Fluctuations of Real Random Matrices and Second-Order Freeness - PowerPoint PPT Presentation

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Definition The nth mixed moment of (classical) random variables X 1 , . . . , X n is an n -linear function defined to be the expectation of their product: a n ( X 1 , . . . , X n ) := E ( X 1 · · · X n ) . Let P ( n ) be the set of partitions of n elements. Definition We define the cumulants k i to satisfy the moment-cumulant formula : � � a n ( X 1 , . . . , X n ) = k r ( X i 1 , . . . , X i r ) . π ∈P ( n ) V = { i 1 ,..., i r }∈ π Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The first four cumulants are: k 1 ( X ) = E ( X ) k 2 ( X , Y ) = E ( XY ) − E ( X ) E ( Y ) k 3 ( X , Y , Z ) = E ( XYZ ) − E ( X ) E ( YZ ) − E ( XY ) E ( Y ) − E ( XY ) E ( Z ) + 2 E ( X ) E ( Y ) E ( Z ) k 4 ( X , Y , Z , W ) = E ( XYZW ) − E ( X ) E ( YZW ) − E ( XZW ) E ( Y ) − E ( XYW ) E ( Z ) − E ( XYZ ) E ( W ) − E ( XY ) E ( ZW ) − E ( XZ ) E ( YW ) − E ( XW ) E ( YZ ) + 2 E ( XY ) E ( Z ) E ( W ) + 2 E ( XZ ) E ( Y ) E ( W ) + 2 E ( XW ) E ( Y ) E ( Z ) + 2 E ( X ) E ( YZ ) E ( W ) + 2 E ( X ) E ( YW ) E ( Z ) + 2 E ( X ) E ( Y ) E ( ZW ) − 6 E ( X ) E ( Y ) E ( Z ) E ( W ) . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Say we wish to calculate � � � � �� XY 1 XY 2 X T Y 3 XY 4 X T Y 5 X T Y 6 XY 7 XY 8 E tr tr . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Say we wish to calculate � � � � �� XY 1 XY 2 X T Y 3 XY 4 X T Y 5 X T Y 6 XY 7 XY 8 E tr tr . The traces of products are a sum over X i 1 j 1 Y (1) j 1 i 2 X i 2 j 2 Y (2) j 3 i 3 Y (3) i 3 i 4 X i 4 j 4 Y (4) j 5 i 5 Y (5) j 6 i 6 Y (6) i 6 i 7 X i 7 j 7 Y (7) j 7 i 8 X i 8 j 8 Y (8) j 2 j 3 X T j 4 j 5 X T i 5 i 1 X T j 8 j 6 . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We construct the faces: Y 1 i 2 j 1 j 2 X Y 2 Y 6 X i 6 j 3 X T j 6 i 7 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X j 5 i 4 Y 4 j 4 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We use a result called the Wick formula. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We use a result called the Wick formula. There are three pairings on 4 elements: Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We use a result called the Wick formula. There are three pairings on 4 elements: If X 1 , X 2 , X 3 , X 4 are components of a multivariate Gaussian random variable, then E ( X 1 X 2 X 3 X 4 ) = E ( X 1 X 2 ) E ( X 3 X 4 ) + E ( X 1 X 3 ) E ( X 2 X 4 ) + E ( X 1 X 4 ) E ( X 2 X 3 ) . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Let P 2 ( n ) be the set of pairings on n elements. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Let P 2 ( n ) be the set of pairings on n elements. Theorem Let { f λ : λ ∈ Λ } , for some index set Λ , be a centred Gaussian family of random variables. Then for i 1 , . . . , i n ∈ Λ , � � E ( f i 1 · · · f i n ) = E ( f i k f i l ) . P 2 ( n ) { k , l }∈P 2 ( n ) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Let P 2 ( n ) be the set of pairings on n elements. Theorem Let { f λ : λ ∈ Λ } , for some index set Λ , be a centred Gaussian family of random variables. Then for i 1 , . . . , i n ∈ Λ , � � E ( f i 1 · · · f i n ) = E ( f i k f i l ) . P 2 ( n ) { k , l }∈P 2 ( n ) Here, for a pairing π ∈ P 2 ( n ): � 1 , if i k = i l and j k = j l for all { k , l } ∈ π � E ( f i k j k f i l j l ) = . 