Random gluing polygons Sergei Chmutov joint work with Boris Pittel - PowerPoint PPT Presentation

Random gluing polygons Sergei Chmutov joint work with Boris Pittel Ohio State University, Mansfield Stochastic Topology and Thermodynamic Limits Workshop at ICERM Wednesday, October 19, 2016 9:00 — 9:45 a.m. Sergei Chmutov Random gluing polygons

Polygons. Notations. n := # (oriented) polygons N := total (even) number of sides � n j = n , � jn j = N n j := # j -gons, [ N ] := { 1 , 2 , . . . , N } α ∈ S N is a permutation of [ N ] cyclically permutes edges of polygons according to their orientations. Example. 1 5 α = ( 1234 )( 5678 ) 2 8 6 4 3 7 n j equals the number of cycles of α of length j . Sergei Chmutov Random gluing polygons

Gluing polygons. Permutations. β ∈ S N is an involution without fixed points; β has N / 2 cycles of length 2. α ( β ( e 2 ))=: e 3 Σ α,β ) β ( e e 3 2 α ( β ( e 1 ) ) = : e 2 e 2 e 1 γ := αβ β ( e 1 ) 1 e # vertices of Σ α,β = # cycles of γ . # connected components of Σ α,β = # orbits of the subgroup generated by α and β . Sergei Chmutov Random gluing polygons

Gluing polygons. Example. n = 2, N = 8, α = ( 1234 )( 5678 ) 1 5 2 8 6 4 3 7 1 5 5 1 6 2 β = ( 15 )( 28 )( 37 )( 46 ) 4 6 4 2 8 8 γ = ( 16 )( 25 )( 38 )( 47 ) 7 3 3 7 Sergei Chmutov Random gluing polygons

Gluing polygons. Example. n = 2, N = 8, α = ( 1234 )( 5678 ) 1 1 2 4 5 6 2 β = ( 15 )( 24 )( 37 )( 68 ) 3 5 3 8 γ = ( 1652 )( 3874 ) 4 8 6 7 7 Sergei Chmutov Random gluing polygons

Gluing polygons. Example. n = 2, N = 8, α = ( 1234 )( 5678 ) 1 3 4 1 2 2 4 β = ( 13 )( 24 )( 57 )( 68 ) 3 5 7 γ = ( 1432 )( 5876 ) 8 5 6 8 6 7 Sergei Chmutov Random gluing polygons

Gluing polygons. Example. n = 2, N = 8, α = ( 1234 )( 5678 ) 1 3 4 1 2 2 4 β = ( 13 )( 24 )( 56 )( 78 ) 3 5 γ = ( 1432 )( 57 )( 6 )( 8 ) 6 5 8 6 7 8 7 Sergei Chmutov Random gluing polygons

Gluing polygons. Example. n = 2, N = 8, α = ( 1234 )( 5678 ) 1 2 1 2 4 3 4 β = ( 12 )( 34 )( 56 )( 78 ) 3 5 γ = ( 13 )( 2 )( 4 )( 57 )( 6 )( 8 ) 6 5 8 6 7 8 7 Sergei Chmutov Random gluing polygons

Gluing polygons. Example. n = 2, N = 8, α = ( 1234 )( 5678 ) There are 7 !! = 105 possibilities for choosing β . T 2 + S 2 S 2 T 2 2 T 2 2 S 2 surface Σ α,β # gluings 36 60 1 4 4 Sergei Chmutov Random gluing polygons

Random gluing. n := { n j } is a partition of n = � n j . Let C n be the conjugacy class of α , all permutations in S N with the cycle structure n . Let C N / 2 be the conjugacy class of β , all permutations in S N with all cycles length 2. A random surface is the surface Σ α,β obtained by gluing according to the permutations α and β that are independently chosen uniformly at random from the conjugacy classes C n and C N / 2 respectively. Sergei Chmutov Random gluing polygons

Harer-Zagier formula. n = 1. n = 1, α = ( 123 . . . N ) . 1 Example: N = 6 6 2 6 2 1 3 5 4 5 3 4 1 4 5 6 2 6 1 3 2 5 3 4 Sergei Chmutov Random gluing polygons

Harer-Zagier formula. n = 1, N = 6. n = 1, N = 6 V n =# vertices of Σ α,β . |C N / 2 | = 5 !! = 15. V n = 4 V n = 2 Generating function: T N ( y ) := � T 6 ( y ) = 5 y 4 + 10 y 2 . y V n . β Sergei Chmutov Random gluing polygons

