We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a $d$-dimensional hypercubic lattice starting from a fixed initial position ${\vec{R}}_0$ and resetting to ${\vec{R}}_0$ with a rate $r$. Previously known results in the continuous space are recovered in the scaling limit $r\to 0, R_0=\left|{\vec{R}}_0\right|\to \infty$ with the product $\sqrt{r}{R}_0$ fixed. However, our formula is valid for any $r$ and any ${\vec{R}}_0$ that enables us to explore a much wider region of the parameter space that are inaccessible in the continuum limit. For example, we have shown that the MFPT, as a function of $r$ for fixed ${\vec{R}}_0$, diverges in the two opposite limits $r\to 0$ and $r\to \infty$ with a unique minimum in between, provided the starting point is not a nearest neighbor of the target. In this case, the MFPT diverges as a power law $\sim r^{\varphi }$ as $r\to \infty$, but very interestingly with an exponent $\varphi =\left(\right|m_1|+|m_2|+...+|m_d\left|\right)-1$ that depends on the starting point ${\vec{R}}_0=a(m_1,m_2,...,m_d)$ where $a$ is the lattice spacing and $m_i$'s are integers. If, on the other hand, the starting point happens to be a nearest neighbor of the target, then the MFPT decreases monotonically with increasing $r$, approaching a universal limiting value 1 as $r\to \infty$, indicating that the optimal resetting rate in this case is infinity. We provide a simple physical reason and a simple Markov-chain explanation behind this somewhat unexpected universal result. These interesting results on a lattice are not captured by the continuum theory. Our analytical predictions are verified in numerical simulations on lattices up to 50 dimensions. Finally, in the absence of a target, we also compute exactly the position distribution of the walker in the nonequlibrium stationary state that also displays interesting lattice effects not captured by the continuum theory.

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• Random walks
• Lattices
• Square lattice
• Green's function methods
• Monte Carlo methods