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 1 / 23

Outline Definable graphs and asymptototic classes Graphons as limits of graphs Regularity lemmas Applications: difference expander polynomials 2 / 23

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 µ X : S → Q , 2. a definable function d X : S → N , 3. a constant C X > 0 , so that, for every F ∈ C and every s ∈ S ( F ) , � � | X s ( F ) | − µ X ( s ) | F | d X ( s ) � � ≤ C X | F | d X ( s ) − 1 / 2 . � � A class C is an asymptotic class, if we have a slightly weaker growth estimate. 3 / 23

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 . 4 / 23

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 = A 1 , and the edge relation is E ( x, y ) ≡ ∃ z x − y = z 2 . The graphs Γ ( F q ) , ( q odd) are called Paley graphs (image is from [3]) 5 / 23

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 = A 1 , and the edge relation is E ( x, y ) ≡ ∃ z x − y = z 2 . The graphs Γ ( F q ) , ( q odd) are called Paley graphs. The incidence matrices for q = 5 , 9 , 13 , 29 , 41 . They seem to tend to p = 1 2 6 / 23

Examples: definable graphs over finite fields Example Γ = ( A 1 , A 1 , E ) with the edge relation E ( x, y ) ≡ ∃ z xy = z 2 . Consider U 1 = A 1 \ U 0 U 0 ( x ) ≡ ∃ z x = z 2 and For q odd, the incidence matrices look like U 1 0 1 U 0 1 0 U 0 U 1 7 / 23

Examples: definable graphs over finite fields Example Γ = ( A 1 , A 1 , E ) with the edge relation E ( x, y ) ≡ ∃ z x + y = z 3 . The ‘limit’ incidence matrices of Γ ( F q ) look like: ◮ If 3 | q − 1 , p = 1 3 ◮ if 3 � | q − 1 , p = 1 8 / 23

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 W ( x, y ) dx dy � . � � S,T ⊆ [0 , 1] � S × T 2. The space of graphons is W 0 = { W ∈ W : 0 ≤ W ≤ 1 } . 9 / 23

Graphs as graphons ◮ A stepfunction is a kernel W such that there exist partitions [0 , 1] = � n i =1 U i and [0 , 1] = � m j =1 V j so that W is constant on each U i × V j . ◮ 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]): 10 / 23

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 | . 11 / 23

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 U i and V = � m j =1 V j so that Γ ↾ U i × V j is ǫ -homogeneous with density w ij . Let W be the stepfunction determined by the weights w ij . Then d � (Γ , W ) ≤ ǫ. Motto: Regularisable means ‘close to a stepfunction’. 12 / 23

Kernel operators Definition The kernel operator T W : L 1 [0 , 1] → L ∞ [0 , 1] associated to a kernel W ∈ W is � 1 ( T W f )( x ) = W ( x, y ) f ( y ) dy. 0 Fact The restriction T W : L 2 [0 , 1] → L 2 [0 , 1] , is a Hilbert-Schmidt operator: ◮ it is a compact operator; ◮ has a singular value decomposition; ◮ finite Hilbert-Schmidt norm. 13 / 23

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. 14 / 23

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 /ǫ 3 ) (1 /ǫ 2 ) steps such that, writing W 6 = W ◦ W ∗ ◦ W ◦ W ∗ ◦ W ◦ W ∗ , � W 6 − W ′ � ∞ ≤ 2 ǫ 2 . � � Idea of proof: Let T = T W 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 . 15 / 23

The spectral proof of the regularity lemma Step 2: major improvements using CDM (a) Proof of Step 1 finds all potential regularisations of Γ 6 ; 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 Γ 6 . 16 / 23

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. 17 / 23

Fields with Frobenius as a CDM-class Consider the difference field K q = (¯ F q , Frob q ) . These are infinite structures, so they cannot literally constitute a CDM-class. However, according to Ryten-T, if we formally set | K q | = 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. 18 / 23

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 ⊂ F q with | A || B | ≫ q 2 − 1 / 8 . 19 / 23

Difference expander polynomials Let X 1 , . . . , X n , Y be finite-dimensional difference varieties. A morphism of difference schemes f : X 1 × · · · × X n → Y is a moderate asymmetric expander, if there exist constants c, C > 0 such that: ◮ for every K q , and ◮ every choice of A i ⊆ X i ( K q ) with | A i | ≥ Cq 1 − c , we have | f s ( A 1 , . . . , A n ) | ≥ C − 1 | Y ( K q ) | . 20 / 23

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. 21 / 23

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. 22 / 23

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 23 / 23