Shared Dynamically-Small Points for Polynomials on Average

Given two rational maps $f,g:{\mathbb{P}}^1\to {\mathbb{P}}^1$ of degree $d$ over $\mathbb{C}$, DeMarco-Krieger-Ye [DKY22] has conjectured that there should be a uniform bound $B=B(d)>0$ such that either they have at most $B$ common preperiodic points or they have the same set of preperiodic points. We study their conjecture from a statistical perspective and prove that the average number of shared preperiodic points is zero for monic polynomials of degree $d\ge 6$ with rational coefficients. We also investigate the quantity ${\mathrm{lim\hspace{0.17em}inf}}_{x\in \bar{\mathbb{Q}}}({\widehat{h}}_f(x)+{\widehat{h}}_g(x))$ for a generic pair of polynomials and prove both lower and upper bounds for it.