0 , otherwise { k , l } Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Putting indices which must be equal next to each other, we get a surface gluing: Y 1 i 2 j 2 j 1 X Y 2 X Y 6 i 6 j 3 i 7 X T j 6 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X j 5 i 4 Y 4 j 4 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We note that if one term is from X and the other from X T , the edge identification is untwisted: Y 1 i 2 j 1 j 2 X Y 2 X Y 6 i 6 j 3 X T j 6 i 7 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X T j 5 i 4 Y 4 j 4 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations If both terms are from X or from X T , the edge identification is twisted: Y 1 i 2 j 1 j 2 X Y 2 X Y 6 i 6 j 3 X T j 6 i 7 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X T j 5 i 4 Y 4 j 4 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The following vertex appears on the surface: i 3 i 6 i 4 Y T i 7 Y 6 3 i 1 i 2 Y T Y 1 5 j 1 i 5 Y T j 7 7 i 8 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The following vertex appears on the surface: i 3 i 6 i 4 Y T i 7 Y 6 3 i 1 i 2 Y T Y 1 5 j 1 i 5 Y T j 7 7 i 8 If a corner appears upside-down, it is the transpose of that matrix which appears. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The following vertex appears on the surface: i 3 i 6 i 4 Y T i 7 Y 6 3 i 1 i 2 Y T Y 1 5 j 1 i 5 Y T j 7 7 i 8 If a corner appears upside-down, it is the transpose of that matrix which appears. It contributes � � Y 1 Y T 3 Y 6 Y T 5 Y T Tr . 7 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The same vertex viewed from the opposite side contributes the same value: i 3 i 6 i 4 i 7 Y 3 Y T 6 i 1 i 2 Y T Y 5 1 i 5 j 1 Y 7 i 8 j 7 � � � � Y 7 Y 5 Y T 6 Y 3 Y T Y 1 Y T 3 Y 6 Y T 5 Y T = Tr Tr . 1 7 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Each vertex gives us a trace, and hence a factor of N when normalized. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Each vertex gives us a trace, and hence a factor of N when normalized. Highest order terms are those with the highest Euler characteristic (typically spheres or collections of spheres). Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Each vertex gives us a trace, and hence a factor of N when normalized. Highest order terms are those with the highest Euler characteristic (typically spheres or collections of spheres). Crossings require handles, so highest order terms typically correspond to noncrossing diagrams with untwisted identifications. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Each vertex gives us a trace, and hence a factor of N when normalized. Highest order terms are those with the highest Euler characteristic (typically spheres or collections of spheres). Crossings require handles, so highest order terms typically correspond to noncrossing diagrams with untwisted identifications. Highest order terms must have a relative orientation of the faces in which none of the edge-identifications are twisted. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The permutation γ encodes face information (cycles enumerate edges in order). Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The permutation γ encodes face information (cycles enumerate edges in order). A pairing π , taken as a permutation, encodes edge information on an orientable surface. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations The permutation γ encodes face information (cycles enumerate edges in order). A pairing π , taken as a permutation, encodes edge information on an orientable surface. The permutation π − 1 γ − 1 encodes vertex information. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Consider the map: 5 2 6 3 1 4 7 12 9 11 8 10 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Consider the map: 5 2 6 3 1 4 7 12 9 11 8 10 The vertex information can be encoded in a permutation σ = (1 , 2 , 3 , 4) (5 , 6) (7 , 8) (9 , 10) (11 , 12) . