That's an interesting observation. I'd suggest modelling the LLM's behaviour in that situation as selecting between different simple strategies, each of which has its own transition function. Some of the strategies will be far more common than others. Some of them may be very simple and obey the detailed balance condition (meaning they are reversible Markov chains), but others, and the overall transition function does not.

The definition of the detailed balance condition is very strict and it's obvious that it won't be met in general by most probabilistic programs (sets of rules with probabilistic output) even if you consider only those where all possible outputs have non-zero probability (as required by detailed balance).

And the LLM+agent is only a Markov chain because of the limited state space of the agent. While an LLM is adding to its context window without reaching the window size limit, it is not a Markov chain, as I explained here: https://news.ycombinator.com/item?id=45124761

And, agreed that better optimisation would be incredible. (I would describe it as a search problem.) I'm not sure how feasible it is improve without changing the architecture, e.g. to a diffusion language model. But LLMs already predict many tokens ahead at once which is why beam search is surprisingly unnecesarr. That's how they're able to write coherent sentences (and rhymes), they've already largely determined at the beginning what they're going to write. (See Anthropic mech interp work.) So maybe if we could tap into that we search over vaguely-formed next blocks of text rather than next words.