Consider simple random walk in $\mathbb{R}^d$, let $\sigma=\inf\{t>0:X_t=0\}$ be the return time. How to estimate the asymptotic lower bound of $P(\sigma=2k)$ given $X_0=0$?
I've got the upper bound $P(\sigma=2k)\le P(S_{2k}=0)=O(k^{-d/2})$, and I tried to construct a bijection between some trajectories that pass through the origin multiple times and orbits that do not pass through the origin.
eg. If a orbit turns at the origin at time $t$, we can change $X_{t-1}$ and $X_t$ so that the new orbit doesn't pass the origin. And if a orbit passes the origin several times with different direction at time $t$ and $s$, we can flip $X_t, X_{t+1},... X_{s-1}$ so the new orbit turns at time $t$ and $s$.
But it's hard to make it clear when an orbit passes the origin several times with the same direction. And I don't know whether my trial is correct.
And I suppose the lower bound is also $O(k^{-d/2})$ since the random walk is strongly transient iff $d\ge 5$.