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Consider the map: 5 2 6 3 1 4 7 12 9 11 8 10 The vertex information can be encoded in a permutation σ = (1 , 2 , 3 , 4) (5 , 6) (7 , 8) (9 , 10) (11 , 12) . The edge information can be encoded in another permutation α = (1 , 2) (3 , 5) (4 , 12) (6 , 7) (8 , 9) (10 , 11) . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations 5 2 6 1 3 4 7 12 9 11 8 10 The face information is encoded in ϕ := σ − 1 α − 1 = (1) (2 , 4 , 11 , 9 , 7 , 5) (3 , 6 , 8 , 10 , 12) . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations This construction works equally well with oriented hypermaps: 5 4 1 2 3 6 7 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations This construction works equally well with oriented hypermaps: 5 4 1 2 3 6 7 σ = (1 , 2 , 3) (4 , 5) (6 , 7) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations This construction works equally well with oriented hypermaps: 5 4 1 2 3 6 7 σ = (1 , 2 , 3) (4 , 5) (6 , 7) α = (1 , 6 , 5) (2 , 7 , 3) (4) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations This construction works equally well with oriented hypermaps: 5 4 1 2 3 6 7 σ = (1 , 2 , 3) (4 , 5) (6 , 7) α = (1 , 6 , 5) (2 , 7 , 3) (4) ϕ = σ − 1 α − 1 = (1 , 4 , 5 , 7) (2) (3 , 6) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations To extend this construction to unoriented surfaces, we construct the orientable two-sheeted covering space (the surface experienced by someone on the surface rather than within it). Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations To extend this construction to unoriented surfaces, we construct the orientable two-sheeted covering space (the surface experienced by someone on the surface rather than within it). We do this by constructing a front and back side of each face. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations To extend this construction to unoriented surfaces, we construct the orientable two-sheeted covering space (the surface experienced by someone on the surface rather than within it). We do this by constructing a front and back side of each face. An untwisted edge-identification connects front to front and back to back, while a twisted edge-identification connects front to back and back to front. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Y 1 i 2 j 1 j 2 X Y 2 Y 6 X i 6 j 3 X T j 6 i 7 i 1 X Y 8 X T Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X j 5 i 4 Y 4 j 4 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Y 1 X Y 2 Y 6 X X T X Y 8 X T Y 5 X Y 7 X T Y 3 X Y 4 Y T 4 X T Y T 3 Y 7 T X X Y T Y T 5 X T X 8 X T X T Y T 6 Y T 2 X T Y T 1 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We label the front sides with positive integers and the corresponding back sides with negative integers. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We label the front sides with positive integers and the corresponding back sides with negative integers. Let δ : k �→ − k . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We label the front sides with positive integers and the corresponding back sides with negative integers. Let δ : k �→ − k . A permutation π describing something in this surface should satisfy π = δπ − 1 δ . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We label the front sides with positive integers and the corresponding back sides with negative integers. Let δ : k �→ − k . A permutation π describing something in this surface should satisfy π = δπ − 1 δ . We let γ + = γ , and γ − = δγδ . