Computing the moments of the GOE bijectively Olivier Bernardi (MIT) - PowerPoint PPT Presentation

Computing the moments of the GOE bijectively Olivier Bernardi (MIT) 4 8 2 1 λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 Probability Seminar at MIT, February 2011

Computing the moments of the GOE bijectively Olivier Bernardi (MIT) n 2+3=5; 4 8 2 1 λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 Probability Seminar at MIT, February 2011

A combinatorial problem

Surfaces from a polygon We consider the different ways of gluing the sides of a 2 n -gon in pairs.

Surfaces from a polygon We consider the different ways of gluing the sides of a 2 n -gon in pairs. The gluing of two sides can either be orientable (giving a cylinder) or non-orientable (giving a M¨ obius strip). Orientable gluing Non-orientable gluing The surface obtained is orientable if and only if each gluing is orientable.

Surfaces from a polygon We consider the different ways of gluing the sides of a 2 n -gon in pairs. There are (2 n − 1)!! = (2 n − 1)(2 n − 3) · · · 3 ways of obtaining an orientable surface. There are 2 n (2 n − 1)!! ways of obtaining a general surface.

Surfaces from a polygon We consider the different ways of gluing the sides of a 2 n -gon in pairs. There are (2 n − 1)!! = (2 n − 1)(2 n − 3) · · · 3 ways of obtaining an orientable surface. There are 2 n (2 n − 1)!! ways of obtaining a general surface. Question: How many ways are there to obtain each surface (consid- ered up to homeomorphism) ?

Surfaces from a polygon We consider the different ways of gluing the sides of a 2 n -gon in pairs. There are (2 n − 1)!! = (2 n − 1)(2 n − 3) · · · 3 ways of obtaining an orientable surface. There are 2 n (2 n − 1)!! ways of obtaining a general surface. Question: How many ways are there to obtain each surface (consid- ered up to homeomorphism) ? Example: The number of ways of getting the sphere is the Catalan � 2 n 1 � number Cat ( n ) = . n +1 n

Surfaces from a polygon We consider the different ways of gluing the sides of a 2 n -gon in pairs. Question: How many ways are there to obtain a surface of type t ? Type: 0 1 2 3 4 By the Euler relation, the type of the surface is t = n + 1 − # vertices .

The Gaussian Orthogonal Ensemble

The GOE Let S p be the set of real symmetric matrices of dimension p × p . s i,j = s j,i S = We define a random variable S in S p by choosing the entries s i,j for i ≤ j to be independent centered Gaussian variables with variance 2 if i = j and variance 1 if i < j (and then setting s i,j = s j,i for i > j ).

The GOE Let S p be the set of real symmetric matrices of dimension p × p . s i,j = s j,i S = We define a random variable S in S p by choosing the entries s i,j for i ≤ j to be independent centered Gaussian variables with variance 2 if i = j and variance 1 if i < j (and then setting s i,j = s j,i for i > j ). Hence, the distribution γ of S over S p has density κ exp( − tr ( S 2 ) / 4) with respect to the Lebesgue measure dS := � i ≤ j ds i,j . The GOE is the probability space ( S p , γ ) .

Eigenvalues of the GOE Let λ 1 ≤ λ 2 ≤ · · · ≤ λ p be the eigenvalues of S . s i,j = s j,i S = λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 Question: What is the distribution of λ := λ U , with U uniform in [ p ] ?

Eigenvalues of the GOE Let λ 1 ≤ λ 2 ≤ · · · ≤ λ p be the eigenvalues of S . s i,j = s j,i S = λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 Question: What is the distribution of λ := λ U , with U uniform in [ p ] ? p � � 1 = 1 � λ n p � tr ( S n ) � . The n th moment of λ is i p i =1 Remark. Odd moments are 0 by symmetry.

