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We explore how the characteristics of the cross-correlation functions betweenthe 21cm emission from the spin-flip transition of neutral hydrogen (HI) andearly Lyman-$\alpha$ (Ly$\alpha$) radiation emitting galaxies (Ly$\alpha$emitters, LAEs) depend on the reionisation history and topology and thesimulated volume. For this purpose, we develop an analytic expression for the21cm-LAE cross-correlation function and compare it to results derived fromdifferent Astraeus and 21cmFAST reionisation simulations covering a physicallyplausible range of scenarios where either low-mass ($<10^{9.5}M_\odot$) ormassive ($>10^{9.5}M_\odot$) galaxies drive reionisation. Our key findings are:(i) the negative small-scale ($<2$ cMpc) cross-correlation amplitude scaleswith the intergalactic medium's (IGM) average HI fraction($\langle\chi_\mathrm{HI}\rangle$) and spin-temperature weighted overdensity inneutral regions ($\langle1+\delta\rangle_\mathrm{HI}$); (ii) the inversionpoint of the cross-correlation function traces the peak of the sizedistribution of ionised regions around LAEs; (iii) the cross-correlationamplitude at small scales is sensitive to the reionisation topology, with itsanti-correlation or correlation decreasing the stronger the ionising emissivityof the underlying galaxy population is correlated to the cosmic web gasdistribution (i.e. the more low-mass galaxies drive reionisation); (iv) therequired simulation volume to not underpredict the 21cm-LAE anti-correlationamplitude when the cross-correlation is derived via the cross-power spectrumrises as the size of ionised regions and their variance increases. Our analyticexpression can serve two purposes: to test whether simulation volumes aresufficiently large, and to act as a fitting function when cross-correlatingfuture 21cm signal Square Kilometre Array and LAE galaxy observations.

On the general nature of 21cm-Lyman-$\alpha$ emitters cross-correlations during reionisation