  The three exceptional lattices, $E_6$, $E_7$, and $E_8$, have attracted much
attention due to their anomalously dense and symmetric structures which are of
critical importance in modern theoretical physics. Here, we study the
electronic band structure of a single spinless quantum particle hopping between
their nearest-neighbor lattice points in the tight-binding limit. Using Markov
chain Monte Carlo methods, we numerically sample their lattice Green's
functions, densities of states, and random walk return probabilities. We find
and tabulate a plethora of Van Hove singularities in the densities of states,
including degenerate ones in $E_6$ and $E_7$. Finally, we use brute force
enumeration to count the number of distinct closed walks of length up to eight,
which gives the first eight moments of the densities of states.
