  Let $F$ be a totally real field and $p$ be an odd prime which splits
completely in $F$. We prove that the eigenvariety associated to a definite
quaternion algebra over $F$ satisfies the following property: over a boundary
annulus of the weight space, the eigenvariety is a disjoint union of countably
infinitely many connected components which are finite over the weight space; on
each fixed connected component, the ratios between the $U_\mathfrak{p}$-slopes
of points and the $p$-adic valuations of the $\mathfrak{p}$-parameters are
bounded by explicit numbers, for all primes $\mathfrak{p}$ of $F$ over $p$.
Applying Hansen's $p$-adic interpolation theorem, we are able to transfer our
results to Hilbert modular eigenvarieties. In particular, we prove that on
every irreducible component of Hilbert modular eigenvarieties, as a point moves
towards the boundary, its $U_p$ slope goes to zero. In the case of eigencurves,
this completes the proof of Coleman-Mazur's `halo' conjecture.
