Detailed Balance in Large Language Model-Driven Agents

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

We conducted experiments on three different models, including GPT-5 Nano, Claude-4, and Gemini-2.5-flash. Each model was prompted to gener- ate a new word based on a given prompt word such that the sum of the letter indices of the new word equals 100. For example, given the prompt “WIZ- ARDS(23+9+26+1+18+4+19=100)”, the model needs to generate a new word whose letter indices also sum to 100, such as “BUZZY(2+21+26+26+25=100)”