Role of dimensions in first passage of a diffusing particle under stochastic resetting and attractive bias

Recent studies in one dimension have revealed that the temporal advantage rendered by stochastic resetting to diffusing particles in attaining first passage, may be annulled by a sufficiently strong attractive potential. We extend the results to higher dimensions. For a diffusing particle in an attractive potential $V({R})=k {R}^n$, in general $d$ dimensions, we study the critical strength $k = k_c$ above which resetting becomes disadvantageous. The point of continuous transition may be exactly found even in cases where the problem with resetting is not solvable, provided the first two moments of the problem without resetting are known. We find the dimensionless critical strength $\kappa_{c,n}(k_c)$ exactly when $d/n$ and $2/n$ take positive integral values. Also for the limiting case of a box potential (representing $n \to \infty$), and the special case of a logarithmic potential $k \ln\big(\frac{R}{a}\big)$, we find the corresponding transition points $\kappa_{c,\infty}$ and $\kappa_{c,l}$ exactly for any dimension $d$. The asymptotic forms of the critical strengths at large dimensions $d$ are interesting. We show that for the power law potential, for any $n \in (0,\infty)$, the dimensionless critical strength $\kappa_{c,n} \sim d^{\frac{1}{n}}$ at large $d$. For the box potential, asymptotically, $\kappa_{c,\infty} \sim (1 - \ln(\frac{d}{2})/d)$, while for the logarithmic potential, $\kappa_{c,l} \sim d$.