Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limit

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence results for the counting measure of types of components of each $k$-skeleton of a random geometric complex. As a consequence, in the case of the $1$-skeleton (i.e. for random geometric graphs) we show that the empirical spectral measure associated to the normalized Laplace operator converges to a deterministic measure.

1. Introduction 1.1. Random geometric complexes. The subject of random geometric complexes has recently attracted a lot of attention, with a special focus on the study of expectation of topological properties of these complexes [Kah11b, Pen03, NSW08, YSA17, BM15, BK18] 1 (e.g. number of connected components, or more generally Betti numbers). In a recent paper [ALL18], Auffinger, Lerario and Lundberg have imported methods from [NS16,SW19] for the study of finer properties of these random complexes, namely the distribution of the homotopy types of the connected components of the complex. Before moving to the content of the current paper, let us discuss the main ideas from [ALL18] and introduce some terminology.
On a compact Riemannian manifold (M, g) of dimension m, we denote byB(x, r) ⊂ M the Riemannian ball centered at x of radius r > 0 and we construct a random M -geometric complex in the thermodynamic regime as follows. We let {p 1 , . . . , p n } be a set of points independently sampled from the uniform distribution on M , we fix a positive number α > 0, and we consider: (p k , r) where r := αn −1/m . (1.1) The choice of such r is what defines the so-called critical or thermodynamic regime 2 and it's the regime where topology is the richest [ALL18,Kah11a]. We say that U n is a random M -geometric complex and the name is motivated by the fact that, for n large enough, U n is homotopy equivalent to itsČech complex, as we shall see in Lemma 2.4 below. This list is by no mean complete! 2 Quoting the Introduction from [ALL18]: random geometric complexes are studied within three main phases or regimes based on the relation between density of points and radius of the neighborhoods determining the complex: the subcritical regime (or "dust phase") where there are many connected components with little topology, the critical regime (or "thermodynamic regime") where topology is the richest (and where the percolation threshold appears), and the supercritical regime where the connectivity threshold appears. The thermodynamic regime is seen to have the most intricate topology. Figure 1. The unknot and the trefoil knot are homotopy equivalent but they are not isotopic.
Auffinger, Lerario and Lundberg [ALL18] proved that, in the case when vol M = 1, the normalized counting measure of connected components of such complexes, counted according to homotopy type, converges in probability to a deterministic measure. That is, where the sum is over all connected components u of U n , [u] denotes their homotopy type and b 0 is the zero-th Betti number, therefore b 0 (U n ) is the number of connected components of U n . In (1.2) the measureΘ n is a random probability measure on the countable set of all possible homotopy types of connected geometric complexes and the convergence is in probability with respect to the total variation distance (see Section 4 for more precise definitions). The support of the limiting deterministic measureΘ equals the set of all homotopy types for Euclidean geometric complexes of dimension m = dim M . Roughly speaking, (1.2) tells that, for every fixed homotopy type [u] of connected geometric complexes, denoting by N n ([u]) the random variable "number of connected components of U n which are in the homotopy equivalence class [u]", there is a convergence of the random variable N n ([u])/b 0 (U n ) to a constant c [u] as n → ∞ (the convergence is in L 1 and c [u] > 0 if and only if [u] contains a R m -geometric complex).
1.2. Isotopy classes of geometric complexes. Let us move now to the content of the current paper. Our first goal is to include the results of [ALL18] into a more general framework which allows to make even more refined counts (e.g. according to the type of the embedding of the components, or on the structure of their skeleta, or on the property of containing a given motif 3 ). The first result that we prove is that (1.2) still holds if we consider isotopy classes instead of homotopy classes: intuitively, two complexes are isotopic if the vertices of one can be moved continuously to the vertices of the other without ever changing the combinatorics of the intersection of the corresponding balls (see Definition 2.5). From now on we will always make the assumption that our complexes are nondegenerate, i.e. that the boundaries of the balls defining them intersect transversely (see Definition 2.3); our random geometric complexes will be nondegenerate with probability one, and the notion of isotopic nondegenerate complexes is the one from differential topology. In Theorem 4.2 we show that 3 A motif in a graph (or more generally in a complex) is a recurrent and statistically significant sub-graph or pattern.
