Collision-count second moment for independent path weights ($\mathbb{E}{Z}_t^2={\mathbb{E}}_{{\xi }^{(1)},{\xi }^{(2)}}(1+{\epsilon }^2{)}^{I_t}$)

Here $I_t={\sum }_{j=1}^t{1}_{\{{\xi }_j^{(1)}={\xi }_j^{(2)}\}}$ counts simultaneous collisions of two independent walks. A shared environment sign contributes $\mathbb{E}(1+\epsilon h{)}^2=1+{\epsilon }^2$; distinct signs contribute one. If the difference walk returns to zero with probability $\rho <1$, its total number of positive-time returns has law $\mathbb{P}(I_{\infty }=k)=(1-\rho ){\rho }^k$. Therefore the second moments are bounded by $(1-\rho )/(1-\rho (1+{\epsilon }^2))$ whenever $\rho (1+{\epsilon }^2)<1$. Convergence in L2 then preserves the limit's mean one.