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Simple-model spectral events as conditional pairing counts

Statement

Let $n$ and $d$ be integers with $n \geq 2$ , $0 < d < n$ and $n d$ even. Let $Ω$ be the uniform labelled pairing space and $S$ its simplicity event. Let $G$ instead denote a uniform element of the nonempty set $G_{d} (n)$ . For each $g \in G_{d} (n)$ , let $A (g)$ be its adjacency matrix in the vertex order $1, \dots, n$ . For real $t$ , put $B_{t} = {g \in G_{d} (n) : ρ_{0} (A (g)) \geq t}$ , where $ρ_{0}$ is the centered spectral quantity. If $G (ω)$ is the collapsed pairing graph, then

$P (G \in B_{t}) = \frac{∣ B_{t} ∣}{∣ G_{d} (n) ∣} = \frac{P ({ω \in S : G (ω) \in B_{t}})}{P (S)} .$

Thus the simple-model spectral event is exactly a conditional pairing event. A theorem already stated for uniform $G_{d} (n)$ can be applied directly; no unconditioned pairing estimate or asymptotic simplicity estimate is needed for that application.

Statement references

Pairings, simple-graph conditioning and model transfer

Feasible labelled simple regular graph orders

Ordered eigenvalues and the centered regular norm

The adjacency matrix of a finite simple graph

Facts & Assumptions

[F1]

Pairings, simple-graph conditioning and model transfer gives the uniform finite laws, conditional-probability formula and the constant number $(d!)^{n}$ of pairings realizing each simple regular graph.

[F2]

Feasible labelled simple regular graph orders ensures $G_{d} (n)$ is nonempty and $P (S) > 0$ .

[F3]

Ordered eigenvalues and the centered regular norm defines the centered spectral threshold on each simple graph.

[F4]

The adjacency matrix of a finite simple graph defines $A (g)$ in the fixed vertex order $1, \dots, n$ ; its entries are $1$ for edges and $0$ otherwise, so it is a real symmetric adjacency matrix to which [F3] applies.

Proof

Given: The finite spaces, collapse map and real threshold in the statement.

Goal: Compute the spectral event probability in the two ensembles.

1.1F1F2F3F4given

The spectral condition defines a subset $B_{t}$ of the finite graph set by [F3] and [F4]. Every subset is an event in its finite uniform space, so its probability is $∣ B_{t} ∣ / ∣ G_{d} (n) ∣$ . By [F2] the denominator is nonzero.

2.1F1F2step 1.1algebra∎

Each graph in $B_{t}$ has exactly $(d!)^{n}$ pairing preimages by [F1], and these preimages are disjoint for distinct graphs. The numerator pairing event therefore contains $∣ B_{t} ∣ (d!)^{n}$ outcomes. The event $S$ contains $∣ G_{d} (n) ∣ (d!)^{n}$ outcomes. Uniform pairing weights cancel in their probability ratio, giving the claimed equality. The conditional-probability formula applies by [F2]; since the graph law itself is uniform, applying a theorem stated in that law requires no further model transfer estimate.

Depends on

Used by · 1

Friedman spectral expansion in the uniform simple model

Dependency tree · 2 levels

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