where Θ n is defined in a similar way asΘ n above, with isotopy classes instead of homotopy classes 4 . Interestingly, the limiting measure depends only on the volume of M and its dimension. To appreciate the difference with the results from [ALL18]: the unknot and the trefoil knot in R 3 (Figure 1) are homotopy equivalent but they are not isotopic, and with positive probability there are connected M -geometric complexes whose embedding looks like these two knots (see Proposition 4.5 below); Theorem 4.2 is able to distinguish between them, whereas the construction from [ALL18] is not.
1.3. A cascade of measures. Theorem 4.2 contains in a sense the richest possible information on the topological structure of our geometric complexes and the convergence of many other counting measures can be deduced from it. To explain this idea, let us consider the space (G/ ∼ =) := {isotopy classes of connected geometric complexes} and let us put an equivalence relation ρ on G/ ∼ = (the relation can be for example: two isotopy classes are the same if their k-skeleta are isomorphic, or if they contain the same number of a given motif). Then the natural map ψ : (G/ ∼ =) → (G/ ∼ =) /ρ defines the random pushforward measure ψ * Θ n on (G/ ∼ =) /ρ and Theorem 4.2 implies that ψ * Θ n → ψ * Θ.
This idea can be used to produce a "cascade" of random relevant measures. Consider in fact the following diagram of maps and spaces: where the spaces are: and the maps are the natural "forgetful" maps. For example, the map ϕ takes the isotopy class of a nondegenerate complex and associates to it its homotopy class; the map ϕ (k) associates to it the isomorphism class of its k-skeleton (it is well defined since isotopic complexs have isomorphicČech complexes). Then for all the pushforward measures defined by these maps we have convergence in probability with respect to the total variation distance (see Section 4) and as n → ∞ 4 It is natural to normalize the volume of M , however in this paper we don't to it to emphasize that the limiting measures do have some dependence on the geometry of M .
1.4. Random geometric graphs. A case of special interest is the case of random geometric graphs: vertices of a random M -geometric graph Γ n are the points {p 1 , . . . , p n } and we put an edge between p i and p j if and only if i = j andB(p i , r) ∩B(p j , r) = ∅. Using the above language, a random M -geometric graph is the 1-skeleton of theČech complex associated to U n . To every such random graph Γ n we can associate the measure: where the sum is over all connected components of Γ n and γ denotes their isomorphism class (as graphs). There is an interesting fact regarding the random variable b 0 (Γ n ) appearing in (1.3): it is the same random variable as b 0 (U n ) (the number of components of the random graph and of the random complex are the same), and in Theorem 4.11 we will prove that there exists a constant β (depending on the parameter α in (1.1) such that: We note that for a related model of random graphs (the Poisson model on R m , see Section 1.6 below) Penrose [Pen03] has proved that there exists a constant β (depending on the parameter α in (1.1)) such that the normalized component count converges to a constant in L 2 . In fact, as we will see below, related to our M -geometric model there is a way to construct a corresponding R m -geometric model, which is in a sense the rescaled limit of the Riemannian one, and the limit constants for the two models are the same (up to the volume of M ).
In fact the limit measure ϕ (1) * Θ also comes from the rescaled Euclidean limit and it is supported on connected R m -geometric graphs. For a given m, the set of such graphs is not easy to describe, but in the case m = 1 they can be characterized by a result of Roberts [Rob69], and from this result we can deduce a description of the support of the limit measure in (1.3) (see Corollary 5.5 and Section 5 for more details).
