234 R. Brooks
of bottle necks. There are also some more problems we have to worry about
as well, as evidenced in the following picture:
But the heart of the idea is to use the result of Theorem 7.1 to see that
these problems only arise late in the game, when the (logarithmically many)
bottlenecks may be large compared to the number of remaining edges. But if
we have gotten to that late in the game, then the LHT path is already quite
long.
The constant 1/πshould be thought of as (1/3)·(3π) , where the second
term arises because each fundamental domain has areaπ/3. The term 1/ 3
should be thought of as the 2/3 arising in the spaghetti model argument,
divided by 2 since each edge has two adjoining LHT segments.
The constant is not sharp for a variety of reasons: first of all, because
we are measuring the expected length starting from a given LHT segment,
rather than the expected length of the longest LHT path. Second, when we
are counting the length of the LHT path we are building, we assume that at
each step the length increases by one this is what happens in the spaghetti
model. But in fact, at each step we increase the length not just of the LHT
path we are measuring, but also another LHT path. So as a point of fact the
length of the LHT path we are building is really increasing much faster than
we are counting.
This completes the sketch of the proof of Theorem 7.2.
12.9 An Annotated Bibliography
As mentioned in the introduction, the work discussed in these notes is spread
out over a large number of papers. This happened because my thinking about
this topic underwent a rather long development, during which different facets
of the picture emerged. In what follows, I give an annotated guide to the
papers I have written on the subject, together with coauthors. I have decided
not to give a comprehensive bibliography on the subject here. All of these
papers are available at my website, http://www.math.technion.ac.il/ rbrooks,
except when they are old.
- [1]. contains an announcement of the results contained in [2], [3], and early
versions of [4]. - [5]. This paper gives the theorem connecting the bottom of the spectrum
of a covering and the Cheeger constant of the corresponding graph. - [6]. This paper gives the compactification techniques using the Ahlfors–
Schwarz lemma. It also gives a presentation of the Platonic graphs and
their corresponding surfaces. - [4]. This paper studies the process of building a surface at random by
choosing a three-regular graph at random. This paper has gone through a
number of different versions, the latest version includes results concerning
the expected genus and the expected largest embedded ball of a randomly
chosen Riemann surface.