What is the non-asymptotic upper bound for the leading eigenvector of the random matrix?

Question

Fix a Gaussian random matrix $A$ with $E[A_{ij}]=0$ for $i, j=1,\dots n$ and $E[A_{ij}^2]=\frac{1}{n}$. Let $v_1$ be the leading eigenvector of $A$. What is the non-asymptotic upper bound for $v_1$, that is something like $$ P(\|v_1\|_2\ge t)\le e^{-\alpha t} $$

Is there any reference for this tail probability?