Attractor States Emerge in Multi-Turn LLM Conversations

Large language models are increasingly used in multi-agent settings, but the long-run dynamics of model--model interaction remain poorly understood. We study whether open-ended LLM discussions exhibit \textit{attractor-like behavior}: stable regions in conversational behavior space. We run controlled dyadic debates among 7 language models on 20 controversial topics, comparing self-play and mixed-play under minimal prompting, and track each conversation with sentence embeddings, topic-centered PCA, LLM-judged discourse traits, and stance annotations. We find that self-play trajectories move toward stable endpoint regions that are specific to each model rather than to the topic, while mixed-play endpoints usually fall between the corresponding self-play regions, revealing asymmetric influence between partners. These results suggest that open-ended LLM discussions converge to predictable conversational regimes governed more by model identity than by topic, with implications for the design and control of multi-model agent systems.