  In this paper we focus on dynamical properties of (real) convex projective
surfaces. Our main theorem provides an asymptotic formula for the number of
free homotopy classes with roughly the same renormalized Hilbert length for two
distinct convex real projective structures. The correlation number in this
asymptotic formula is characterized in terms of their Manhattan curve. We show
that the correlation number is not uniformly bounded away from zero on the
space of pairs of hyperbolic surfaces, answering a question of Schwartz and
Sharp. In contrast, we provide examples of diverging sequences, defined via
cubic rays, along which the correlation number stays larger than a uniform
strictly positive constant. In the last section, we extend the correlation
theorem to Hitchin representations.