1.5. The spectrum of a random geometric graph. When talking about a graph, a natural associated object to look at is its Laplace operator, see Section 6. It is known that the spectrum of the (symmetric) normalized Laplace operator for graphs encodes important information about the graphs [Chu97]. For example, it tells us how many connected components a graph has; it tells whether a graph is bipartite and whether it is complete; it tells us how difficult it is to partition the vertex set of a graph into two disjoint sets V 1 and V 2 such that the number of edges between V 1 and V 2 is as small as possible and such that the volume of both V 1 and V 2 , i.e. the sum of the degrees of their vertices, is as big as possible. Therefore, the Laplace operator gives a partition of graphs into families and isospectral graphs share important common features. Since, furthermore, the computation of the eigenvalues can be performed with tools from linear algebra, such operator is a very powerful and used tool in graph theory and data analytics.
In this context, the convergence of the counting measure in (1.3) can be used to deduce the existence of a limit measure for the spectrum of the Laplace operator for random geometric graphs. More specifically, we define the empirical spectral measure of a graph as the normalized counting measure of eigenvalues of the normalized Laplace operator and we prove that there exists a universal deterministic measure µ on the real line such that (Theorem 7.4) Here, λ 1 (Γ n ), . . . , λ n (Γ n ) are the eigenvalues of the normalized Laplace operator of Γ n and the convergence in (1.4) means that for every continuous function f : [0, 2] → R we have: The measure µ in (1.4) is far from trivial and we don't have yet a clear understanding of it: we know it is supported on the interval [0, 2], but for example it is not absolutely continuous with respect to Lebesgue measure (in fact µ({0}) = β).
Remark 1.1. In [GJLS16], Gu, Jost, Liu and Stadler introduce the notion of spectral class of a family of graphs. Given a Radon measure ρ on [0, 2] and a sequence (Γ n ) n∈N of graphs, with #(V (Γ n )) = n, they say that this sequence belongs to the spectral class ρ if µ Γn * ρ as n → ∞. We can interpret (1.4) as saying that our family of random geometric graphs (Γ n ) n belongs to the spectral class µ.
1.6. The Euclidean Poisson model. As we already observed, in [ALL18], the proof of (1.2) is based on a rescaling limit idea. Namely, one can fix R > 0 and a point p ∈ M , and study the limit structure of the random complexes inside the ballB(p, Rn −1/d ). The random geometric complex obtained as n → ∞ can then be described as follows. For the random complex P R , one can define completely analogue measures, where now the parameter is R > 0, and all the above discussion applies also to this model (this is discussed throughout the paper).
Structure of the paper. This paper is structured as follows. In Section 2 we talk about (deterministic) M -geometric complexes and, in particular, we define and see some properties of the set G/ ∼ = of isotopy classes of connected, nondegenerate M -geometric complexes. In Section 3 we talk about random M -geometric complexes; in Section 4 we prove (1.2). Morevorer, in Section 5 we define and see some properties of geometric graphs; in Section 6 we recall the definition of the normalized Laplace operatorL for graphs and we prove some properties of the empirical spectral measure in the case of geometric graphs. Finally, in Section 7 we prove (1.4).

Geometric complexes
Throughout this paper we fix a Riemannian manifold (M, g) of dimension m.
Remark 2.2. In order to avoid unnecessary complications, in the sequel we will always assume that the injectivity radius This requirement ensures that for every point p ∈ M the set is smooth (in fact it is the image of the sphere of radius r in the tangent space at p under the exponential map, which is a diffeomorphism on B TpM (0, inj(M ))). Observe also that for r ≤ inj(M ) the ball B(p, r) is contractible, but not necessarily geodesically convex.
