  We study the regularity of segregated profiles arising from competition -
diffusion models, where the diffusion process is of nonlocal type and is driven
by the fractional Laplacian of power $s \in (0,1)$. Among others, our results
apply to the regularity of the densities of an optimal partition problem
involving the eigenvalues of the fractional Laplacian. More precisely, we show
$C^{0,\alpha^*}$ regularity of the density, where the exponent $\alpha^*$ is
explicit and is given by \begin{equation*} \alpha^* = \begin{cases} s &
\text{for $s \in (0,1/2]$}\\ 2s-1 &\text{for $s \in (1/2,1]$}.\end{cases}
\end{equation*} Under some additional assumptions, we then show that solutions
are $C^{0,s}$. These results are optimal in the class of H\"older continuous
functions. Thus, we find a complete correspondence with known results in case
of the standard Laplacian.
