  We introduce the Ungarian Markov chain ${\bf U}_L$ associated to a finite
lattice $L$. The states of this Markov chain are the elements of $L$. When the
chain is in a state $x\in L$, it transitions to the meet of $\{x\}\cup T$,
where $T$ is a random subset of the set of elements covered by $x$. We focus on
estimating $\mathcal E(L)$, the expected number of steps of ${\bf U}_L$ needed
to get from the top element of $L$ to the bottom element of $L$. Using direct
combinatorial arguments, we provide asymptotic estimates when $L$ is the weak
order on the symmetric group $S_n$ and when $L$ is the $n$-th Tamari lattice.
When $L$ is distributive, the Markov chain ${\bf U}_L$ is equivalent to an
instance of the well-studied random process known as last-passage percolation
with geometric weights. One of our main results states that if $L$ is a trim
lattice, then $\mathcal E(L)\leq\mathcal E(\text{spine}(L))$, where
$\text{spine}(L)$ is a specific distributive sublattice of $L$ called the spine
of $L$. Combining this lattice-theoretic theorem with known results about
last-passage percolation yields a powerful method for proving upper bounds for
$\mathcal E(L)$ when $L$ is trim. We apply this method to obtain uniform
asymptotic upper bounds for the expected number of steps in the Ungarian Markov
chains of Cambrian lattices of classical types and the Ungarian Markov chains
of $\nu$-Tamari lattices.
