Random Latin Squares and 2-dimensional Expanders Roy Meshulam - PowerPoint PPT Presentation

Random Latin Squares and 2-dimensional Expanders Roy Meshulam Technion – Israel Institute of Technology joint work with Alex Lubotzky Applied and Computational Algebraic Topology Bremen, July 2013

Plan Expansion in Graphs and Complexes ◮ Expander Graphs ◮ Cohomological Expansion of Complexes ◮ The Topological Overlap Property

Plan Expansion in Graphs and Complexes ◮ Expander Graphs ◮ Cohomological Expansion of Complexes ◮ The Topological Overlap Property Latin Square Complexes ◮ A Model for Random 2-Complexes ◮ Spectral Gaps and 2-Expansion ◮ Large Deviations for Latin Squares ◮ Random LS-Complexes are 2-Expanders ◮ Related Questions and Open Problems

The Graphical Cheeger Constant Edge Cuts For a graph G = ( V , E ) and S ⊂ V , S = V − S let e ( S , S ) = |{ e ∈ E : | e ∩ S | = 1 }| . S S

The Graphical Cheeger Constant Edge Cuts For a graph G = ( V , E ) and S ⊂ V , S = V − S let e ( S , S ) = |{ e ∈ E : | e ∩ S | = 1 }| . S S Cheeger Constant e ( S , S ) h ( G ) = min . | S | 0 < | S |≤ | V | 2

Expander Graphs ( d , ǫ )-Expanders A family of graphs { G n = ( V n , E n ) } n with | V n | → ∞ with two seemingly contradicting properties: ◮ High Connectivity: h ( G n ) ≥ ǫ . ◮ Sparsity: max v deg G n ( v ) ≤ d .

Expander Graphs ( d , ǫ )-Expanders A family of graphs { G n = ( V n , E n ) } n with | V n | → ∞ with two seemingly contradicting properties: ◮ High Connectivity: h ( G n ) ≥ ǫ . ◮ Sparsity: max v deg G n ( v ) ≤ d . Pinsker: Random 3 ≤ d -regular graphs are ( d , ǫ )-expanders.

Expander Graphs ( d , ǫ )-Expanders A family of graphs { G n = ( V n , E n ) } n with | V n | → ∞ with two seemingly contradicting properties: ◮ High Connectivity: h ( G n ) ≥ ǫ . ◮ Sparsity: max v deg G n ( v ) ≤ d . Pinsker: Random 3 ≤ d -regular graphs are ( d , ǫ )-expanders. Margulis: Explicit construction of expanders.

Expander Graphs ( d , ǫ )-Expanders A family of graphs { G n = ( V n , E n ) } n with | V n | → ∞ with two seemingly contradicting properties: ◮ High Connectivity: h ( G n ) ≥ ǫ . ◮ Sparsity: max v deg G n ( v ) ≤ d . Pinsker: Random 3 ≤ d -regular graphs are ( d , ǫ )-expanders. Margulis: Explicit construction of expanders. Lubotzky-Phillips-Sarnak, Margulis: Ramanujan Graphs - an ”optimal” family of expanders.

Spectral Gap Laplacian Matrix G = ( V , E ) a graph, | V | = n . The Laplacian of G is the V × V matrix L G :  deg( u ) u = v  L G ( u , v ) = − 1 uv ∈ E  0 otherwise .

Spectral Gap Laplacian Matrix G = ( V , E ) a graph, | V | = n . The Laplacian of G is the V × V matrix L G :  deg( u ) u = v  L G ( u , v ) = − 1 uv ∈ E  0 otherwise . Eigenvalues of L G 0 = µ 1 ( G ) ≤ µ 2 ( G ) ≤ · · · ≤ µ n ( G ) . µ 2 ( G ) = Spectral Gap of G .

Expansion and Spectral Gap Theorem (Alon-Milman, Tanner): For all ∅ � = S � V e ( S , S ) ≥ | S || S | µ 2 . n In particular h ( G ) ≥ µ 2 2 .

