Slides for SPP2265 - Münster

Slides (PDF) ↗ - The slides from the presentation.

Slides Source Code (GitHub) ↗ - This includes SIRG generation scripts for the visuals I used in my slides. I give no formal guarantee of the script's correctness, but I believe it is a decent starting point.

Overview

This presentation will be given during my talk at the University of Münster. Below I go slide by slide covering technical details and references.

Title Slide

Paper in the works with Pim van der Hoorn ↗, while the thesis was entirely my own work.

Motivation

This is a very high level slide so the more technical caveats will be covered later. The type of convergence we care about here is local weak convergence, see van der Hofstad, Section 2.4.5 ↗, and the point process representations are adjacency processes.

Background - Spatial Random Graphs in

The examples shown here are legitimate models that are approachable in our framework, with the SIRG one specifically chosen for contrast against the RGG. For information on SIRGs see van der Hofstad, van der Hoorn, and Maitra ↗ and Sánchez Flores's point-process treatment ↗.

The here are most often given by an i.i.d. mark distribution in some mark space, and represent auxiliary randomness. They don't necessarily have to be i.i.d., but the machinery cannot handle far from the i.i.d. case in any meaningful way.

Local Weak Convergence

The main thing I want to note here is the choice behind using

instead of the more standard treatment of

or something along those lines. The reasoning for this is that we want to treat local weak convergence as a special type of weak convergence (convergence in distribution) within the Benjamini-Schramm space, see Benjamini and Schramm ↗. Importantly, is doing the specialization here. The more general case allows to be any root distribution selected measurably from graph realizations, and it turns out that lifting that general convergence

to

is actually a meaningfully simpler starting point. This is exactly what Theorem 4.1 in the bachelor thesis covers. It is just lifting convergence in distribution from one topology to convergence in distribution in another topology via a continuous-ish construction map.

The specialization to uniform rooting is exactly where the Palm machinery comes in.

The Setup

For more details on the adjacency measure construction see Section 3 of the thesis. Again the convergence lift question is restated here to remind the audience what they should be paying attention to.

Constructing Adjacency Processes

Here, it is best to read the discussion around the start of Section 6 in the thesis, specifically around the role of measurable edge constructions and the vertex -> connection mechanism philosophy.

Continuous-ish here refers to the a.e. continuity of the connection function with respect to a special set dependent on , where gives . This set is often a null set for most models, hence a.e. continuity with respect to the lebesgue measure is sufficient.

The Classical Case - Binomial in

See Section 7.3.2 in the thesis for a discussion on this case, and van der Hofstad, van der Hoorn, and Maitra ↗ for the original result.

Here is the Palm distribution at 0, which we know exists due to homogeneity of the exhausting tori. This is part of why having our exhausting subspaces homogeneous is necessary for Palm theory.

The Main Theorem

This is just Theorem 6.4 of the thesis. All the caveats can be found there and in the surrounding discussion.

Novel Results - More Vertex Processes

These can be found in Section 7 of the thesis. I included the speculative examples in red because Theorem 6.4 is checklist-ish enough to where I consider these examples to be completely manageable. While I haven't explored specific lemmas for them, I am confident that the best approach to tackle them would be from the starting point of Theorem 6.4 rather than through independent investigation.

Novel Results - Non-Euclidian Spaces

These can also be found in Section 7 of the thesis. I didn't include it in the presentation, but atomic spaces such as are also plausible here.

For a short discussion on the hyperbolic problem, see here.

How YOU Can Benefit

This more or less just covers the discussion in Section 8 of the thesis and the general modular philosophy of the research project.

Next Steps

Nothing here is promised for the preprint, but it is an outline of our intentions.