Commuting Involutions in Finite Simple Groups

We prove that if $G$ is a finite simple group and $x,y\in G$ are involutions, then $\mathrm{\mid }{x}^G\cap C_G(y)\mathrm{\mid }\to \mathrm{\infty }$ as $\mathrm{\mid }G\mathrm{\mid }\to \mathrm{\infty }$. This extends results of Guralnick-Robinson and Skresanov. We also prove a related result about $C_G(t)\mathrm{/}O(C_G(t))$ that does not require the classification of finite simple groups.