Expansion and Spectral Gap Theorem (Alon-Milman, Tanner): For all ∅ � = S � V e ( S , S ) ≥ | S || S | µ 2 . n In particular h ( G ) ≥ µ 2 2 . Theorem (Alon, Dodziuk): If G is d -regular then � h ( G ) ≤ 2 d µ 2 .

Expansion and Spectral Gap Theorem (Alon-Milman, Tanner): For all ∅ � = S � V e ( S , S ) ≥ | S || S | µ 2 . n In particular h ( G ) ≥ µ 2 2 . Theorem (Alon, Dodziuk): If G is d -regular then � h ( G ) ≤ 2 d µ 2 . Expanders can thus be defined using the spectral gap.

Why Expanding Graphs? Uses of Expanders ◮ Construction of efficient communication networks. ◮ Randomization reduction in probabilistic algorithms. ◮ Construction of good error correcting (LDPC) codes. ◮ Tools in computational complexity lower bounds.

Why Expanding Graphs? Uses of Expanders ◮ Construction of efficient communication networks. ◮ Randomization reduction in probabilistic algorithms. ◮ Construction of good error correcting (LDPC) codes. ◮ Tools in computational complexity lower bounds. Interactions with Other Areas ◮ Expansion and Kazhdan’s property T. ◮ Expanders as spaces of maximal Euclidean distortion. ◮ Dimension expanders and representation theory. ◮ Expanders on finite simple groups.

What are Expanding Complexes? Three Notions of Expansion ◮ Combinatorial: via the mixing property. ◮ Spectral: via eigenvalues of the higher Laplacians. ◮ Cohomological: via the Hamming weights of coboundaries.

What are Expanding Complexes? Three Notions of Expansion ◮ Combinatorial: via the mixing property. ◮ Spectral: via eigenvalues of the higher Laplacians. ◮ Cohomological: via the Hamming weights of coboundaries. Cohomological Expansion This notion is strongly tied to topology, e.g. : ◮ Linial-M-Wallach: Homology of random complexes. ◮ Gromov: The topological overlap property. ◮ Gundert-Wagner: Laplacians of random complexes. ◮ Dotterrer-Kahle: Expansion of random subcomplexes.

Simplicial Cohomology X a simplicial complex on V , R a fixed abelian group. i -face of σ = [ v 0 , · · · , v k ] is σ i = [ v 0 , · · · , � v i , · · · , v k ]. C k ( X ) = k -cochains = skew-symmetric maps φ : X ( k ) → R . Coboundary Operator d k : C k ( X ) → C k +1 ( X ) given by k +1 � ( − 1) i φ ( σ i ) . d k φ ( σ ) = i =0

Simplicial Cohomology X a simplicial complex on V , R a fixed abelian group. i -face of σ = [ v 0 , · · · , v k ] is σ i = [ v 0 , · · · , � v i , · · · , v k ]. C k ( X ) = k -cochains = skew-symmetric maps φ : X ( k ) → R . Coboundary Operator d k : C k ( X ) → C k +1 ( X ) given by k +1 � ( − 1) i φ ( σ i ) . d k φ ( σ ) = i =0 d − 1 : C − 1 ( X ) = R → C 0 ( X ) given by d − 1 a ( v ) = a for a ∈ R , v ∈ V . Z k ( X ) = k -cocycles = ker( d k ). B k ( X ) = k -coboundaries = Im ( d k − 1 ). k -th reduced cohomology group of X : k ( X ) = ˜ k ( X ; R ) = Z k ( X ) / B k ( X ) . ˜ H H

Cut of a Cochain Cut determined by a k -cochain φ ∈ C k ( X ; R ): supp ( d k φ ) = { τ ∈ X ( k + 1) : d k φ ( τ ) � = 0 } . Cut Size of φ : � d k φ � = | supp ( d k φ ) | .