Computing the 2 n th moment using the Wick formula � We want the expectation of tr ( S 2 n ) = s i 1 ,i 2 s i 2 ,i 3 · · · s i 2 n ,i 1 . i 1 ,i 2 ,...,i 2 n ∈ [ p ] Since the s i,j are Gaussian, the Wick formula gives � � � � � s i 1 ,i 2 s i 2 ,i 3 · · · s i 2 n ,i 1 � = s i k ,i k +1 s i l ,i l +1 . π matching on [2 n ] { k,l }∈ π

Computing the 2 n th moment using the Wick formula � We want the expectation of tr ( S 2 n ) = s i 1 ,i 2 s i 2 ,i 3 · · · s i 2 n ,i 1 . i 1 ,i 2 ,...,i 2 n ∈ [ p ] Since the s i,j are Gaussian, the Wick formula gives � � � � � s i 1 ,i 2 s i 2 ,i 3 · · · s i 2 n ,i 1 � = s i k ,i k +1 s i l ,i l +1 . π matching on [2 n ] { k,l }∈ π � � � tr ( S 2 n ) � � � � = s i k ,i k +1 s i l ,i l +1 . π matching on [2 n ] i 1 ...i 2 n ∈ [ p ] { k,l }∈ π Contribution of matching π ?

Computing the 2 n th moment using the Wick formula � We want the expectation of tr ( S 2 n ) = s i 1 ,i 2 s i 2 ,i 3 · · · s i 2 n ,i 1 . i 1 ,i 2 ,...,i 2 n ∈ [ p ] Since the s i,j are Gaussian, the Wick formula gives � � � � � s i 1 ,i 2 s i 2 ,i 3 · · · s i 2 n ,i 1 � = s i k ,i k +1 s i l ,i l +1 . π matching on [2 n ] { k,l }∈ π � � � tr ( S 2 n ) � � � � = s i k ,i k +1 s i l ,i l +1 . π matching on [2 n ] i 1 ...i 2 n ∈ [ p ] { k,l }∈ π Contribution of matching π ? � � Hint: s i k ,i k +1 s i l ,i l +1 = 0 except if ( i k , i k +1 ) = ( i l , i l +1 ) or ( i l +1 , i l ) . i 1 i 1 i 2 n i 2 i 2 n i 2 i 3 i 3 i 4 i 4 N O i 5 i 5

Computing the 2 n th moment using the Wick formula � � � tr ( S 2 n ) � � � � p = s i k ,i k +1 s i l ,i l +1 π matching on [2 n ] i 1 ...i 2 n ∈ [ p ] { k,l }∈ π � � � = 1 i k = i l ,i k +1 = i l +1 π matching on [2 n ] { k,l }∈ ǫ − 1 ( N ) i 1 ...i 2 n ∈ [ p ] ǫ : π →{ O,N } � × 1 i k = i l +1 ,i k +1 = i l { k,l }∈ ǫ − 1 ( O ) � p # vertices . = gluing of 2 n -gon i 1 i 1 i 2 n i 2 i 2 n i 2 i 3 i 3 i 4 i 4 N O i 5 i 5

Computing the 2 n th moment using the Wick formula � � � tr ( S 2 n ) � � � � p = s i k ,i k +1 s i l ,i l +1 π matching on [2 n ] i 1 ...i 2 n ∈ [ p ] { k,l }∈ π � � � = 1 i k = i l ,i k +1 = i l +1 π matching on [2 n ] { k,l }∈ ǫ − 1 ( N ) i 1 ...i 2 n ∈ [ p ] ǫ : π →{ O,N } � × 1 i k = i l +1 ,i k +1 = i l { k,l }∈ ǫ − 1 ( O ) � p # vertices . = gluing of 2 n -gon Summary: The 2 n th moment of λ in GOE(p) is 1 � p # vertices . p gluings of 2 n -gon