Harer-Zagier formula. J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves , Invent. Math. 85 (1986) 457–485. T N ( y ) := � y V n . β � ∞ T 2 k ( y ) ( 2 k − 1 )!! x k . Generating function: T ( x , y ) := 1 + 2 xy + 2 x k = 1 � 1 + x � y T ( x , y ) = 1 − x —,B.Pittel, JCTA 120 (2013) 102–110: g N = genus of Σ α,β . Asymptotically as N → ∞ , g N is normal N (( N − log N ) / 2 , ( log N ) / 4 ) . Sergei Chmutov Random gluing polygons

Main result. —,B.Pittel, On a surface formed by randomly gluing together polygonal discs , Advances in Applied Mathematics, 73 (2016) 23–42. V n =# vertices of Σ α,β . Theorem. V n is asymptotically normal with mean and variance log N both, V n ∼ N ( log N , log N ) , as N → ∞ , and uniformly on n . Sergei Chmutov Random gluing polygons

Previous results. E [ V n ] ∼ log n Var ( χ ) ∼ log n N. Pippenger, K. Schleich, Topological characteristics of random triangulated surfaces , Random Structures Algorithms, 28 (2006) 247–288. All polygons are triangles. A. Gamburd, Poisson-Dirichlet distribution for random Belyi surfaces , Ann. Probability, 34 (2006) 1827–1848. All polygons have the same number of sides, k . 2 lcm ( 2 , k ) | kn γ is asymptotically uniform on the alternating group A kn . Sergei Chmutov Random gluing polygons

Key Theorem. Depending on the parities of permutations α ∈ C n and β ∈ C N / 2 the permutation γ = αβ is either even γ ∈ A N or odd γ ∈ A c N := S N − A N . The probability distribution of γ is asymptotically uniform (for N → ∞ uniformly in n ) on A N or on A c N . Let P γ be the probability distribution of γ and let U be the uniform probability measure on A N or on A c N . Let � P γ − U � := ( 1 / 2 ) � s ∈ S N | P γ ( s ) − U ( s ) | be the total variation distance between P γ and U . � N − 1 � Theorem. � P γ − U � = O . Sergei Chmutov Random gluing polygons

Ideas of the proof. P . Diaconis, M. Shahshahani, Generating a random permutation with random tranpositions , Z. Wahr. Verw. Gebiete, 57 (1981) 159–179. Using the Fourier analysis on finite groups and the Plancherel Theorem: � � ˆ P ( ρ ) ∗ � � P − U � 2 ≤ 1 P ( ρ )ˆ dim ( ρ ) tr ; 4 ρ ∈ � G , ρ � = id here � G denotes the set of all irreducible representations ρ of G , “id” denotes the trivial representation, dim ( ρ ) is the dimension of ρ , and ˆ P ( ρ ) is the matrix value of the Fourier transform of P P ( ρ ) := � at ρ , ˆ g ∈ G ρ ( g ) P ( g ) . Sergei Chmutov Random gluing polygons

Ideas of the proof. For G = S N , the irreducible representations ρ are indexed by partitions λ ⊢ N , λ = ( λ 1 ≥ λ 2 ≥ . . . ) of N . Let f λ := dim ( ρ λ ) (given by the hook length formula) and χ λ be the character of ρ λ . � � 2 � χ λ ( C n ) χ λ ( C N / 2 ) � P γ − U � 2 ≤ 1 . f λ 4 λ � =( N ) , ( 1 N ) Gamburd used estimate from S. V. Fomin, N. Lulov, On the number of rim hook tableaux , J. Math. Sciences, 87 (1997) 4118–4123, for N = kn , � N 1 / 2 − 1 / ( 2 k ) � | χ λ ( C N / k ) | = O ( f λ ) 1 / k . Sergei Chmutov Random gluing polygons

Ideas of the proof. M. Larsen, A. Shalev, Characters of symmetric groups: sharp bounds and applications , Invent. Math., 174 (2008) 645–687. Extension of the Fomin-Lulov bound for all permutations σ without cycles of length below m , and partitions λ : | χ λ ( σ ) | ≤ ( f λ ) 1 / m + o ( 1 ) , N → ∞ . � P γ − U � 2 = O ( N − 2 ) . Sergei Chmutov Random gluing polygons

Thanks. THANK YOU! Sergei Chmutov Random gluing polygons