Next result is classical and relates the homotopy of a geometric complex to the one of its associatedČech complex.  Definition 2.5 (Isotopy classes of connected geometric complexes). Let p 1 , . . . , p n and q 1 , . . . , q n be points in M and let r 0 , r 1 ≥ 0 such that U 0 := U({p 1 , . . . , p n }, r 0 ) and U 1 := U({p 1 , . . . , p n }, r 1 ) are nondegenerate M -geometric complexes. We say that U 0 and U 1 are (rigidly) isotopic and we write U 0 ∼ = U 1 if there exists an isotopy of diffeomorphisms ϕ t : M → M with ϕ 0 = id M and a continuous function r(t) > 0, for t ∈ [0, 1], such that: Remark 2.6 (Isotopy classes and discriminants). The definition of two complexes being (rigidly) isotopic is very reminiscent of the notion of rigid isotopy from algebraic geometry, where the "regular" deformations are those which do not intersect some discriminant. We can make this analogy more precise. For every n ∈ N consider the smooth manifold together with the discriminant The set Σ n is closed since its complement R n is defined by the (finitely many) transversality conditions (2.1). Adopting this point of view, isotopy classes of nondegenerate M -geometric complexes built using n many balls are labeled by the connected components of R n := H n \Σ n (the complement of a discriminant). In fact, given a nondegenerate complex U({p 1 , . . . , p n }, r), then (p 1 , . . . , p n , r) ∈ R n (because it is nondegenerate) and viceversa every point in R n correspond to a nondegenerate complex. Moreover, a nondegenerate isotopy of two nondegenerate complexes defines a curve between the corresponding points of R n , and this curve is entirely contained in R n ; the two corresponding parameters must therefore lie in the same connected component of R n ; viceversa, because for an open set of a manifold connected and path connected are equivalent, every two points in the same component R n can be joined by an arc all contained in R n and give therefore rise to isotopic complexes. Remark 2.8. Observe that, by definition, each class [U] ∈ G/ ∼ = keeps also the information on the way U is embedded in M . In particular, it might be that to two nondegenerate Mgeometric complexes U 1 and U 2 there correspond isomorphicČech complexesČ(U 1 ) Č (U 2 ), but at the same time the complexes U 1 and U 2 themselves are not rigidly isotopic (see Figure  2).
Proof. Let us first partition G/ ∼ = into countably many sets. For every n ∈ N we consider the set: (G n / ∼ =) := {classes of complexes in G/ ∼ = which are built using n many balls}, and we need to prove that this set is countable. We have already seen (Remark 2.6) that isotopy classes of nondegenerate complexes which are built using n many balls are in one-toone correspondence with the connected components of R n = H n \Σ n (these sets are defined by (2.2) and (2.3)). The function "number of connected components of a nondegenerate complex" is constant on each component of R n and consequently the number of isotopy classes of connected and nondegenerate complexes (i.e. the cardinality of G n / ∼ =) is smaller than the number of components of R n : We are therefore reduced to prove that R n has at most countably many components. To this end we write R n as the disjoint union of its components (each of which is an open set in H n ): (2.4) We cover now the manifold H n with countably many manifold charts {(V j , ϕ j )} j∈N with ϕ j : V j ∼ −→ R m (here N denotes some countable set). For every j ∈ N we consider also the decomposition of the open set R n ∩ V j into its connected components: Since each C α from (2.4) is the union of elements of the form with the index set j running over the countable set N α ⊂ N , it is therefore enough to prove that for every j ∈ N the index set B j is countable, i.e. that the number of connected components of R n ∩ V j is countable. Observe now that, since ϕ j is a diffeomorphism between V j and R m , then the number of connected components of R n ∩ V j is the same of the number of connected components of ϕ j (R n ∩V j ), which is an open subset of R m . Since in each component of ϕ j (R n ∩ V j ) we can pick a point with rational coordinates, it follows that the number of such components is countable, and this concludes the proof.  Remark 2.13. In order to appreciate the difference between the classes in (G/ ∼ =), (G/∼) and G (k) / , let us look at Figure 2. Here, we have three R 3 -geometric complexes, given by the union of balls in R 3 , that have three different shapes. All three complexes are homotopy equivalent to each other and they are all pairwise not isotopic. Moreover, while the first one and the second one have isomorphic 1-skeleta, the 1-skeleton of the third complex is not isomorphic to the other ones.
