  We introduce and study several random combinatorial billiard trajectories.
Such a system, which depends on a fixed parameter $p\in(0,1)$, models a beam of
light that travels in a Euclidean space, occasionally randomly reflecting off
of a hyperplane in the Coxeter arrangement of an affine Weyl group with some
probability that depends on the side of the hyperplane that it hits. In one
case, we (essentially) recover Lam's reduced random walk in the limit as $p$
tends to $0$. The investigation of our random billiard trajectories relies on
an analysis of new finite Markov chains that we call stoned exclusion
processes. These processes have remarkable stationary distributions determined
by well-studied polynomials such as ASEP polynomials, inhomogeneous TASEP
polynomials, and open boundary ASEP polynomials; in many cases, it was
previously not known how to construct Markov chains with these stationary
distributions. Using multiline queues, we analyze correlations in the stoned
multispecies TASEP, allowing us to determine limit directions for reduced
random billiard trajectories and limit shapes for new random growth processes
for $n$-core partitions. Our perspective coming from combinatorial billiards
naturally leads us to formulate a new variant of the ASEP on $\mathbb{Z}$
called the scan ASEP, which we deem interesting in its own right.
