Let $S_n^1,\dots,S^d_n$ be $d$ mutually independent simple one-dimensional random walks on $\mathbb{Z}$ with $x=S_0^1=\dots=S_0^d$. In each time step each of the $d$ random walks moves, independently of the others, either $+1$ or $-1$ with probability $\frac{1}{2}$. Let $T_x$ denote the number of steps before at least one of the $d$ random walks hits $0$ for the first time. What is $E(T_x)$?
For $d=1$ this becomes the standard one-dimensional random walk and thus $E(T_x)=\infty$ even for $x=1$. If $d$ were infinite one could argue that after $x$ steps infinitely many random walks would have a.s. taken $-1$ step $x$-times and thus $E(T_x)=x$. I know that for finite $d\geq 3$ it holds $ax^{2-\epsilon}\leq E(T_x)\leq bx^{2+\epsilon} $ for some constants $a,b,\epsilon>0$ and all sufficiently large $x$. What can we tell about $E(T_x)$ for $d=2$?
