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).
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