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In this work we establish the Central Limit Theorem for the hierarchical O(N) Heisenberg model at criticality (system at inverse critical temperature βc) via partial differential equation (obtained in the local potential approximation (L ↓ 1) of renormalization group transformation) u (N) t = 2 N xu xx + u (N) x − 2x ( u x )2 − γxu x + du − u x (t, 0) with initial condition u(0, x) = − 1 N ln [ Γ(N/2) ( i √ βxN/2 )N/2−1JN/2−1 ( i √ βxN )] , in the limit N → ∞ (continuum (L ↓ 1) hierarchical spherical (N = ∞) model). For simplicity we only treat the d = 4 case but the theorem is still valid for d > 4. By studying a given partial differential equation (PDE) we determine for any d > 2 the critical inverse temperature βc(d) of the continuum hierarchical spherical model, and we show a connection between criticality and the fixed point of PDE. By means of a geometric analysis of the critical trajectory {u x (t, x), β = βc, t ≥ 0} we obtain some informations about Lee-Yang zeros’s dynamics and distribution. Finally, we show u (∞) x (0, x) is in Pick class P of functions; we verify indirectly that the flow u (∞) xt (t, x) preserves that class, i. e., u (∞) x (t, x) ∈ P for all t > 0.

Teorema Central do Limite para o modelo O(N) de Heisenberg hierárquico na criticalidade e o papel do limite N -&gt; infinito na dinâmica dos zeros de Lee-Yang