Remark 2.14. There are natural forgetful maps and In the paper [NS16] Nazarov and Sodin have introduced a powerful tool (the "integral geometry sandwitch") for localizing the count of the number of components of the zero set of random harmonics in a Riemannian manifold. This tool has been used by Sarnak and Wigman [SW19] for the study of distribution of components type of the zero set of random waves on a Riemannian manifold, and it has been extended to geometric complexes in [ALL18]. We recall here this tool, stated in the language of this paper.
Theorem 2.16 (Analogous of the Integral Geometry Sandwich). The following two estimates are true: (1) (The local case) Let U be a generic geometric complex in R m and fix w ∈ G(R m )/ ∼ =.
Then for 0 < r < R (2) (The global case) Let U be a generic geometric complex in a compact Riemannian manifold M and fix w ∈ G(M )/ ∼ =. Then for every ε > 0 there exists η > 0 such that for every r < η: Proof. The proof of both statements is exactly the same as in [SW19], after noticing that the only property needed on the counting functions is that the considered topological spaces only have finitely many components, and these components are counted according to a specific type (here are selected according to isotopy type, but we could consider instead any function that partitions the set of components and count only the components belonging to a given class).
3. Random geometric complexes (thermodynamic regime) We say that U n is a random M -geometric complex. The choice of such r is what defines the so-called critical or thermodynamic regime.  From now on, we will only consider nondegenerate complexes, without further mentioning this assumption. This is not reductive, since our random complexes are nondegenerate with probability one.

Random measures
We fix the following notation. Given a measure A with a fixed sigma algebra (omitted from the notation), we denote by: be its decomposition into connected components. We define Θ U ∈ M 1 (G/ ∼ =) as We also endow M 1 (G/ ∼ =) with the total variation distance: When U is a random geometric complex, Θ U is a random variable with values in the metric space (M 1 (G/ ∼ =), d tv ). In this context, recall the notion of convergence in probability: We shall see the proof of Theorem 4.2 in Section 4.1. As a first corollary we recover the results from [ALL18]. We have that As a second corollary we see that, because Theorem 4.2 keeps track of fine properties of the geometric complex, we can use other forgetful maps and obtain information on the limit distribution of the components type of each k-skeleton.
Corollary 4.4. For each k ∈ N, consider the forgetful map from Remark 2.14: We have that ϕ Proposition 4.5 (Existence of all isotopy types). Let U be a finite geometric complex in R m and let α > 0. Let (U n ) n be a sequence of random M -geometric complexes constructed using α. There exist n 0 , R, c > 0 (depending on Γ and α but not on M ) such that for every p ∈ M and for every n ≥ n 0 : Proof. The proof is the same as the one of [ALL18, Proposition 1.2], where instead of homotopy equivalences we consider isotopy equivalences. This is allowed by the last few lines of the proof of [ALL18, Proposition 6.2].
Example 4.6. Let C 137 be the cycle with 137 vertices and 137 edges. By Proposition 4.5 there exist n 0 , R and c > 0 (depending on C 137 and α but not on M ) such that for every p ∈ M and for every n ≥ n 0 :

Proof of Theorem 4.2.
We split the statement of Theorem 4.2 into two parts. The first part, Theorem 4.8, states that the random measure Θ R converges in probability to a universal deterministic measure Θ ∈ M 1 (G/ ∼ =) supported on the set G(R m )/ ∼ =. The second part, Theorem 4.12, states that also the random measure Θ n converges in probability to Θ.  Since P R ⊂ B(0, R), this implies that We now have to prove that c w > 0. Since w ∈ G(R m )/ ∼ =, given U ∈ w there exist β > 0 and y 1 , . . . , y n ∈ R m such that U = U({y 1 , . . . , y n }, βn −1/m ).
