# How should the proof of the XYZ theorem be understood?

The XYZ Theorem of Shepp  states that for a given poset $$P$$. Consider the probability space of all the linear extensions of $$P$$, where each possible extension is equally likely. Then for any three elements $$x, y, z \in P$$, we have $$\mathbb{P}[x \le y \textrm{ and } x \le z] \ge \mathbb{P}[x \le y] \cdot \mathbb{P}[x \le z].$$

Shepp's proof uses FKG inequality on a distributive lattice on $$[m]^n$$, in which $$x \le y$$ iff $$x_1>y_1$$ and $$x_i-x_1 \le y_i-y_1$$ for $$i \ge 2$$, and the meet and join are defined as: $$(x \wedge y)_i=\min\{x_i-x_1, y_i-y_1\}+\max\{x_1, y_1\},$$ $$(x \vee y)_i=\max\{x_i-x_1, y_i-y_1\}+\min\{x_1, y_1\}.$$

I can verify that this gives a distributive lattice and thus FKG applies. But what is the motivation behind it, i.e. what makes Shepp believe that this is even a distributive poset?

 L.A. Shepp, The XYZ conjecture and the FKG inequality, Ann. Probab. 10 (1982) 824--827, doi:10.1214/aop/1176993791

• In what sense are the $x_i$ elements of $P$? – darij grinberg Mar 7 '19 at 3:36
• (I made a slight edit replacing $\wedge$ by "and" because I was extremely confused about the claimed result at first- parsing $y \wedge x$ as the meet of $y$ and $x$.) – Sam Hopkins Mar 7 '19 at 4:26