Zero-one laws for existential first order sentences of bounded quantifier depth

For any fixed positive integer $k$, let ${\alpha }_k$ denote the smallest $\alpha \in (0,1)$ such that the random graph sequence $\left{G\left(n, n^{-\alpha}\right)\right}$doesnotsatisfythezero-onelawfortheset$$\mathcal{E}${k}$ of all existential first order sentences that are of quantifier depth at most $k$. This paper finds upper and lower bounds on ${\alpha }_k$, showing that as $k\to \mathrm{\infty }$, we have $\alpha{k} = \left(k - 2 - t(k)\right)^{-1}$forsomefunction$t(k) = \Theta(k^{-2})$. We also establish the precise value of ${\alpha }_k$ when $k=4$.