The weak and strong disorder regimes in the continuous random field Ising model

We present a nonperturbative analysis of the weak- and strong-disorder regimes of the continuous random-field Ising model using the distributional zeta-function method. By performing the quenched-disorder average at the level of the effective action, we derive exact quadratic and interaction terms. In the weak-disorder limit, we show that the infrared structure of the two-point correlation functions yields a decomposition of the physical field into correlated components with distinct scaling dimensions. This mechanism exhibits the characteristic $1/p^4$ behavior, which shifts the upper critical dimension to $d_c^{+}=6$. The universal critical behavior of the RFIM near this dimension is governed by a minimal infrared effective action. In the strong-disorder regime, we obtain an exact diagonal quadratic action with a discrete spectrum of massive modes. Here, the absence of massless modes implies the absence of conventional criticality. The resulting spectral representation of correlation functions converges rapidly and remains well controlled in the infrared regime.