  Let $\mathcal{M}_g$ be the moduli space of compact hyperbolic surfaces of
genus $g$. Given $X\in \mathcal{M}_g$, let $\ell_X(\gamma_1)$ and
$i(\gamma_1,\gamma_2)$ denote the length and geometric intersection number of a
pair of closed geodesics on $X$, and let $$ I(X):=\sup
\limits_{\gamma_1,\gamma_2}
\frac{i(\gamma_1,\gamma_2)}{\ell_X(\gamma_1)\ell_X(\gamma_2)}. $$ We refer to
$I(X)$ as the {\em interaction strength} on $X$, since it controls the best
upper bound on $i(\gamma_1,\gamma_2)$ in terms of
$\ell_X(\gamma_1)\ell_X(\gamma_2)$.
  It is easy to see that $I(X)<\infty$ and $I(X) \to \infty$ as $X \to \infty$
in $\mathcal{M}_g$. Our main result describes the exact asymptotic behaviour of
$I(X)$ on $\mathcal{M}_g$: we show $$ I(X) \sim
\frac{1}{2\operatorname{sys}(X)\log(\frac{1}{\operatorname{sys}(X)})}, $$ as $X
\to \infty$ in $\mathcal{M}_g$. Here $\operatorname{sys}(X):=\inf
\limits_{\gamma} \ell_X(\gamma)$ denotes the {\em systole} of $X$. We obtain a
similar result for finite volume hyperbolic surfaces, provided we restrict
attention to closed geodesics in a fixed compact subset of $X$. We also show:
$$ \min \limits_{X \in \mathcal{M}_g} I(X) \asymp \frac{1}{(\log g)^2}; $$ in
particular, the minimum of $I(X)$ over $\mathcal{M}_g$ tends to zero as $g \to
\infty$.
