Finite orbits in random subshifts of finite type

For each $n, d \in \mathbb{N}$ and $0 < \alpha < 1$, we define a random subset of $\mathcal{A}^{\{1, 2, \dots, n\}^d}$ by independently including each element with probability $\alpha$ and excluding it with probability $1-\alpha$, and consider the associated random subshift of finite type. Extending results of McGoff and of McGoff and Pavlov, we prove there exists $\alpha_0 = \alpha(d, |\mathcal{A}|) > 0$ such that for $\alpha < \alpha_0$ and with probability tending to $1$ as $n \to \infty$, this random subshift will contain only finitely many elements. In the case $d = 1$, we obtain the best possible such $\alpha_0$, $1/|\mathcal{A}|$.