Cut of a Cochain Cut determined by a k -cochain φ ∈ C k ( X ; R ): supp ( d k φ ) = { τ ∈ X ( k + 1) : d k φ ( τ ) � = 0 } . Cut Size of φ : � d k φ � = | supp ( d k φ ) | . Example: 0 0 0 σ 1 σ 2 1 1 0 � d 1 φ � = |{ σ 1 , σ 2 }| = 2

Hamming Weight of a Cochain The Weight of a k -cochain φ ∈ C k ( X ; R ): � [ φ ] � = min { | supp ( φ + d k − 1 ψ ) | : ψ ∈ C k − 1 ( X ; R ) } .

Hamming Weight of a Cochain The Weight of a k -cochain φ ∈ C k ( X ; R ): � [ φ ] � = min { | supp ( φ + d k − 1 ψ ) | : ψ ∈ C k − 1 ( X ; R ) } . Example: � φ � = 3 but � [ φ ] � = 1 A 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 B 1 1 φ − d 0 1 A , B φ d 0 1 A , B

Expansion of a Complex Expansion of a Cochain The expansion of φ ∈ C k ( X ; R ) − B k ( X ; R ) is � d k φ � � [ φ ] �

Expansion of a Complex Expansion of a Cochain The expansion of φ ∈ C k ( X ; R ) − B k ( X ; R ) is � d k φ � � [ φ ] � k -Cheeger Constant � � d k φ � � � [ φ ] � : φ ∈ C k ( X ; R ) − B k ( X ; R ) h k ( X ; R ) = min .

Expansion of a Complex Expansion of a Cochain The expansion of φ ∈ C k ( X ; R ) − B k ( X ; R ) is � d k φ � � [ φ ] � k -Cheeger Constant � � d k φ � � � [ φ ] � : φ ∈ C k ( X ; R ) − B k ( X ; R ) h k ( X ; R ) = min . Remarks: ◮ h k ( X ; R ) > 0 ⇔ ˜ H k ( X ; R ) = 0 . ◮ In the sequel: h k ( X ) = h k ( X ; F 2 ) .

Cheeger Constants of a Simplex ∆ n − 1 = the ( n − 1)-dimensional simplex on V = [ n ]. Claim [M-Wallach, Gromov]: n h k − 1 (∆ n − 1 ) = k + 1 .

Cheeger Constants of a Simplex ∆ n − 1 = the ( n − 1)-dimensional simplex on V = [ n ]. Claim [M-Wallach, Gromov]: n h k − 1 (∆ n − 1 ) = k + 1 . Example: [ n ] = ∪ k n i =0 V i , | V i | = k +1 φ = 1 V 0 ×···× V k − 1 n k +1 ) k � [ φ ] � = ( k +1 ) k +1 n � d k − 1 φ � = (

The Affine Overlap Property Number of Intersecting Simplices For A = { a 1 , . . . , a n } ⊂ R k and p ∈ R k let γ A ( p ) = |{ σ ⊂ [ n ] : | σ | = k + 1 , p ∈ conv { a i } i ∈ σ }| .

The Affine Overlap Property Number of Intersecting Simplices For A = { a 1 , . . . , a n } ⊂ R k and p ∈ R k let γ A ( p ) = |{ σ ⊂ [ n ] : | σ | = k + 1 , p ∈ conv { a i } i ∈ σ }| . Theorem [B´ ar´ any]: There exists a p ∈ R k such that � � 1 n − O ( n k ) . f A ( p ) ≥ ( k + 1) k k + 1

The Topological Overlap Property Number of Intersecting Images For a continuous map f : ∆ n − 1 → R k and p ∈ R k let γ f ( p ) = |{ σ ∈ ∆ n − 1 ( k ) : p ∈ f ( σ ) }| .

The Topological Overlap Property Number of Intersecting Images For a continuous map f : ∆ n − 1 → R k and p ∈ R k let γ f ( p ) = |{ σ ∈ ∆ n − 1 ( k ) : p ∈ f ( σ ) }| . Theorem [Gromov]: There exists a p ∈ R k such that � � 2 k n − O ( n k ) . γ f ( p ) ≥ ( k + 1)!( k + 1) k + 1

Topological Overlap and Expansion Number of Intersecting Images For a continuous map f : X → R k and p ∈ R k let γ f ( p ) = |{ σ ∈ X ( k ) : p ∈ f ( σ ) }| .