The ag-ensemble probability bound

Let $(\Gamma,\nu)$ be the ag-ensemble, with parameters $\alpha$, $\gamma$, and $D$, and let $X^i_n$ denote the relevant random variable for the ensemble. The ag-ensemble probability bound. For any $\alpha$, $\gamma$, and $D$, there exists a constant $c>0$ such that

$$\nu(\{X^i_n>0\})\leq \frac{c}{n}(1+o(1)).$$

This bound is the missing ingredient needed to extend the almost-sure quenched Hausdorff and spectral dimension results for the ag-model from the proven parameter ranges to the full range of $\alpha$, $\gamma$, and $D$.