p.4 -4: t_p must be R - r_p

--- done

p.10 -12: A_u may also contain points with distance > R from u, I think this
should be made
clear (but this causes no problems of course).

--- done

p.10 What is B_{i_{u_0}} in Step 1 of Phase 1? [You probably mean
B_{C_\lambda}^{i_0}]

--- done

p.10 Step 3: there is probably no vertex with type t_{j_0-1}; here you take
u_1 as an
auxiliary point to continue the process - this should be made clearer. Maybe
you could
also write more concisely: Let u_1 = (t_{j_0-1}, theta_u_0 + Thetahat).
Similar in Step 5.

--- done

Phase I of the exploration process should be defined more clearly, as it is
crucial for the
rest of the analysis. Also the notation is quite heavy. For example, you
could avoid using
u_0 and u_1 (since u_2, u_3,... is not needed) and the associated indexing,
and use just
u and v instead. Alternatively, as it is convinient later in the proofs, you
may want to
index the sequence of generated points by u_0, u_1, u_2, ...

--- changed the indices to v,w

To increase readability you should move Def 3.2 to an earlier place in the
section - then
the algorithm makes immediately more sense. Moreover, you could already say
at this point
that the "width" of such a bounding path essentially determined the number
of vertices
in the component.

--- done

p.12 The definition of Theta(u) should be repeated.

--- done

p.13 -8. Here it not clear p_t^2N vanishes, add this a few lines above when
you
define p_t^{(i_0)} (as you do later for example with p_II)

--- changed order of paragraphs to make this clear

p.16 before (3.4). I don't see formally where the bound on E[|B|] comes
from. Conditionally
on the type being \le t_0 the probability that \Theta(u) is large might
increase.

--- removed the condition of having type \leq t_0 -- this was not necessary.

The ``slicing'' of the disc in Section 4, does it allow you to get a bound
on the diameter
of the largest component?

--- not directly, there will be vertices in the same component that we do not account for. We only get a bound on the diameter of some subgraph of the largest component, which we use to bound the size of the largest one.

p.21 Denoting (4.20) as the event {\cal N} is somehow confusing, since you
haev the random
set {\cal N}_i.

--- chaged notation

After (4.28): why is the change bounded by \theta^{(i)}/\ell_{i-1}?

--- added a sentence to make this clear

p. 37 A stronger assumption seems necessary: \nu > 24\pi. Please check this
(especially when
bounding E[Y_1^2]), this might have some implication to the main result.

--- changed proof to correct assumption (\nu>20\pi suffices)