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The bright central galaxies (BCGs) dominate the inner portion of the diffusecluster light, but it is still unclear where the intracluster light (ICL) takesover. To investigate the BCG-ICL transition, we stack the images of${\sim}3000$ clusters between $0.2{<}z{<}0.3$ in the SDSS $gri$ bands, andmeasure their BCG+ICL stellar surface mass profile $\Sigma_{*}^{\texttt{B+I}}$down to $3{\times}10^4\,M_{\odot}/\mathrm{kpc}^{2}$ at$R{\simeq}1\,\mathrm{Mpc}$ (${\sim}32$ mag/arcsec$^2$ in the $r$-band). Wedevelop a physically-motivated method to decompose $\Sigma_{*}^{\texttt{B+I}}$into three components, including an inner de Vaucouleurs' profile, an outer ICLthat follows the dark matter distribution measured from weak lensing, and anintriguing transitional component between 70 and 200 kpc. To investigate theorigin of this transition, we split the clusters into two subsamples by theirBCG stellar mass $M_*^{\mathrm{BCG}}$ (mass enclosed roughly within 50 kpc)while making sure they have the same distribution of satellite richness. The$\Sigma_{*}^{\texttt{B+I}}$ profiles of the two subsamples differ by more thana factor of two at $R{<}50\,\mathrm{kpc}$, consistent with their 0.34 dexdifference in $M_*^{\mathrm{BCG}}$, whereas on scales beyond 400 kpc the twoprofiles converge to the same amplitudes, suggesting a satellite-strippingorigin of the outer ICL. Remarkably, however, the discrepancy between the two$\Sigma_{*}^{\texttt{B+I}}$ profiles persist at above $50\%$ level on allscales below 200 kpc, thereby revealing the BCG sphere of influence with radius$R_{\mathrm{SOI}}{\simeq}$ 200 kpc. Finally, we speculate that the surprisinglylarge sphere of influence of the BCG is tied to the elevated escape velocityprofile within $r_s$, the characteristic radius of the dark matter haloes.

The Sphere of Influence of the Bright Central Galaxies in the Diffuse Light of SDSS Clusters