  We continue our study of a new boundedness condition for affine permutations,
motivated by the fruitful concept of periodic boundary conditions in
statistical physics. We focus on bounded affine permutations of size $N$ that
avoid the monotone decreasing pattern of fixed size $m$. We prove that the
number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$
times an explicit constant as $N\to\infty$. For instance, the number of bounded
affine permutations of size $N$ that avoid $321$ is asymptotically equal to
$4^N (N/4\pi)^{1/2}$. We also prove a permuton-like result for the scaling
limit of random permutations from this class, showing that the plot of a
typical bounded affine permutation avoiding $m\cdots1$ looks like $m-1$ random
lines of slope $1$ whose $y$ intercepts sum to $0$.
