One example I enjoy is that if you add a list of numbersadd a list of numbers, the carries form a markov chain
If $n$ integers in base $b$ with digits chosen uniformly random, the carries form a markov chain
1 12021 01111 11111 11111 11011 10111 01111 11111 21011 1112.
43935 23749 58561 74916 62215 47448 33196 51990 19807 27075
48537 53642 77448 32760 14421 72142 82116 37225 43300 51498
33618 41327 41561 16257 43616 55134 82714 63369 87142 45607
-------------------------------------------------------------
1 26091 18719 77571 23934 20253 74725 98027 52585 50250 24180
Another important example is random walk on the cube [0,1]n and it's projection is the Ehrenfest Urn
- start at $\vec{v} = (0,\dots, 0) \in \{ 0,1\}^n$ and at each time step change $0 \leftrightarrow 1$ for one of the coordinates.
- If we take the inner product $\vec{v} \cdot (1,\dots, 1) = v_1 + \dots v_n \in \mathbb{N}$ this is also a Markov chain.
- We could have spots $0, 1, \dots, n$ and if $X_t = k$ we can jump to the left with probability $\frac{k}{n}$ and to the right with probability $\frac{n-k}{n}$ and this is a Markov chain.
I had to verify my logic was correct it's in Markov Chains and Mixing Times.