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The time it takes a random walker in a lattice to reach the origin fromanother vertex $x$, has infinite mean. If the walker can restart the walk at$x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized overrestarting strategies) is finite; it was called the ``grade'' of $x$ byDumitriu, Tetali and Winkler. They showed that, in a more general setting, thegrade (a variant of the ``Gittins index'') plays a crucial role in controlproblems involving several Markov chains. Here we establish several conjecturesof Dumitriu et al on the asymptotics of the grade in Euclidean lattices. Inparticular, we show that in the planar square lattice, $T(x,0)$ is asymptoticto $2|x|^2\log|x|$ as $|x| \to \infty$. The proof hinges on the local varianceof the potential kernel $h$ being almost constant on the level sets of $h$. Wealso show how the same method yields precise second order asymptotics forhitting times of a random walk (without restarts) in a lattice disk.

Hitting times for random walks with restarts