Towards Hyperbolic SIRGs - Refining Exhausting Subspaces

These notes detail an exploration into the first steps that are necessary to extend the framework developed in my bachelor thesis to allow it to handle spatial inhomogeneous random graphs (SIRGs) on Hyperbolic spaces natively, specifically focusing on the immediate changes that must be made to the exhaustive subspaces approach to accommodate hyperbolic spaces. 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.

Brief Overview of the Subspaces Approach

I will briefly recall the classical torus SIRG setting of this paper. Focusing on the one dimensional case, the authors place points uniformly in and connect them according to a SIRG kernel depending only on interpoint distance. For example, an RGG is a SIRG kernel that connects two vertices whenever their Euclidean distance is at most some fixed .

This produces finite random graphs , viewed in the Benjamini-Schramm space. The authors show thatlocally weakly where is the corresponding RGG/SIRG constructed from the unit-intensity Palm-Poisson process on .

Equivalently, increasingly large observation windows approximate the infinite model: the graph seen from a uniformly chosen finite vertex converges in distribution to the graph seen from the Palm root at the origin. If instead one places vertices in each box, the vertex process is exactly a unit-intensity Poisson process on restricted to that box.

My thesis replaces boxes by homogeneous compact approximants, such as tori, because these interact cleanly with Palm theory. In the Euclidean case the box and torus constructions should be asymptotically equivalent, since boundary-wrapping effects become negligible as the volume grows (since it is a Følner style exhaustion of ), although I haven't formally shown this.

The thesis then generalizes this construction through what I call a -admissible exhaustion. Roughly, an exhaustion of a proper metric space must satisfy:

1. exhausts set-wise.
2. The inclusions are measurable.
3. Each is compact and homogeneous under a transitive isometric group action.
4. The metrics on converge suitably to the ambient metric, together with a radial compactness condition.

Under these assumptions, Section 6 establishes local weak convergence of suitable SIRGs on to the corresponding SIRG on .

This extends the torus construction to non-Euclidean examples such as -adic balls in . However, no analogous admissible exhaustion exists for hyperbolic space . Consequently, an important class of hyperbolic random graph models remains outside the framework. The purpose of this article is to sketch an alternative characterization of such approximating sequences that better highlights what gaps still remain in the search for a -compatible theory.

A Cleaner Group-Theoretic Formulation

In the thesis, a distinction was made between -admissible exhaustions (which don't yet care about isometries) and -admissible exhaustions (which do). However, a common theory would likely need to use isometric structure for a few reasons that will be outlined shortly. It is also cleaner for the sake of this article to unite the two cases, so I will present what I expect to be the correct reformulation rather than a set-wise nesting. Below I present a sketch of compact quotient exhaustions:

Given an unbounded¹ proper metric space to serve as the ambient space, a sequence of quotients of subspaces of , given for each by , is called a compact quotient exhaustion if it satisfies the following properties:

1. For all compact , eventually in ;
2. For all , , and these groups are nice² enough to ensure that is a compact metric quotient of ;
3. It holds that .

By injectivity radius here we mean the smallest scale at which the quotient identifications become visible, i.e. the radius before one encounters the nearest non-trivial "wraparound''. More precisely, for we take

with whenever every equivalence class is of the form , as then the quotient is isomorphic to the original space.

Effectively, this is the type of procedure that generates us a sequence of "torus-like" quotients on which to place our finite approximating graphs. To recover the case of the tori in , we use

and to recover the case of the -adic balls we use

For hyperbolic space, we just need to find a sufficiently nice sequence of isometry subgroups. In for example, we can consider satisfying

where

These are discrete, cocompact, properly discontinuous subgroups of the isometry group of , representing principal congruence subgroups of the norm-one group of an order in a quaternion division algebra, embedded as arithmetic Fuchsian groups in . Thus the sequence given by

is a compact quotient exhaustion of .

Notes on the Conditions

This will not be an enumerative list of every role the conditions play, but a quick collection of insights. Let's go one by one. The first condition (in combination with unboundedness of the space) ensures that the sequence grows far enough from the root for boundary effects to not exist. The second condition ensures that the quotients formed are compact metric quotients, which is essential for the SIRG connection rules to work appropriately and be resilient under Palm theory.

The third condition ensures that, asymptotically, balls of any given radius around any point in the quotient space are indistinguishable from balls in the ambient space of the same radius. This is what ensures that quotients can be compared with vertex processes that live in . We can eventually identify large regions of the quotient space with the ambient space and use that to analyze convergence ambiently.

However it should be noted hyperbolic space has no Følner exhaustion, so we cannot asymptotically equate hyperbolic compact quotient exhaustions with their corresponding Dirichlet fundamental domains in the same way we did in . The two will almost certainly have distinct local weak limits.

But Wait! Not Yet!

There is still one large glaring issue. This reformulation does not give us hyperbolic SIRGs. Not even close. The issue is that there is no compact quotient exhaustion of where the quotients are themselves homogeneous spaces, thanks to Proposition 7.7 here, which states that every compact hyperbolic surface has a finite isometry group and therefore cannot admit a transitive isometric action.

I think it is important to view this as something that shortens our work in the other cases. If we instead consider a compact homogeneous quotient exhaustion, essentially adding the condition that every is homogeneous, we find that the tori and -adic balls still fit perfectly fine in this more restricted class. This is why we are able to leverage all of the benefits of homogeneity for Palm theory in this case.

It is likely that the next steps are thus to find a way to transfer uniform root selection from the quotients to the ambient space. Because injectivity radius tends to infinity, our boundary effects diminish, and thus it is possible that some averaged Palm distribution against the intensity of point occurrences or other approach could still give us a fixed base root in the ambient space.

Alternatively, perhaps some sort of kernel-level convergence of the Palm laws

would suffice (under some identification of points , possibly via fundamental domains), where plausibly exists everywhere due to homogeneity of .

Even still, it is clear that this formulation naturally fits hyperbolic spaces better than the original. I am not guaranteeing that it is the correct formulation until I have done the proof work, but it is a better direction to work from.

Footnotes

1. This is needed so that grows away from any center . ↩

2. Typically requires a discrete cocompact properly discontinuous isometry subgroup or something along those lines. ↩