L$是有限分配格，$μ: L \to {\mathbb R}^{+}$是对数超模函数.对于函数$k: L \to {\mathbb R}^{+}$设$$E_μ (k; q) \defeq \sum_{x\in L} k(x) μ(x) q^{{\mathrm rank}(x)} \in {\mathbb R}^{+}[q].$$我们证明了对于任何单调增函数对$g,h: L\to {\mathbb R}^{+}$，有$$E_μ (g; q)\cdot E_μ (h; q) \ll E_μ (1; q)\cdot E_μ (gh; q), $$，其中"$ \ll $"表示实多项式的系数不等式. Fortuin，Kasteleyn和Ginibre（1971）的FKG不等式是通过专门研究$q=1$得到的实数不等式。多项式FKG不等式可以应用于$f$ -向量的连接和交叉的单纯复形，贝蒂数的交叉的某些舒伯特品种，并以下列类型的相关不等式的权力系列加权的杨tableaux。令$Y$是所有整数分区的集合。给定函数$k, μ: Y \rarr \R^+$和参数$0\le s\le t$，定义形式幂级数$$F_μ(k ; z) \defeq \sum_{\la\in Y} k(\la) μ(\la) (f_{\la})^t \frac{z^{|\la|}}{(|\la| !)^s} \in \R^+ [[z]], $$ % \sum_{\la\in Y} k（\la）μ（\la）（f_{\la}）^t \frac{z^{|\la|}}{|\la| !} \in \R^+ [[z]]，$$ where $f_{\la}$ is the number of standard Young tableaux of shape $\la$. Assume that $μ: Y\rarr \R^+$ is log-supermodular, and that $g, h: Y \rarr \R^+$ are monotonely increasing with respect to containment order of partition shapes. Then $$ F_μ（g;z）\cdot F_μ（h;z）\ll F_μ（1;z）\cdot F_μ（gh;z）。$$
英文摘要：
Let $L$ be a finite distributive lattice and $μ: L \to {\mathbb R}^{+}$ a log-supermodular function. For functions $k: L \to {\mathbb R}^{+}$ let $$E_μ (k; q) \defeq \sum_{x\in L} k(x) μ(x) q^{{\mathrm rank}(x)} \in {\mathbb R}^{+}[q].$$ We prove for any pair $g,h: L\to {\mathbb R}^{+}$ of monotonely increasing functions, that $$E_μ (g; q)\cdot E_μ (h; q) \ll E_μ (1; q)\cdot E_μ (gh; q), $$ where ``$ \ll $'' denotes coefficientwise inequality of real polynomials. The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) is the real number inequality obtained by specializing to $q=1$. The polynomial FKG inequality has applications to $f$-vectors of joins and intersections of simplicial complexes, to Betti numbers of intersections of certain Schubert varieties, and to the following kind of correlation inequality for power series weighted by Young tableaux. Let $Y$ be the set of all integer partitions. Given functions $k, μ: Y \rarr \R^+$, and parameters $0\le s\le t$, define the formal power series $$F_μ(k ; z) \defeq \sum_{\la\in Y} k(\la) μ(\la) (f_{\la})^t \frac{z^{|\la|}}{(|\la| !)^s} \in \R^+ [[z]], $$ %\sum_{\la\in Y} k(\la) μ(\la) (f_{\la})^t \frac{z^{|\la|}}{|\la| !} \in \R^+ [[z]],$$ where $f_{\la}$ is the number of standard Young tableaux of shape $\la$. Assume that $μ: Y\rarr \R^+$ is log-supermodular, and that $g, h: Y \rarr \R^+$ are monotonely increasing with respect to containment order of partition shapes. Then $$F_μ(g;z) \cdot F_μ(h;z) \ll F_μ(1;z) \cdot F_μ(gh;z). $$
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