Let R 1 be such that U ⊂ B(0, R 1 ) and choose r such that rβ = α, where α is the constant that we use for constructing the random complex P. We can then rescale U in B(0, r · R 1 ) so that it is constructed on radius αn −1/m . Now, since U is nondegenerate, there exists ε > 0 such that, if for every i we have that y i − y i < ε, then the complex U ({y 1 , . . . , y n }, αn −1/m ) is isotopic to U. Now, observe that for R → ∞, R >> r · R 1 . Consider O(vol(B(0, R)))-many disjoint balls {B(y j , r · R 1 )} j=1,...,O(vol(B(0,R))) inside B(0, R). Then since the random variables N (P R , B(y j , r · R 1 ); w) are i.i.d. by the fact that the balls are disjoint. Now, O(vol(B(0, R))) vol(B(0, R)) is bigger than a certain constant c > 0 and E N (P R , B(y, r · R 1 ); w) > 0. Therefore, c w > 0.
Proposition 4.7 allows us to deduce the following theorem.  Proof. The proof is the same as the one of [ALL18, Theorem 3.1], with the following differences: -In point (3), instead of considering the homotopy equivalence between the unions of the balls, we consider the isotopy equivalences between their k-skeleta. This is allowed because, at the end of the proof of point (3), it is proved that the combinatorics of the covers are the same. -After assuming point (1), point (2) and the modified point (3), we can say that the kskeleta of the two unions of balls are isotopic and also the unions of all the components entirely contained in B(0, R) (respectivelyB(p, Rn −1/m )) have isotopic k-skeleta. In particular, the number of components of a given isotopy class w is the same for both sets with probability at least 1 − δ.

Geometric graphs
We specialize the previous discussion to the case k = 1, and consider G (1) = {isomorphism classes of connected, nondegenerate M -geometric graphs}.
Remark 5.1. The set of M -geometric graphs defined using closed balls equals the set of M -geometric graphs defined using open balls. To see this, assume that a geometric graph Γ = (V (Γ), E(Γ)) is defined using closed balls of radius r. Then, for each pair of distinct vertices (p i , p j ), Now, choose ε ≥ 0 small enough that, for each pair of distinct vertices (p i , p j ), Therefore, Γ can be constructed as a M -geometric graph using open balls of radius r + ε. The inverse implication is analogous.
In the case M = R m , the problem of describing the set G (1) (R m ) is equivalent to asking which graphs are realizable as R m -geometric graphs in a given dimension m. There is a vast literature about this problem and, commonly, geometric graphs realizable in dimension m are called m-sphere graphs while the minimal dimension m such that a given graph is a m-sphere graph is called its sphericity. In [Mae84] it is proved that every graph has finite sphericity; in [KM12] the authors prove that the problem of deciding, given a graph Γ, whether Γ is a msphere is NP-hard for all m > 1. We can also observe that, for each m > 0, there are graphs that are not m-sphere graphs. To see this, consider the kissing number k(m) in dimension m, defined as the number of non-overlapping unit spheres that can be arranged such that they each touch a common unit sphere. Consider the star graph with a central vertex connected to n external vertices, where n > k(m). In order to have a realization of dimension m of this graph, we need a central sphere that touches n spheres which do not touch each other. Since n > k(m), this is not possible.
Example 5.2. In dimension 2, the kissing number is 6, as shown in Figure 3. Therefore, any star graph S n on n + 1 vertices, with n > 6, is not realizable in dimension 2 as a sphere graph. In the particular case of m = 1, 1-sphere graphs are called indifference graphs, unit interval graphs or proper interval graphs and there are many characterizations of such graphs [LB63,Rob69,Jac92,GOn96,Weg04,Mer08]. A classical characterization is due to Roberts [Rob69] and it characterizes unit distance graphs by the absence of certain forbidden subgraphs, this is recalled in Theorem 5.4. As a consequence, we get the following corollary.
Corollary 5.5. The support of the measure ϕ (1) * Θ defined in Corollary 4.4 is given by all graphs that are chordal and do not contain any of the graphs shown in Figure 4 as induced subgraph.

