Graphons, Taos regularity and difference polynomials Ivan Tomai c - - PowerPoint PPT Presentation

Graphons, Tao’s regularity and difference polynomials

Ivan Tomaši´ c (joint work with Mirna Džamonja)

Queen Mary University of London

FPS Leeds, 09/04/2018

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Outline

Definable graphs and asymptototic classes Graphons as limits of graphs Regularity lemmas Applications: difference expander polynomials

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Classes of finite structures

Definition (Macpherson-Steinhorn, Ryten, Elwes. . . )

A class of finite structures C is a CDM-class, if, for every definable function X → S (X is a definable family with parameters from S), there exist

• 1. a definable function µX : S → Q,
• 2. a definable function dX : S → N,
• 3. a constant CX > 0,

so that, for every F ∈ C and every s ∈ S(F),

• |Xs(F)| − µX(s)|F|dX(s)
• ≤ CX|F|dX(s)−1/2.

A class C is an asymptotic class, if we have a slightly weaker growth estimate.

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Definable graphs over asymptotic classes

Let C be an asymptotic class, and let Γ = (U, V, E ⊆ U × V) be a definable (bipartite) graph.

Motivating question

Describe the limit behaviour of finite graphs Γ(F) = (U(F), V(F), E(F)) for F ∈ C.

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Examples: definable graphs over finite fields

Let C be the class of finite fields (in the language of rings). Let Γ = (U, V, E) where U = V = A1, and the edge relation is E(x, y) ≡ ∃z x − y = z2. The graphs Γ(Fq), (q odd) are called Paley graphs (image is from [3])

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Examples: definable graphs over finite fields

Let C be the class of finite fields (in the language of rings). Let Γ = (U, V, E) where U = V = A1, and the edge relation is E(x, y) ≡ ∃z x − y = z2. The graphs Γ(Fq), (q odd) are called Paley graphs. The incidence matrices for q = 5, 9, 13, 29, 41. They seem to tend to p = 1 2

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Examples: definable graphs over finite fields

Example

Γ = (A1, A1, E) with the edge relation E(x, y) ≡ ∃z xy = z2. Consider U0(x) ≡ ∃z x = z2 and U1 = A1 \ U0 For q odd, the incidence matrices look like

1 1

U0 U0 U1 U1

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Examples: definable graphs over finite fields

Example

Γ = (A1, A1, E) with the edge relation E(x, y) ≡ ∃z x + y = z3. The ‘limit’ incidence matrices of Γ(Fq) look like:

◮ If 3|q − 1,

p = 1 3

◮ if 3 |q − 1,

p = 1

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Kernels and Graphons

Definition

• 1. The space of kernels is

W = L∞([0, 1]2), the space of essentially bounded measurable functions [0, 1]2 → R, with cut distance δ that comes from the cut norm W = sup

S,T⊆[0,1]

• S×T

W(x, y) dx dy

• .
• 2. The space of graphons is

W0 = {W ∈ W : 0 ≤ W ≤ 1}.

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Graphs as graphons

◮ A stepfunction is a kernel W such that there exist partitions

[0, 1] = n

i=1 Ui and [0, 1] = m j=1 Vj so that W is constant

• n each Ui × Vj.

◮ Let Γ = (U, V, E ⊆ U × V, w) be a finite (weighted) bipartite

• graph. The associated stepfunction

W(Γ) is the incidence matrix of Γ linearly scaled down to the square [0, 1] × [0, 1].

◮ Now, if we have a sequence of finite graphs, it makes

sense to ask whether it has a limit in the space of graphons (figure is from [2]):

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Cut metric and regularity/homogeneity

Definition

Let Γ = (U, V, E ⊆ U × V ) be a finite bipartite graph and ǫ > 0.

• 1. We say that Γ is ǫ-homogeneous of density w ∈ [0, 1] if, for

every A ⊆ U and B ⊆ V , |E ∩ (A × B)| − w|A||B|| ≤ ǫ|U||V |.

• 2. We say that Γ is ǫ-regular of density w ∈ [0, 1] if, for every

A ⊆ U with |A| > ǫ|U| and B ⊆ V with |B| > ǫ|V |, |E ∩ (A × B)| − w|A||B|| ≤ ǫ|A||B|.

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Regularisation and cut distance

Remark

Let Γ = (U, V, E ⊆ U × V ) be a finite bipartite graph, ǫ > 0. Suppose that there exist partitions U = n

i=1 Ui and

V = m

j=1 Vj so that

Γ ↾Ui×Vj is ǫ-homogeneous with density wij. Let W be the stepfunction determined by the weights wij. Then d(Γ, W) ≤ ǫ.

Motto:

Regularisable means ‘close to a stepfunction’.

