= Collision-count second moment for independent path weights
{title2=$\mathbb E Z_t^2=\mathbb E_{\xi^{(1)},\xi^{(2)}}(1+\varepsilon^2)^{I_t}$}
Here $I_t=\sum_{j=1}^t\mathbf1_{\{\xi_j^{(1)}=\xi_j^{(2)}\}}$ counts simultaneous collisions of two <independent> walks. A shared environment sign contributes $\mathbb E(1+\varepsilon h)^2=1+\varepsilon^2$; 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 $(1-\rho)/(1-\rho(1+\varepsilon^2))$ whenever $\rho(1+\varepsilon^2)<1$. <Convergence in L2> then preserves the limit's <mean> one.
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