Diffusion with stochastic resetting on a lattice

Abstract

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 {\em 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 $r\to 0$, $R_0=|{\vec{R}}_0|\to \infty$ 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 is 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 neighbour 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 $\varphi =(|m_1|+|m_2|+\dots +|m_d|)-1$ that depends on the starting point ${\vec{R}}_0=a(m_1,m_2,\dots ,m_d)$ 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 neighbour 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. 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.

17 pages, 7 figures, data gnuplot files for plots available at https://doi.org/10.57782/VGCHTI

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Cited by in corpus (2)

• Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols
• Tethering effects on first-passage variables of lattice random walks in linear and quadratic focal point potentials