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We label the front sides with positive integers and the corresponding back sides with negative integers. Let δ : k �→ − k . A permutation π describing something in this surface should satisfy π = δπ − 1 δ . We let γ + = γ , and γ − = δγδ . Vertex information is given by γ − 1 + π − 1 γ − . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations In the example, π = (1 , − 7) (7 , − 1) (2 , − 4) (4 , − 2) (3 , − 6) (6 , − 3) (5 , 8) ( − 8 , − 5) . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations In the example, π = (1 , − 7) (7 , − 1) (2 , − 4) (4 , − 2) (3 , − 6) (6 , − 3) (5 , 8) ( − 8 , − 5) . The vertices are given by the cycles of (1 , − 3 , 6 , − 5 , − 7) (7 , 5 , − 6 , 3 , − 1) (2 , − 8 , − 4) (4 , 8 , − 2) . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations In the example, π = (1 , − 7) (7 , − 1) (2 , − 4) (4 , − 2) (3 , − 6) (6 , − 3) (5 , 8) ( − 8 , − 5) . The vertices are given by the cycles of (1 , − 3 , 6 , − 5 , − 7) (7 , 5 , − 6 , 3 , − 1) (2 , − 8 , − 4) (4 , 8 , − 2) . This diagram contributes the term: � � � � �� N − 2 E Y 1 Y T 3 Y 6 Y T 5 Y T Y 2 Y T 8 Y T tr tr 7 4 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Let: ◮ tr := 1 N Tr , ◮ n 1 , . . . , n r positive integers, n := n 1 + · · · + n r , ◮ A (1) = A , A ( − 1) = A T , ◮ [ n ] = { 1 , . . . , n } , ◮ ε : [ n ] → { 1 , − 1 } , ◮ δ ε : k �→ ε ( k ) k . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations � � For γ = ( c 1 , . . . , c n 1 ) · · · c n 1 + ··· + n r − 1 , . . . , c n ∈ S n , we define: � � � � Tr γ ( A 1 , . . . , A n ) := Tr A c 1 · · · A c n 1 · · · Tr A c n 1+ ··· + nr − 1 · · · A c n . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations � � For γ = ( c 1 , . . . , c n 1 ) · · · c n 1 + ··· + n r − 1 , . . . , c n ∈ S n , we define: � � � � Tr γ ( A 1 , . . . , A n ) := Tr A c 1 · · · A c n 1 · · · Tr A c n 1+ ··· + nr − 1 · · · A c n . Then � Tr γ ( A 1 , . . . , A n ) = A i 1 i γ (1) · · · A i n i γ ( n ) . 1 ≤ i 1 ,..., i n ≤ N Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations For example: Tr (1 , 2 , 3 , 4 , 5 , 6)(7 , 8 , 9 , 10) ( A 1 , . . . , A 10 ) = Tr ( A 1 A 2 A 3 A 4 A 5 A 6 ) Tr ( A 7 A 8 A 9 A 10 ) N � A (1) i 1 , i 2 A (2) i 2 , i 3 A (3) i 3 , i 4 A (4) i 4 , i 5 A (5) i 5 , i 6 A (6) i 6 , i 1 A (7) i 7 , i 8 A (8) i 8 , i 9 A (9) i 9 , i 10 A (10) = i 10 , i 1 i 1 ,..., i 6 =1 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We wish to calculate expressions of the form � � �� X ( ε (1)) Y 1 · · · X ( ε ( n )) Y n E tr γ Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We wish to calculate expressions of the form � � �� X ( ε (1)) Y 1 · · · X ( ε ( n )) Y n E tr γ � � Y (1) · · · Y ( n ) � N − #( γ ) − n E = ι − ε (1) ι ε ( γ (1)) ι − ε ( n ) ι ε ( γ ( n )) n 1 γ (1) γ ( n ) 1 ≤ ι + 1 ,...,ι + n ≤ M 1 ≤ ι − 1 ,...,ι − n ≤ N � � 1 · · · f ι + E f ι + 1 ι − n ι − n Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We wish to calculate expressions of the form � � �� X ( ε (1)) Y 1 · · · X ( ε ( n )) Y n E tr γ � � Y (1) · · · Y ( n ) � N − #( γ ) − n E = ι − ε (1) ι ε ( γ (1)) ι − ε ( n ) ι ε ( γ ( n )) n 1 γ (1) γ ( n ) 1 ≤ ι + 1 ,...,ι + n ≤ M 1 ≤ ι − 1 ,...,ι − n ≤ N � � 1 · · · f ι + E f ι + 1 ι − n ι − n � � Y (1) · · · Y ( n ) � � N − #( γ ) − n E = . ι − ε (1) ι ε ( γ (1)) ι − ε ( n ) ι ε ( γ ( n )) n 1 γ (1) γ ( n ) 1 ≤ ι + 1 ,...ι + π ∈P 2 ( n ) n ≤ M ι ± k = ι ± 1 ≤ ι − 1 ,...,ι − l : { k , l }∈ π n ≤ N Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Reversing the order of summation, � � Y (1) · · · Y ( n ) � � N − #( γ ) − n E ι − ε (1) ι ε ( γ (1)) ι − ε ( n ) ι ε ( γ ( n )) n 1 γ (1) γ ( n ) 1 ≤ ι + π ∈P 2 ( n ) 1 ,...