Linear theory of random textures of $^3$He-A in an aerogel

The spatial variation of the orbital part of the order parameter of $^3$He-A in an aerogel has been represented as random walk of a unit vector $\mathbf{l(r)}$ over a sphere under the action of random anisotropy created by the system of strands of the aerogel. The statistical properties of the resulting random texture have been studied. For distances at which the variation of $\mathbf{l}$ is much smaller than its magnitude, the average square $\langle\delta\mathbf{l}^2\rangle$ of variation of $\mathbf{l}$ has been expressed in terms of the correlation function of the component of the random anisotropy tensor. Under simplifying assumptions on the structure of this correlation function, an analytical dependence of $\langle\delta\mathbf{l}^2\rangle$ on $r$ has been obtained for isotropic and axially anisotropic aerogels. The average values of the squares of the projections of $\mathbf{l}$ on the axes of anisotropy for an anisotropic aerogel have been represented in terms of the parameters of the aerogel. The characteristic scale at which the long-range order is broken, as well as the magnitude of global anisotropy sufficient for the recovery of the long-range order, has been numerically estimated within a simple model. The values obtained have been compared to other estimates.