Phase transitions for the noisy transformer model in arbitrary dimension

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≥2. There is a unique β_*(d)>0 such that beginequation* fracId/2+1(β_*(d))Id/2(β_*(d))=frac1d, endequation* where I_nu is the modified Bessel function of the first kind. For 0<β≤ β_*(d), the uniform density remains the unique global minimizer up to the linear-stability threshold beginequation* K_#(d)(β)=fracβd/22d/2Gamma(d/2)Id/2(β), endequation* 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 Kc<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.