In $1d$ and $2d$, the probability that a simple random walk of length $n$ never returns to the origin asymptotes to $0$ as $n \to \infty$. What is the leading asymptotic behavior of this decay at large $n$?
A nice proof of recurrence in $1d$ and $2d$ is given here, where it's shown that the expected number of returns to the origin is, up to multiplicative constants, $\sim \sqrt{n}$ and $\sim \log(n)$ in $1d$ and $2d$ respectively. With an asymptotically infinite number of returns, the probability of returning must be $1$.
I'm tempted that these expected number of returns roughly point to the asymptotic probability of never returning to the origin in $n$ steps. That is, I anticipate that, up to multiplicative constants, the probability of never returning in a walk of $n$ steps in $1d$ and $2d$ is $\sim \frac{1}{\sqrt{n}}$ and $\sim \frac{1}{\log(n)}$ respectively. This is just a guess, but I show below the $1d$ guess is in fact correct.
One can show that in $1d$, the guess of $\sim \frac{1}{\sqrt{n}}$ up to multiplicative factors is correct for the probability of no returns in $n$ steps. One can find the entire distribution of the number of returns in time $2n$ in $1d$; the probability of $k$ returns is $\frac{\binom{2n-k}{n-k}}{2^{2n}}2^k$. The probability of no returns in $2n$ steps is then $\frac{\binom{2n}{n}}{2^{2n}}$ (which is curiously the probability of returning on exactly the $2n$th step); this probability goes as $\sim \frac{1}{\sqrt{\pi n}}$, confirming the guess.
Thus, only the $2d$ case is left to investigate. At one point, I found a somewhat complicated recursion relation for the exact probability of never returning to the origin in $2n$ steps; I'm going to try to dig that up. However, I anticipate there are simpler methods of estimating the asymptotic probability of no return up to multiplicative constants that won't need to first pass through calculating the exact probability of never returning.
Best Answer
It seems your hunch is correct! One of the result in "Some Problems on Random Walk in Space" by Dvoretzky and Erdös gives a precise estimate. They look at the probability $\gamma_d(n)$ of no returns to the origin in $n$-step random walks in $d$ dimensions. For $d=2$, they compute it precisely and get $$\gamma_2(n) = \frac{\pi}{\log n}+O\left(\frac{\log\log n}{\log^2 n}\right),$$
which grows as you suggested (up to multiplicative constant).