These notes detail an exploration into the steps that are necessary to extend the framework developed in my bachelor thesis to reduce the complexity of the radial tightness condition. They have been designed with care but I do not claim full formal correctness of each and every detail. This document should be seen as a sketch into a possible direction that could be formally pursued at a later point.
A Fix For Radial Tightness
As originally laid out in the thesis, radial tightness has this form:
For every , , there exists a compact set and an index , such that for every it holds that
This is a fine starting point, but it has a few issues. Firstly, it conflates genuine non-escape of graph neighbourhoods with tightness of the root point. Specifically, once we have vague convergence
we already know that must be a tight sequence (in the sense of Prokhorov). Hence, what we really care about is that the radius required to contain spatial vertices remains tight. Specifically, we need the following:
For every , the sequence
is tight, where
essentially the smallest spatial radius for which the spatial ball of the root contains the radius graph ball of the root.
Necessarily, once vague convergence and hence tightness of is assumed, this condition is equivalent to the old radial tightness, as we can union the necessary compact set for root containment with the necessary compact set for radial containment at every and .
We can for now call the new condition inter-radial tightness, as it is tightness of the radial relation between spatial and graph forms, but perhaps it can assume the old name if incorporated into the published version of the paper.
What is interesting is that now, we can have sequences that do not converge vaguely still being radially-tight. Indeed, I believe this is clearer. For example, in the local weak setup, it is perfectly reasonable for
to be inter-radially tight, even though the sequence generally has no vague limit. It would even be plausible to conjecture that one could show that local weak convergence occurs for the sequence under this inter-radial tightness, without necessarily constructing the corresponding local weak limit object. In that case, the convergence issue would be separated from the limit representation issue. The former could exist perfectly fine without the Palm machinery developed in the thesis, and it gives a clearer explanation as to how the Palm machinery is useful representationally.
Simplifying Further - Bounds and Conditions
I want to go more in depth on the uniform rooting case of the sequence
This case is very useful because, currently speculatively, it may be possible to reduce the tightness of this inter-radial observable to the tightness of two far simpler observables. These are specified below. The first is the local spatial population within radius , represented by
This counts the number of vertices contained within a spatial ball of radius at the origin. The second observable is the edge distance observable, given by
which determines the length of the longest edge incident to the root. I would be willing to conjecture that something along the lines of the following may hold:
If
is tight for every fixed , and
is tight, then for every , it holds that
If this did indeed hold, then it would greatly reduce the conditions needed to evaluate inter-radial tightness. This is because the local spatial population is effectively tight by default, since our approximating sequences always remain boundedly-finite in the point process space. Specifically , without considering the root, already gives tightness of this when the members and the limit are boundedly finite and simple, under suitable stationarity conditions. That effectively means that we would only need to check tightness of the edge distance distribution around the root point, which is dramatically easier to check than the original radial tightness condition, especially in SIRG-type models.
Specifically, I expect that the radial tightness condition around the Palm root could be reduced from its current form of
eventually uniformly in to something similar to
eventually uniformly in , where denotes the intensity measure of the reduced Palm vertex process. For the notation details see Theorem 6.4 in the Thesis and the surrounding context. As a small note, I expect that the fibre-finiteness condition from there can be dropped entirely as I am now of the opinion that radial tightness (or inter-radial tightness) effectively guarantee local finiteness of the limiting graph object.
This condition would suffice because the intensity measure is sufficient to compute the edges incident to the root, whereas before the second factorial moment measure was needed to compare edges incident to every possible point. It would then be immediately clear that bounding
would give the standard form
where , i.e. the volume occupied by a radius ball with respect to the ambient left Haar measure on the homogeneous space , which immediately reduces to the familiar
bound for Euclidian models, with .
Why I Believe This Conjecture
The main difficulty is that tightness of only controls long edges incident to a uniformly sampled vertex. It does not immediately rule out the possibility that a very small collection of exceptional vertices has extremely long edges and that such vertices are nevertheless likely to occur somewhere inside the graph neighborhood of a typical root. However, it appears to me that the local population assumption rules out precisely this obstruction.
Specifically, we can consider the event that a vertex is "bad", and collect these vertices into a set denoted by
Then tightness of immediately gives that
goes to in . So we know that for a sufficiently long length , an arbitrary small amount of vertices are expected to have an edge longer than . However we still need to show that a typical root is also unlikely to lie near one of these rare bad vertices.
This is where a bound such as the following comes in, for every and :
This effectively says the probability that the root is within distance of the bad set is bounded by the probability of the root having an abnormally dense -neighborhood, plus times the proportion of bad vertices (throughout the entire collection of vertices).
Specifically, for any two vertices within distance of the same bad vertex , we know by triangle inequality that , and thus all vertices within of must live within a ball of respectively. This is why we care about the density of the neighborhood of the uniformly chosen root.
To visualize this, imagine a realization , so that the uniform root selection is the only remaining randomness. Then we can split the possible roots into two classes. The first would be , which we for now give up trying to control, and this is reflected by the first probability. The second would be , in which case a single bad vertex can be within of at most roots, since by the earlier triangle inequality any two such roots have .
Thus at most low-density roots can be within of a bad vertex in this realization. Dividing by the total number of vertices gives
and since the root is uniform it holds that
The bound is then achieved by summing over these mutually exclusive cases, in a sort of union bound style.
I expect that inter-radial tightness could then be derived from this bound via induction over the graph radii , with the tightness properties of and as used in the bound letting us make the probability of encountering an -bad vertex inside the current spatially bounded graph neighborhood arbitrarily small. This would then control the additional spatial displacement incurred when passing from radius to radius , allowing tightness of to propagate inductively to .
Of course, this is not a formal argument, and is more a sketch of my intuition, but I do think it is plausible enough to devote time to attacking as a conjecture. Definitely will be considered for developments in the preprint.