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We give a deterministic $\tilde{O}(\log n)$-space algorithm for approximatelysolving linear systems given by Laplacians of undirected graphs, andconsequently also approximating hitting times, commute times, and escapeprobabilities for undirected graphs. Previously, such systems were known to besolvable by randomized algorithms using $O(\log n)$ space (Doron, Le Gall, andTa-Shma, 2017) and hence by deterministic algorithms using $O(\log^{3/2} n)$space (Saks and Zhou, FOCS 1995 and JCSS 1999).  Our algorithm combines ideas from time-efficient Laplacian solvers (Spielmanand Teng, STOC `04; Peng and Spielman, STOC `14) with ideas used to show thatUndirected S-T Connectivity is in deterministic logspace (Reingold, STOC `05and JACM `08; Rozenman and Vadhan, RANDOM `05).

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space