Collective Alignment in LLM Multi-Agent Systems: Disentangling Bias from Cooperation via Statistical Physics

We investigate the emergent collective dynamics of LLM-based multi-agent systems on a 2D square lattice and present a model-agnostic statistical-physics method to disentangle social conformity from intrinsic bias, compute critical exponents, and probe the collective behavior and possible phase transitions of multi-agent systems. In our framework, each node of an $L\!\times\!L$ lattice hosts an identical LLM agent holding a binary state ($+1$/$-1$, mapped to yes/no) and updating it by querying the model conditioned on the four nearest-neighbor states. The sampler temperature $T$ serves as the sole control parameter. Across three open-weight models (llama3.1:8b, phi4-mini:3.8b, mistral:7b), we measure magnetization and susceptibility under a global-flip protocol designed to probe $\mathbb{Z}_2$ symmetry. All models display temperature-driven order-disorder crossovers and susceptibility peaks; finite-size scaling on even-$L$ lattices yields effective exponents $\gamma/\nu$ whose values are model-dependent, close to but incompatible with the 2D Ising universality class ($\gamma/\nu=7/4$). Our method enables the extraction of effective $\beta$-weighted couplings $\tilde{J}(T)$ and fields $\tilde{h}(T)$, which serve as a measure of social conformity and intrinsic bias. In the models we analyzed, we found that collective alignment is dominated by an intrinsic bias ($\tilde{h}\gg\tilde{J}$) rather than by cooperative neighbor coupling, producing field-driven crossovers instead of genuine phase transitions. These effective parameters vary qualitatively across models, providing compact collective-behavior fingerprints for LLM agents and a quantitative diagnostic for the reliability of multi-agent consensus and collective alignment.