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Kernel operators

Definition

The kernel operator TW : L1[0, 1] → L∞[0, 1] associated to a kernel W ∈ W is (TW f)(x) = 1 W(x, y)f(y)dy.

Fact

The restriction TW : L2[0, 1] → L2[0, 1], is a Hilbert-Schmidt operator:

◮ it is a compact operator; ◮ has a singular value decomposition; ◮ finite Hilbert-Schmidt norm.

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Algebraic regularity in the language of graphons

Theorem (Tao’s algebraic regularity lemma)

Let Γ = (U, V, E) be a definable bipartite graph on a CDM class of finite structures C. There exists a constant M = M(Γ) > 0 and a definable stepfunction W such that for every F ∈ C, d(Γ(F), W(F)) ≤ M|F|−1/12. We say that W is a definable regularisation of Γ.

Corollary

In the space of graphons, the set of accumulation points of the family of realisations of a definable bipartite graph over the structures ranging in an asymptotic class is a finite set of stepfunctions.

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The spectral proof of the regularity lemma

Step 1: weak regularity for graphons

Let W be a graphon. For every ǫ > 0 there exists a stepfunction W ′ with n(ǫ) ≤ (5/ǫ3)(1/ǫ2) steps such that, writing W 6 = W ◦ W ∗ ◦ W ◦ W ∗ ◦ W ◦ W ∗,

• W 6 − W ′
• ∞ ≤ 2ǫ2.

Idea of proof: Let T = TW be the kernel operator associated with W. Using singular value decomposition, write TT ∗TT ∗TT ∗ = A + B, where A is a low (finite) rank operator, and B is a small ‘error’ wrt ǫ.

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The spectral proof of the regularity lemma

Step 2: major improvements using CDM

(a) Proof of Step 1 finds all potential regularisations of Γ6; use discreteness of CDM-growth rates to choose the appropriate ǫ for Step 1, and to show that the regularisation is definable. (b) ‘Self-improvement’ using CDM to obtain a regularisation of Γ from that of Γ6.

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Advantages of our proof

◮ well-founded functional analysis on the space of graphons; ◮ an explicit construction of the definable regularisation; ◮ a detailed treatment of the parameter space; ◮ no ‘bounded complexity’ statements.

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Fields with Frobenius as a CDM-class

Consider the difference field Kq = (¯ Fq, Frobq). These are infinite structures, so they cannot literally constitute a CDM-class. However, according to Ryten-T, if we formally set |Kq| = q, and only consider finite-dimensional definable sets, most CDM-style proofs work. In particular:

Algebraic regularity lemma for fields with Frobenius

Any finite-dimensional definable graph (in the language of difference rings) over fields with Frobenius can be definably regularised.

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Expander polynomials

Theorem (Tao)

Let f(x, y) be a polynomial which is not of the form

◮ f(x, y) = p(r(x) + s(y)), or ◮ f(x, y) = p(r(x) · s(y)).

Then f is a moderate expander, |f(A, B)| ≫ q, whenever A, B ⊂ Fq with |A||B| ≫ q2−1/8.

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Difference expander polynomials

Let X1, . . . , Xn, Y be finite-dimensional difference varieties. A morphism of difference schemes f : X1 × · · · × Xn → Y is a moderate asymmetric expander, if there exist constants c, C > 0 such that:

◮ for every Kq, and ◮ every choice of Ai ⊆ Xi(Kq) with |Ai| ≥ Cq1−c,

we have |fs(A1, . . . , An)| ≥ C−1|Y (Kq)|.

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Solving the algebraic constraint in dimension 1

Theorem

Let X, Y , Z be difference varieties of dimension 1, and let f : X ×S Y → Z be a difference morphism. Then at least one of the following holds:

• 1. f is definably isogenous to the additive or multiplicative

group law;

• 2. f is definably isogenous to the addition law on an elliptic

curve;

• 3. f is a moderate asymmetric expander.

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Idea of proof

Very general principles:

◮ Tao: a non-expander yields an algebraic constraint, the

morphism (x, x′, y, y′) → (f(x, y), f(x, y′), f(x′, y), f(x′, y′)) is not dominant.

◮ Hrushovski: this gives a group configuration, so

non-expansion is related to a group law.

◮ Kowalski-Pillay: ‘group configuration’ in ACFA, the group is

isogenous to an algebraic group.

◮ In dimension 1, there are few choices of algebraic groups.

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Bibliography

Mirna Džamonja, Ivan Tomaši´

• c. Graphons arising from

graphs definable over finite fields. arXiv:1707.06296 Daniel Glasscock. What is. . . a Graphon? Notices of the AMS, vol 62, no 1. http://www.ams.org/notices/201501/rnoti-p46.pdf Wolfram MathWorld. Paley graph resource. http://mathworld.wolfram.com/PaleyGraph.html

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