A finite $n$-state Markov chain with transition matrix $P$ can be said to be ergodic, if there exists a power $a \in \mathbb{N}$ such that $P^a > 0$ entry-wise (the matrix is regular).
Are there any obvious bounds on the value of $a$ (depending on $n$), or can I construct examples for which $a$ is arbitrarily large ?
From irreducibility, you can reach state $j$ from state $i$ in $<n$ steps.
From aperiodicity, we have two coprime periods $p_1,p_2\leq n$ in our class. For any state $i$, we find paths to and from these cycles (each $<n$), so we have three coprime-length cycles (but not-necessarily pairwise coprime) starting from $i$:
These have lengths $<3n$, $<4n$, $<4n$ respectively. So we can represent all integers $\geq c(n)$ with positive integer multiples of these three numbers, so $p_{ii}^{(m)}>0$ for all $m\geq c(n)$. We can bound $c(n)\leq2(2n-1)^2-1$ (Lewin bound on the Frobenius number).
So $a<c(n)+n<8n^2$.