Counting social interactions for discrete subsets of the plane - PowerPoint PPT Presentation

Counting social interactions for discrete subsets of the plane Samantha Fairchild University of Washington skayf@uw.edu

Overview The Golden L and holonomy vector population density 1 Counting closed geodesics via Γ-orbits 2 Expected populations on n -street 3 Few nearby neighbors 4 BREAK 5 Higher moments of the Siegel–Veech transform 6 Proof ideas: Orbit decomposition and counting orbits 7

The Golden L

Holonomy Vectors on the Golden L

Veech ’98 Set of closed geodesics are finite union of H 5 orbits. Λ 5 = H 5 · e 1 ⊔ H 5 · ue 1 √ u = 1 + 5 2 � � �� � � �� π 0 1 1 2 cos q H q = , − 1 0 0 1

Looking at one orbit �� � � � � �� π 0 1 1 2 cos q V = H 5 · e 1 H q = , − 1 0 0 1

Looking at one orbit � � �� � � �� π 0 1 1 2 cos V = H 5 · e 1 H q = , q − 1 0 0 1

Our friend the Torus � � �� � � �� π 0 1 1 2 cos V = H 3 · e 1 H q = , q − 1 0 0 1

Population Density on the torus Assuming Riemann Hypothesis (Wu, 2002) # { V ∩ B (0 , R ) } = 6 221 π 2 ( π R 2 ) + O ( R 304 + ǫ )

Population density on the Golden L Theorem (BNRW 2019) # { V ∩ B (0 , R ) } = 10 4 3 π 2 · π R 2 + O ( R 3 )

Population density on the Golden L Theorem (BNRW 2019) # { V ∩ B (0 , R ) } = 10 4 3 π 2 · π R 2 + O ( R 3 ) Theorem (Burrin-F., Coming soon!) Ω bounded Jordan measurable domain E (# { V ∩ R · Ω } ) = 10 3 π 2 · | Ω | R 2 + O ( R c ) where c = max { 4 3 , 2 s 1 } .

Theorem (Burrin-F., Coming soon!) Ω bounded Jordan measurable domain E (# { V ∩ R · Ω } ) = 10 3 π 2 · | Ω | R 2 + O ( R c ) where c = max { 4 3 , 2 s 1 } . Big proof idea: Count pairs of vectors in V ! � � v 2 v 1 , w 2 Given v , w ∈ V ∩ B (0 , 30) with | v ∧ w | < 30 plot w 1

Population density on n th street Counting Pairs by determinant (F. 2019) 3 π 2 · π 2 E ( { v , w ∈ V ∩ B (0 , R ) : | v ∧ w | = n } ) ∼ 10 n · ϕ ( n ) · R 2

Population density on n th street

Density of nearby neighbors Corollary to F.2019, Coming soon! For all δ > 0, there exists ǫ > 0 so that # { v ∈ V ∩ B (0 , R ) : ∃ w ∈ V ∩ B ( v , ǫ ) } lim sup < δ. R 2 R →∞

Density of nearby neighbors Corollary to F.2019, Coming soon! For all δ > 0, there exists ǫ > 0 so that # { v ∈ V ∩ B (0 , R ) : ∃ w ∈ V ∩ B ( v , ǫ ) } lim sup < δ. R 2 R →∞ v , w ∈ V ∩ B (0 , 50) | v ∧ w | = 1 || w ∈ B ( v , 1 / 2)

Break

Siegel–Veech Integral Formula Γ < SL (2 , R ) non-uniform lattice Non-uniform : SL (2 , R ) / Γ not compact Lattice : Γ is discrete with c (Γ) def = vol( SL (2 , R ) / Γ) < ∞ . V = Γ · e 1

Siegel–Veech Integral Formula Γ < SL (2 , R ) non-uniform lattice Non-uniform : SL (2 , R ) / Γ not compact Lattice : Γ is discrete with c (Γ) def = vol( SL (2 , R ) / Γ) < ∞ . V = Γ · e 1 Theorem (Veech ’98) For f ∈ B c ( R 2 ) define the Siegel–Veech transform � f : SL (2 , R ) / Γ → R � � f ( g ) = f ( gv ) v ∈ V

Siegel–Veech Integral Formula Γ < SL (2 , R ) non-uniform lattice Non-uniform : SL (2 , R ) / Γ not compact Lattice : Γ is discrete with c (Γ) def = vol( SL (2 , R ) / Γ) < ∞ . V = Γ · e 1 Theorem (Veech ’98) For f ∈ B c ( R 2 ) define the Siegel–Veech transform � f : SL (2 , R ) / Γ → R � � f ( g ) = f ( gv ) v ∈ V the Siegel–Veech mean value formula � � 1 � f ( g ) d µ ( g ) = R 2 f ( x ) dx . c (Γ) SL (2 , R ) / Γ

Siegel–Veech Integral Formula Theorem (Veech ’98) For f ∈ B c ( R 2 ) define the Siegel–Veech transform � f : SL (2 , R ) / Γ → R � � f ( g ) = f ( gv ) v ∈ V the Siegel–Veech mean value formula � � 1 � f ( g ) d µ ( g ) = R 2 f ( x ) dx . c (Γ) SL (2 , R ) / Γ 1 c (Γ) · π R 2 # { V ∩ B (0 , R ) } ∼

Higher moments for general Γ Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + f g f g d η ( g ) 0 c (Γ) n SL (2 , R ) n ∈ N (Γ)

Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + f g f g d η ( g ) 0 n c (Γ) SL (2 , R ) n ∈ N (Γ) f ) k for all k ∈ N . (F. 2019) integral formula for ( �

Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + f g f g d η ( g ) 0 n c (Γ) SL (2 , R ) n ∈ N (Γ) N (Γ) is set of possible determinants. N (Γ) = { n ∈ R : ∃ v 1 , v 2 ∈ V s.t. | v 1 ∧ v 2 | = n } .

Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + f g f g d η ( g ) 0 c (Γ) n SL (2 , R ) n ∈ N (Γ) �� �� 1 h 1 Maximal parabolic Γ 0 = stab σ − 1 Γ σ ( e 1 ) = . 0 1

Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + f g f g d η ( g ) 0 c (Γ) n SL (2 , R ) n ∈ N (Γ) �� �� 1 h 1 Maximal parabolic Γ 0 = stab σ − 1 Γ σ ( e 1 ) = . 0 1 2 � �� � �� � � � � �� � � � � m ∗ ∗ � � � � ϕ ( n ) = ∈ V : 0 ≤ m < h | n | � = Γ 0 γ Γ 0 : γ = ∈ Γ � . � � � � � � ∗ n n

Sketch of Proof Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + f g f g d η ( g ) c (Γ) 0 n SL (2 , R ) n ∈ N (Γ) Note � f : SL (2 , R ) / Γ → R � � f ( g ) = f ( gv ) v ∈ V Implies � � 2 ( g ) = � � f f ( gv 1 ) f ( gv 2 ) ( v 1 , v 2 ) ∈ V × V

Sketch of Proof Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + f g f g d η ( g ) c (Γ) 0 n SL (2 , R ) n ∈ N (Γ) Decompose V × V into SL (2 , R )-orbits: V × V = { ( v , v ) : v ∈ V } ⊔ { ( v , − v ) : v ∈ V }⊔ � { ( v , w ) ∈ V × V : | v ∧ w | = n } n ∈ N (Γ)

Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + d η ( g ) f g f g 0 n c (Γ) SL (2 , R ) n ∈ N (Γ) Decompose V × V into SL (2 , R )-orbits: V × V = { ( v , v ) : v ∈ V } ⊔ { ( v , − v ) : v ∈ V }⊔ � { ( v , w ) ∈ V × V : | v ∧ w | = n } n ∈ N (Γ)

Theorem (Fairchild ’19) � � � 2 ( g ) d µ ( g ) � f SL (2 , R ) / Γ � 1 = R 2 f ( x ) f ( x ) + f ( x ) f ( − x ) dx c (Γ) � � �� � � �� � � ϕ ( n ) 1 1 + d η ( g ) f g f g 0 n c (Γ) SL (2 , R ) n ∈ N (Γ) Decompose V × V into SL (2 , R )-orbits: V × V = { ( v , v ) : v ∈ V } ⊔ { ( v , − v ) : v ∈ V }⊔ � { ( v , w ) ∈ V × V : | v ∧ w | = n } n ∈ N (Γ)

Reduction to Γ orbits of D n Lemma � � f ( gv 1 ) f ( gv 2 ) d µ ( g ) SL (2 , R ) / Γ ( v 1 , v 2 ) ∈ D n � � �� � � �� � = ϕ ( n ) 1 1 f g f g d η ( g ) 0 c (Γ) n SL (2 , R ) For n ∈ N (Γ) define D n = { ( v , w ) ∈ V × V : | v ∧ w | = n } Want to use � � � 1 f ( g γ v 1 ) f ( g γ v 2 ) d µ ( g ) = f ( gv 1 ) f ( gv 2 ) d η ( g ) . c (Γ) SL (2 , R ) / Γ SL (2 , R ) γ ∈ Γ

ϕ is number of Γ orbits of D n Lemma � � � 1 j D n = Γ · 0 n 1 ≤ j ≤ h | n | ( j , n ) T ∈ V Thus there are ϕ ( n ) orbits. Each has a contribution of � � �� � � �� � 1 1 j d η ( g ) f g f g 0 c (Γ) n SL (2 , R )

ϕ is number of Γ orbits of D n Lemma � � � 1 j D n = Γ · 0 n 1 ≤ j ≤ h | n | ( j , n ) T ∈ V Thus there are ϕ ( n ) orbits. Each has a contribution of � � �� � � �� � 1 1 j d η ( g ) f g f g 0 c (Γ) n SL (2 , R ) Theorem � � f ( gv 1 ) f ( gv 2 ) d µ ( g ) SL (2 , R ) / Γ v 1 , v 2 ∈ D n � � �� � � �� � = ϕ ( n ) 1 1 d η ( g ) f g f g 0 n c (Γ) SL (2 , R )

From integrals to asymptotics � � �� � � �� � � ϕ ( n ) 1 1 d η f g f g 0 n c (Γ) SL (2 , R ) n ∈ N (Γ) � � 1 = R 2 f ( x ) f ( y ) ω ( | x ∧ y | ) dx dy c (Γ) R 2 where � ϕ ( n ) ω ( t ) = n 3 n ≥ t n ∈ N (Γ)