Conformal theory of the two-dimensional Ising model with homogeneous boundary conditions and with disordred boundary fields

The critical Ising model defined on the upper half plane is studied with conformal-invariance methods. For homogeneous free and fixed-spin boundary conditions we derive all the many-point correlation functions 〈σ₁{{{σ}}}₂$. . .${{{σ}}}ₙ$〉 and 〈${{{ε}}}₁ε₂. . .εₙ〉 of the spin {σ} and the energy density {ε}. We also treat the case of inhomogeneous boundary fields that orient the spins on the x axis up for ζ₁x{∞}, down for ζ₂xζ₁, up for ζ₃xζ₂, etc. Exact expressions for the correlation functions 〈{σ}〉_ζ1{{{ζ}}}₂$. . .${{{ζ}}}ₘ$, 〈${{{σ}}}₁σ₂{〉}_{{{ζ}}1}ζ₂. . .ζₘ, and 〈ε₁{{{ε}}}₂$. . .${{{ε}}}ₙ〉_ζ1{{{ζ}}}₂$. . .${{{ζ}}}ₘ$ are obtained. Examples of droplet shapes for random ${{{ζ}}}ᵢ$ are shown.