Laplacian of a graph and its spectrum
We now fix a graph Γ on n vertices v 1 , . . . , v n and we recall the definition of the (symmetric) normalized Laplace matrix, together with other common matrices associated to graphs. We shall then define the spectrum and the empirical spectral measure associated to these matrices, and show some properties. Definition 6.2 (Spectrum of a matrix). Given Q ∈ Sym(n, R) let spec(Q) be the spectrum of Q, i.e. the collection of its eigenvalues repeated with multiplicity, We define the empirical spectral measure of Q as Definition 6.3 (Spectrum of a graph). We define spectrum of Γ, spec(Γ), as the spectrum ofL and we write it as λ 1 (Γ) ≤ . . . ≤ λ n (Γ).
Theorem 6.4. Let (Γ 1,n ) n and (Γ 2,n ) n be two sequences of graphs such that, for every n, (Γ 1,n ) n and (Γ 2,n ) n are two graphs on n nodes that differ at most by c edges. Denote by µ 1,n and µ 2,n the empirical measures associated to one of the matrices A, D, L,L. Then We shall prove Theorem 6.4 in Section 6.1.
Remark 6.5. In the case ofL, we have convergence in total variation distance for "connected sum" of complete graphs, but not for paths, as we shall see in Section 6.2.
Remark 6.6. Theorem 2.8 from [GJLS16] tells that, if two families (Γ 1,n ) n and (Γ 2,n ) n differ by at most c edges and their corresponding spectral measures have weak limits, then they belong to the same spectral class (the notion of spectral class of a family of graph is recalled in Remark 1.1). In this sense our previous Theorem 6.4 can be considered as an analogue of [GJLS16, Theorem 2.8]: the difference of the spectral measure of two families of graphs (Γ 1,n ) n and (Γ 2,n ) n differing by at most a finite number c of edges, goes to zero weakly (without the assumption that the corresponding spectral measures have weak limits).
6.1.1. Preliminaries. Given Q ∈ Sym(n, R), we define the 1-Shatten norm of Q as The Weilandt-Hoffman inequality [Tao12, Exercise 1.36] holds: Proposition 6.7. Let Q 1 , Q 2 ∈ Sym(n, R) such that Then, for each f ∈ C 0 c (R, R) and for each ε > 0, there exists δ > 0 such that Proof. Denote by {λ the eigenvalues of Q 1 and Q 2 respectively. Then Now, since f ∈ C 0 c (R, R), f is uniformly continuous and given ε > 0 there exists δ = δ(f ) such that Therefore, since by Equation 6.1.1 and by hypothesis we have that Therefore, we have the following chain of inequalities: 6.1.2. Applications to graphs.
Lemma 6.8. Let Γ 1 , Γ 2 be two graphs with V (Γ 1 ) = V (Γ 2 ) that differ by at most C-many edges. Then, Proof. Observe that any of the matrices consists of all zeros except for at most 4C entries, all of which entries are bounded by a constant (it is 1 for ∆ 1 and C for ∆ 2 and ∆ 3 ). Therefore, each ∆ i ∈ Sym(n, R) for i = 1, 2, 3 has rank at most 2C and all its eigenvalues are zero, except for at most 2C of them. It follows that, for i = 1, 2, 3, Similarly, ∆ 4 :=L 1 −L 2 consists of all zeros except for at most 2C(n − 1) entries, all of which entries are bounded by 1, and it has rank at most 4C. Therefore, As a corollary, we can prove Theorem 6.4.
Proof of Theorem 6.4. We prove that, for each f ∈ C 0 c (R, R) and for each ε > 0, By Lemma 6.8, we have that ∆ i S 1 ≤ c i for each i = 1, 2, 3. By Proposition 6.7, there exists δ > 0 such that Similarly, by Lemma 6.8 we have that By Proposition 6.7, there exists δ > 0 such that Therefore, lim n→∞ R f dµ 1,n − R f dµ 2,n ≤ ε 6.2. Strong convergence for complete graphs.