ι + n ≤ M 1 ≤ ι − 1 ,...,ι − n ≤ N ι ± k = ι ± l : { k , l }∈ π Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Reversing the order of summation, � � Y (1) · · · Y ( n ) � � N − #( γ ) − n E ι − ε (1) ι ε ( γ (1)) ι − ε ( n ) ι ε ( γ ( n )) n 1 γ (1) γ ( n ) 1 ≤ ι + π ∈P 2 ( n ) 1 ,...ι + n ≤ M 1 ≤ ι − 1 ,...,ι − n ≤ N ι ± k = ι ± l : { k , l }∈ π N # ( γ − 1 � � − δ ε πδπδ ε γ + ) / 2 − #( γ ) − n E � = − δ ε πδπδ ε γ + / 2 ( Y 1 , . . . , Y n ) tr γ − 1 . π ∈P 2 ( n ) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Real Ginibre matrices are square matrices Z := X with M = N . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Real Ginibre matrices are square matrices Z := X with M = N . Thus � � �� Z ( ε (1)) Y 1 , . . . , Z ( ε ( n )) Y n E tr γ � � � N χ ( γ,δ ε πδ ε ) − #( γ ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) . π ∈{ ρδρ : ρ ∈P 2 ( n ) } Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Real Ginibre matrices are square matrices Z := X with M = N . Thus � � �� Z ( ε (1)) Y 1 , . . . , Z ( ε ( n )) Y n E tr γ � � � N χ ( γ,δ ε πδ ε ) − #( γ ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) . π ∈{ ρδρ : ρ ∈P 2 ( n ) } This is a sum over all gluings compatible with the edge directions given by the transposes. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations 1 � X + X T � If we expand out the GOE matrix T := √ , we get 2 E ( tr γ ( TY 1 , . . . , TY n )) 1 � � �� � X ( ε (1)) Y 1 · · · X ( ε ( n )) Y n = 2 n / 2 E tr γ . ε : { 1 ,..., n }→{ 1 , − 1 } Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations If we collect terms, this is equivalent to summing over all edge-identifications. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations If we collect terms, this is equivalent to summing over all edge-identifications. Thus E ( tr γ ( TY 1 , . . . , TY n )) � � � N χ ( γ,π ) − #( γ ) E = − πγ + / 2 ( Y 1 , . . . , Y n ) tr γ − 1 . π ∈ PM ( ± [ n ]) ∩P 2 ( ± [ n ]) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations With Wishart matrices W := X T D k X , we can collapse the edges corresponding to each matrix to a single edge. We can think of the connecting blocks as (possibly twisted) hyperedges. j 2 j 3 Y 1 i 2 X D 1 i 1 X T i 3 D 2 X T i 4 j 1 X Y 3 j 6 X j 4 X T Y 2 D 3 j 5 j 7 j 10 i 6 i 5 Y 5 X T X i 10 i 7 D 4 D 5 i 8 i 9 X X T Y 4 j 8 j 9 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations With Wishart matrices W := X T D k X , we can collapse the edges corresponding to each matrix to a single edge. We can think of the connecting blocks as (possibly twisted) hyperedges. j 2 j 3 Y 1 i 2 X D 1 i 1 X T i 3 D 2 X T i 4 j 1 X Y 3 j 6 X j 4 X T Y 2 D 3 j 5 j 7 j 10 i 6 i 5 Y 5 X T X i 10 i 7 D 4 D 5 i 8 i 9 X X T Y 4 j 8 j 9 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations With Wishart matrices W := X T D k X , we can collapse the edges corresponding to each matrix to a single edge. We can think of the connecting blocks as (possibly twisted) hyperedges. Y 1 W 1 1 W 2 Y 3 W 3 Y 2 Y 4 W 4 W 5 Y 5 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations Thus: E ( tr γ ( W 1 Y 1 , · · · , W n Y n )) � N χ ( γ,π ) − #( γ ) tr π − 1 / 2 ( D 1 , . . . , D n ) = π ∈ PM ([ n ]) � � − πγ + / 2 ( Y 1 , . . . , Y n ) E tr γ − 1 . Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We note that all of the matrix ensembles satisfy � � X ( ε (1)) Y 1 , · · · , X ( ε ( n )) �� E tr γ Y n λ 1 λ n � � � N χ ( γ,δ ε πδ ε ) − 2#( γ ) f c ( π ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) π ∈ PM c ( ± [ n ]) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We note that all of the matrix ensembles satisfy � � X ( ε (1)) Y 1 , · · · , X ( ε ( n )) �� E tr γ Y n λ 1 λ n � � � N χ ( γ,δ ε πδ ε ) − 2#( γ ) f c ( π ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) π ∈ PM c ( ± [ n ]) ◮ PM c ( ± I ) is a subset of the premaps on ± I , Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We note that all of the matrix ensembles satisfy � � X ( ε (1)) Y 1 , · · · , X ( ε ( n )) �� E tr γ Y n λ 1 λ n � � � N χ ( γ,δ ε πδ ε ) − 2#( γ ) f c ( π ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) π ∈ PM c ( ± [ n ]) ◮ PM c ( ± I ) is a subset of the premaps on ± I , ◮ f c : � I ⊆ N , | I | < ∞ PM c ( ± I ) → C Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We note that all of the matrix ensembles satisfy � � X ( ε (1)) Y 1 , · · · , X ( ε ( n )) �� E tr γ Y n λ 1 λ n � � � N χ ( γ,δ ε πδ ε ) − 2#( γ ) f c ( π ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) π ∈ PM c ( ± [ n ]) ◮ PM c ( ± I ) is a subset of the premaps on ± I , ◮ f c : � I ⊆ N , | I | < ∞ PM c ( ± I ) → C ◮ for any J ⊆ I , the π ∈ PM c ( ± I ) which do not connect ± J and ± ( I \ J ) are the product of a π 1 ∈ PM c ( ± J ) and π 2 ∈ PM c ( ± ( I \ J )) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We note that all of the matrix ensembles satisfy � � X ( ε (1)) Y 1 , · · · , X ( ε ( n )) �� E tr γ Y n λ 1 λ n � � � N χ ( γ,δ ε πδ ε ) − 2#( γ ) f c ( π ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) π ∈ PM c ( ± [ n ]) ◮ PM c ( ± I ) is a subset of the premaps on ± I , ◮ f c : � I ⊆ N , | I | < ∞ PM c ( ± I ) → C ◮ for any J ⊆ I , the π ∈ PM c ( ± I ) which do not connect ± J and ± ( I \ J ) are the product of a π 1 ∈ PM c ( ± J ) and π 2 ∈ PM c ( ± ( I \ J )) ◮ lim N →∞ f c ( π ) exists Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations We note that all of the matrix ensembles satisfy � � X ( ε (1)) Y 1 , · · · , X ( ε ( n )) �� E tr γ Y n λ 1 λ n � � � N χ ( γ,δ ε πδ ε ) − 2#( γ ) f c ( π ) E = tr γ − 1 − δ ε πδ ε γ + / 2 ( Y 1 , . . . , Y n ) π ∈ PM c ( ± [ n ]) ◮ PM c ( ± I ) is a subset of the premaps on ± I , ◮ f c : � I ⊆ N , | I | < ∞ PM c ( ± I ) → C ◮ for any J ⊆ I , the π ∈ PM c ( ± I ) which do not connect ± J and ± ( I \ J ) are the product of a π 1 ∈ PM c ( ± J ) and π 2 ∈ PM c ( ± ( I \ J )) ◮ lim N →∞ f c ( π ) exists ◮ if π ∈ PM c ( I ) does not connect ± J and ± ( I \ J ), then � � � � f c ( π ) = f c π | ± J π | ± ( I \ J ) f c Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations It is possible to mix ensembles in an expression. Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations It is possible to mix ensembles in an expression. � � � � � � �� Z 3 W ( λ 2 ) W ( λ 3 ) W ( λ 6 ) 3 W ( λ 8 ) W ( λ 9 ) Z T 3 Z T Z T E tr tr tr 2 1 3 2 2 1 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations It is possible to mix ensembles in an expression. � � � � � � �� Z 3 W ( λ 2 ) W ( λ 3 ) W ( λ 6 ) 3 W ( λ 8 ) W ( λ 9 ) Z T 3 Z T Z T E tr tr tr 2 1 3 2 2 1 W ( λ 3 ) Z T W ( λ 6 ) Z 1 2 Z T W ( λ 2 ) Z T W ( λ 8 ) W ( λ 9 ) 2 2 1 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations It is possible to mix ensembles in an expression. � � � � � � �� Z 3 W ( λ 2 ) W ( λ 3 ) W ( λ 6 ) 3 W ( λ 8 ) W ( λ 9 ) Z T 3 Z T Z T E tr tr tr 2 1 3 2 2 1 W ( λ 3 ) Z T W ( λ 6 ) Z 1 2 Z T W ( λ 2 ) Z T W ( λ 8 ) W ( λ 9 ) 2 2 1 γ = (1 , 2) (3 , 4 , 5) (6 , 7 , 8 , 9) Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations W ( λ 3 ) Z T W ( λ 6 ) Z 1 2 Z T W ( λ 2 ) Z T W ( λ 8 ) W ( λ 9 ) 2 2 1 Emily Redelmeier Second-Order Freeness

Introduction The Matrix Models Genus Expansion Cumulants Asymptotic Freeness Matrix Calculations W ( λ 3 ) Z T W ( λ 6 ) Z 1 2 Z T W ( λ 2 ) Z T W ( λ 8 ) W ( λ 9 ) 2 2 1 π 1 = (3) ( − 3) (9) ( − 9) Emily Redelmeier Second-Order Freeness