Computing the 2 n th moment using the Wick formula � � � tr ( S 2 n ) � � � � p = s i k ,i k +1 s i l ,i l +1 π matching on [2 n ] i 1 ...i 2 n ∈ [ p ] { k,l }∈ π � � � = 1 i k = i l ,i k +1 = i l +1 π matching on [2 n ] { k,l }∈ ǫ − 1 ( N ) i 1 ...i 2 n ∈ [ p ] ǫ : π →{ O,N } � × 1 i k = i l +1 ,i k +1 = i l { k,l }∈ ǫ − 1 ( O ) � p # vertices . = gluing of 2 n -gon Summary: The 2 n th moment of λ in GOE(p) is 1 � p # vertices . p gluings of 2 n -gon λ Remark. When p → ∞ the 2 n th moment of which is √ p p # vertices − n − 1 tends to Cat ( n ) . � ⇒ Semi-circle law. gluings of 2 n -gon

Back to the polygon

Surfaces from a polygon: state of the art Question: How many ways of gluing 2 n -gon into a surface of type t ? Type: 0 1 2 3 4

Surfaces from a polygon: state of the art Question: How many ways of gluing 2 n -gon into a surface of type t ? v = # vertices 1 1 1 η v ( n ) 5 5 2 41 52 22 5 n = # edges 509 690 374 93 14

Surfaces from a polygon: state of the art Question: How many ways of gluing 2 n -gon into a surface of type t ? v = # vertices 1 1 1 η v ( n ) 5 5 2 41 52 22 5 n = # edges 509 690 374 93 14 [Harer, Zagier 86] Results for orientable surfaces : v ǫ v ( n ) p v and recurrence formula for ǫ v ( n ) . Formula for � v = # vertices 1 0 1 ǫ v ( n ) 1 0 2 0 10 0 5 n = # edges 21 0 70 0 14

Surfaces from a polygon: state of the art Question: How many ways of gluing 2 n -gon into a surface of type t ? v = # vertices 1 1 1 η v ( n ) 5 5 2 41 52 22 5 n = # edges 509 690 374 93 14 [Harer, Zagier 86] Results for orientable surfaces : v ǫ v ( n ) p v and recurrence formula for ǫ v ( n ) . Formula for � p � p n ǫ v ( n ) p v = � 2 q − 1 � � � � (2 n − 1)!! q − 1 q v q =1 ( n + 1) ǫ v ( n ) = (4 n − 2) ǫ v − 1 ( n − 1) + ( n − 1)(2 n − 1)(2 n − 3) ǫ v ( n − 2) . Related work : Jackson, Adrianov, Zagier, Mehta, Haagerup-Thorbjornsen, Kerov, Ledoux, Lass, Goulden-Nica, Schaeffer-Vassilieva, Morales-Vassilieva, Chapuy. . .

Surfaces from a polygon: state of the art Question: How many ways of gluing 2 n -gon into a surface of type t ? v = # vertices 1 1 1 η v ( n ) 5 5 2 41 52 22 5 n = # edges 509 690 374 93 14 [Harer, Zagier 86] Results for orientable surfaces : v ǫ v ( n ) p v and recurrence formula for ǫ v ( n ) . Formula for � v η v ( n ) p v . [Goulden, Jackson 97] Formula for � [Ledoux 09] Recurrence formula for η v ( n ) .

Surfaces from a polygon: state of the art Question: How many ways of gluing 2 n -gon into a surface of type t ? v = # vertices 1 1 1 η v ( n ) 5 5 2 41 52 22 5 n = # edges 509 690 374 93 14 [Harer, Zagier 86] Results for orientable surfaces : v ǫ v ( n ) p v and recurrence formula for ǫ v ( n ) . Formula for � v η v ( n ) p v . [Goulden, Jackson 97] Formula for � [Ledoux 09] Recurrence formula for η v ( n ) . [B., Chapuy 10] Asymptotic η t ( n ) ∼ n →∞ c t n 3( t − 1) / 2 4 n ,  2 t − 2 if t odd , √  t − 1 ( t − 1)!! 6 where c t = 3 · 2 t − 2 � t/ 2 − 1 � 2 i 16 − i � if t even . √ √ π t ( t − 1)!! i =1  i 6