Phase transitions for the noisy transformer model in arbitrary dimension

Abstract: We study the McKean--Vlasov free energy on the unit sphere associated with the unnormalized self-attention (USA) model for noisy transformer dynamics. We prove a sharp global-minimizer dichotomy in every dimension $d\ge2$. There is a unique $β^{(d)}>0$ such that \begin{equation} \frac{I{d/2+1}(β*^{(d)})}{I{d/2}(β^{(d)})}=\frac1d, \end{equation} where $Iν$ is the modified Bessel function of the first kind. For $0<β\le β^{(d)}$, the uniform density remains the unique global minimizer up to the linear-stability threshold \begin{equation} K#^{(d)}(β)=\frac{β^{d/2}}{2^{d/2}Γ(d/2)I_{d/2}(β)}, \end{equation*} and the phase transition is continuous. For $β>β*^{(d)}$, the uniform density is not globally minimizing at $K#^{(d)}(β)$, so the critical coupling satisfies $K_c<K_#^{(d)}(β)$ and the transition is discontinuous. This result generalizes the authors' recent $d=2$ work arXiv:2604.16288 to arbitrary dimension. The proof uses the sharp Beckner--Onofri/logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the sphere, together with a Funk--Hecke/Bessel coefficient computation and a degree-two quartic obstruction.