  We provide a deterministic $\tilde{O}(\log N)$-space algorithm for estimating
random walk probabilities on undirected graphs, and more generally Eulerian
directed graphs, to within inverse polynomial additive error
($\epsilon=1/\mathrm{poly}(N)$) where $N$ is the length of the input.
Previously, this problem was known to be solvable by a randomized algorithm
using space $O(\log N)$ (following Aleliunas et al., FOCS 79) and by a
deterministic algorithm using space $O(\log^{3/2} N)$ (Saks and Zhou, FOCS 95
and JCSS 99), both of which held for arbitrary directed graphs but had not been
improved even for undirected graphs. We also give improvements on the space
complexity of both of these previous algorithms for non-Eulerian directed
graphs when the error is negligible ($\epsilon=1/N^{\omega(1)}$), generalizing
what Hoza and Zuckerman (FOCS 18) recently showed for the special case of
distinguishing whether a random walk probability is $0$ or greater than
$\epsilon$.
  We achieve these results by giving new reductions between powering Eulerian
random-walk matrices and inverting Eulerian Laplacian matrices, providing a new
notion of spectral approximation for Eulerian graphs that is preserved under
powering, and giving the first deterministic $\tilde{O}(\log N)$-space
algorithm for inverting Eulerian Laplacian matrices. The latter algorithm
builds on the work of Murtagh et al. (FOCS 17) that gave a deterministic
$\tilde{O}(\log N)$-space algorithm for inverting undirected Laplacian
matrices, and the work of Cohen et al. (FOCS 19) that gave a randomized
$\tilde{O}(N)$-time algorithm for inverting Eulerian Laplacian matrices. A
running theme throughout these contributions is an analysis of "cycle-lifted
graphs", where we take a graph and "lift" it to a new graph whose adjacency
matrix is the tensor product of the original adjacency matrix and a directed
cycle (or variants of one).
