  Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain,
with state space a totally ordered finite alphabet of size $m$. Using
combinatorial constructions and weak invariance principles, we obtain the
limiting shape of the associated RSK Young diagrams as a multidimensional
Brownian functional. Since the length of the top row of the Young diagrams is
also the length of the longest weakly increasing subsequences of $(X_k)_{1\le k
\le n}$, the corresponding limiting law follows. We relate our results to a
conjecture of Kuperberg by providing, under a cyclic condition, a spectral
characterization of the Markov transition matrix precisely characterizing when
the limiting shape is the spectrum of the $m \times m$ traceless GUE. For each
$m \ge 4$, this characterization identifies a proper, non-trivial class of
cyclic transition matrices producing such a limiting shape. However, for $m=3$,
all cyclic Markov chains have such a limiting shape, a fact previously only
known for $m=2$. For $m$ arbitrary, we also study reversible Markov chains and
obtain a characterization of symmetric Markov chains for which the limiting
shape is the spectrum of the traceless GUE. To finish, we explore, in this
general setting, connections between various limiting laws and spectra of
Gaussian random matrices, focusing in particular on the relationship between
the terminal points of the Brownian motions, the diagonal terms of the random
matrix, and the scaling of its off-diagonal terms, a scaling we conjecture to
be a function of the spectrum of the covariance matrix governing the Brownian
motion.