Lemma 6.9. Given N ∈ N + , let K N and K N be two complete graphs on N nodes. Let K N K N be their disjoint union and let Γ c N := K N c edges K N be their union together with c edges (v i , v i ) where v i ∈ K N and v i ∈ K N , for i = 1, . . . , c. Let also µ K N K N and µ Γ c N be the empirical spectral measures of these two graphs. Then, for some a i ∈ (0, 2).
Remark 6.10. In order to prove Lemma 6.9, we make the following observation. It is easy to see that the spectrum of the symmetric normalized Laplacian matrixL = I n − D −1/2 AD −1/2 equals the spectrum of the random walk normalized Laplacian matrixL := I n − D −1 A. Moreover, for a graph with vertex set V ,L can be seen as an operator from the set {f : V → R} to itself. We shall work on this operator for proving Lemma 6.9 and, for a graph Γ, we shall use the simplified notation L Γ in order to indicate the random walk normalized Laplace operator for Γ.
Proof of Lemma 6.9. Since the spectrum of K N K N is given by 0 with multiplicity 2 and N N −1 with multiplicity 2(N − 1), we have that In order to prove the second part of the lemma, we shall find 2(N − 1 − c) functions on V (Γ N ) that are eigenfunctions for the Laplace operator with eigenvalue N N −1 and are orthogonal to each other. In particular, by the symmetry of Γ N , it suffices to find N − 1 − c such functions that are 0 on the vertices of K N .
Observe that K N −1−c is a subgraph of K N \ {v 1 , . . . , v c } that has N − 1 − c eigenfunctions f 1 , . . . , f N −1−c for the largest eigenvalue. These are orthogonal to each other and orthogonal to the constants, therefore Since for complete graphs any function that is orthogonal to the constants is an eigenfunction for N N −1 , we have thatf 1 , . . . ,f N −1−c are (pairwise orthogonal) eigenfunctions for N N −1 . Analogously, for i ∈ {1, . . . , N − 1 − c}, let nowf i be the function on V (Γ N ) that is equal to zero on K N ∪ {v 1 , . . . , v c } and is equal to f i otherwise. It's then easy to see that also these functions are orthogonal to each other and orthogonal to the constants. Now, for each i and for each v ∈ Γ N withf i (v) = 0, we have that v ∈ K N −1−c , therefore This proves that the functionsf i 's are N − 1 − c orthogonal eigenfunctions of the Laplace Operator in Γ N for the eigenvalue N N −1 . Since they are all 0 on K N , by symmetry we can also get N − 1 − c eigenfunctions for N N −1 on Γ N that are 0 on K N and therefore are orthogonal to the first N − 1 − c functions. This implies that the multiplicity of N N −1 for Γ N is at least 2(N − 1 − c). Therefore, for some a i ∈ (0, 2).
Corollary 6.11. The total variation distance between the probability measures µ K N K N and µ Γ c N defined in the previous lemma is In particular, if c = o(N ), the total variation distance tends to zero for N → ∞.
Example 6.12. The previous corollary doesn't hold in general. As a counterexample, take two copies of the path on l edges, A l and A l . Their union via one external edge can be for example the path on 2l + 1 edges, and µ A 2l+1 = 1 2(l + 1) The total variation distance between these two measures does not tend to zero for l → ∞.

Random geometric graphs
Specializing Corollary 4.4 to the case k = 1, we get  Proof. Let us start with point (3): we have that µ({0}) = c 0 where 0 = x 0 and c 0 is the L 1 -limit of c 0 ,R , which is the random variable: c 0 ,R = #{components of Γ R containing 0 in their spectrum} = b 0 (Γ R ), and therefore (3) follows from the definition of β.
Point (1) also follows immediately, since β 0 > 0 and µ charges positively sets of Lebesgue measure zero (hence it cannot be absolutely continuous with respect to Lebesghe measure).
For the proof of point (2) we argue exactly as in the proof of Theorem 7.4, by replacing f with χ [a,b] (the characteristic function of the interval [a, b]) and observing that the only property of f that we have used is its boundedness.