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We study the dynamics of a charged tracer particle (TP) on a two-dimensionallattice all sites of which except one (a vacancy) are filled with identicalneutral, hard-core particles. The particles move randomly by exchanging theirpositions with the vacancy, subject to the hard-core exclusion. In case whenthe charged TP experiences a bias due to external electric field ${\bf E}$,(which favors its jumps in the preferential direction), we determine exactlythe limiting probability distribution of the TP position in terms ofappropriate scaling variables and the leading large-N ($n$ being the discretetime) behavior of the TP mean displacement $\bar{{\bf X}}_n$; the latter isshown to obey an anomalous, logarithmic law $|\bar{{\bf X}}_n| = \alpha_0(|{\bfE}|) \ln(n)$. On comparing our results with earlier predictions by Brummelhuisand Hilhorst (J. Stat. Phys. {\bf 53}, 249 (1988)) for the TP diffusivity $D_n$in the unbiased case, we infer that the Einstein relation $\mu_n = \beta D_n$between the TP diffusivity and the mobility $\mu_n = \lim_{|{\bf E}| \to0}(|\bar{{\bf X}}_n|/| {\bf E} |n)$ holds in the leading in $n$ order, despitethe fact that both $D_n$ and $\mu_n$ are not constant but vanish as $n \to\infty$. We also generalize our approach to the situation with very small butfinite vacancy concentration $\rho$, in which case we find a ballistic-type law$|\bar{{\bf X}}_n| = \pi \alpha_0(|{\bf E}|) \rho n$. We demonstrate that here,again, both $D_n$ and $\mu_n$, calculated in the linear in $\rho$approximation, do obey the Einstein relation.Comment: 25 pages, one figure, TeX, submitted to J. Stat. Phy

Ultraslow vacancy-mediated tracer diffusion in two dimensions: The Einstein relation verified