# Shannon-McMillan-Breiman theorem for expander graphs: rate of convergence?

Is the following uniform SMB theorem for random walks on expander graphs true? For simplicity, I will state it for a finite group $$G=\langle S \rangle$$ and a uniform probability measure $$\mu$$ on the symmetric generating subset $$S$$ with cardinality $$d$$ and spectral gap at least $$c>0$$.

[Shannon-McMillan-Breiman Theorem] For any $$\delta>0$$, there is $$N$$ such that $$n\geq N$$ implies $$\mu^{*n}(\{ x \in G : -\log\mu^{*n}(x) \in [(1-\delta)H(n),(1+\delta)H(n)]\})>1-\delta,$$ where $$H(n)=\int-\log\mu^{*n}(x)\,d\mu^{*n}(x)$$ is the entropy of $$\mu^{*n}$$.

Of course, $$\mu^{*n}$$ converges to the uniform measure on $$G$$ and such $$N$$ exists for each $$(G,\mu)$$, but I want $$N$$ to be independent of $$|G|$$ (but may depend on $$d$$ and $$c$$). By Kaimanovich and Vershik's SMB theorem, the above estimate ought hold at some $$n$$, but I do not have a clue if it ought hold for all $$n$$ large enough. I'm particularly interested when $$n=\Omega(\log|G|)$$.