I am interested in any known results/empirical studies done on the average height of a poset with $N$ elements. Obviously this would depend on how that poset relation was randomly defined, however, at this point, I'll take any reasonable result on the topic regardless of how the poset was formed.
In my case $N$ is a very large number (billions of elements), so I am interested whether the height $h \ll N$ asymptotically.
In Asymptotic Enumeration of Partial Orders on a Finite Set (1975), Kleitman and Rothschild showed that almost all partial orders on an $n$-element set have a simple description: they have three "layers" $L_1$, $L_2$, and $L_3$ of incomparable elements, of size $n/4$, $n/2$, and $n/4$ respectively. Each element of $L_1$ is covered by about half the elements of $L_2$. Likewise for $L_2$ and $L_3$, and in the reverse direction.
So almost all finite posets have height $3$.
(If you don't have the time/effort to read Kleitman and Rothschild's paper, I came across this reference via G. Brightwell, Linear extensions of random orders, Discrete Math. 125 (1994) pp. 87–96. If memory serves, Brightwell's paper presents a good summary of this one.)
