Diffusion with stochastic resetting on a lattice

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 R[over ⃗]_{0} and resetting to R[over ⃗]_{0} with a rate r. Previously known results in the continuous space are recovered in the scaling limit r→0, R_{0}=|R[over ⃗]_{0}|→∞ with the product sqrt[r]R_{0} fixed. However, our formula is valid for any r and any R[over ⃗]_{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 R[over ⃗]_{0}, diverges in the two opposite limits r→0 and r→∞ 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 ∼r^{ϕ} as r→∞, but very interestingly with an exponent ϕ=(|m_{1}|+|m_{2}|+...+|m_{d}|)-1 that depends on the starting point R[over ⃗]_{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→∞, 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.