Graph convergence, Luck approximation mod p and the entropy of cellular automata

I will discuss a conjecture with Balazs Szegedy that asks whether the entropy per site in the image of a fixed cellular automaton is continuous in the Benjamini-Schramm graph convergence topology.
The conjecture connects a new approach to sofic entropy of measure preserving actions (joint work with Benjy Weiss) to the so-called L¨uck Approximation conjecture mod p. Some natural subcases of the conjecture have been established, including the asymptotic behavior of the mod p rank of
random regular graphs.

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