Given two hyperbolic structures m and m’ on a closed orientable surface, how many closed curves have m- and m’-length roughly equal to x, as x gets large? Schwartz and Sharp’s correlation theorem answers this question. Their explicit asymptotic formula involves a term exp(Mx) and 0<M<1 is the correlation number of the hyperbolic structures m and m’.
In this talk, we will show that the correlation number can decay to zero as we vary m and m’, answering a question of Schwartz and Sharp. Then, we extend the correlation theorem to the context of higher Teichmuller theory. We find diverging sequences of SL(3,R)-Hitchin representations along which the correlation number stays uniformly bounded away from zero.
This talk is based on joint work with Xian Dai.