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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.

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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 $\nu$).

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