# How to find the critical exponent of some directional dependent correlation length?

I am working on a two dimensional anisotropic system with correlation length diverging with different critical exponent in different directions. And I am wondering if there is any theoretical prediction on what exponent characterize the divergence of the correlation length in some random direction?

More specifically, if $$\nu_x$$ and $$\nu_y$$ characterize the divergence of the correlation length in the x and y direction, (i.e. $$\xi_x \sim t^{-\nu_x}$$ and $$\xi_y \sim t^{-\nu_y}$$), what critical exponent should I expect if I look at the correlation length in some random direction, $$u=cos(\theta)e_x + sin(\theta) e_y$$?

I searched on internet but I haven't found anything. I'll be happy if someone has any good reference on that type of things too.

When you refer to there being two different correlation lengths in two orthogonal directions, I assume what you mean is that the correlation functions take the form $$G(x,y) = \exp\left[ - x/\xi_x - y/\xi_y \right]$$ at long distances (let me know if you have something else in mind). Now, if you consider the decay of correlations along some direction $$u=cos(\theta)e_x + sin(\theta) e_y$$ in space, then the correlation function will decay as $$G(r,\theta) = \exp\left[ - \left(\cos(\theta)/\xi_x + \sin(\theta)/\xi_y \right) r \right],$$ where I'm considering $$\theta$$ fixed and $$r$$ to be the Euclidean distance between the two points being considered for the correlation function. Then we identify the correlation length in the $$u$$ direction as $$\xi_u = \left(\cos(\theta)/\xi_x + \sin(\theta)/\xi_y \right)^{-1}.$$
Now as we approach the critical point, the correlation lengths diverge as $$\xi_x = c_x t^{-\nu_x}$$ and $$\xi_y = c_y t^{-\nu_y}$$ with some non-universal constants $$c_{x,y}$$. Let's say I've chosen coordinates such that $$\nu_x > \nu_y$$. Then we can write $$\xi_u = \left(\cos(\theta)/\xi_x + \sin(\theta)/\xi_y \right)^{-1} = t^{-\nu_y} \left( c_x^{-1} \cos(\theta) t^{\nu_x - \nu_y} + c_y^{-1} \sin(\theta) \right)^{-1}.$$ Since $$\nu_x > \nu_y$$, the quantity in the parentheses smoothly goes to a constant as $$t \rightarrow 0^+$$, and we find that the correlation length diverges as $$\xi_u = \frac{c_y}{\sin \theta} \ t^{-\nu_y}.$$ So if we consider correlation length in an arbitrary direction, it diverges with the smaller of the two critical exponents $$\nu_x$$ and $$\nu_y$$ (unless it is parallel to the direction with the larger $$\nu$$).