An introduction to free probability 2. Noncrossing partitions and - PowerPoint PPT Presentation

An introduction to free probability 2. Noncrossing partitions and free cumulants Wojtek M� lotkowski (Wroc� law) Villetaneuse, 11.03.2014 WM () Free probability 03.03.2014 1 / 19

Definition. A partition of a set X is a family π of subsets of X such that � π = X and if U , V ∈ π then either U = V or U ∩ V = ∅ . Elements of π are called blocks of π . The class of partitions of the set { 1 , 2 , . . . , n } will be denoted P ( n ). The cardinality of P ( n ) is counted by Bell numbers B n : 1 , 1 , 2 , 5 , 15 , 52 , 203 , 877 , 4140 , 21147 , . . . (sequence A000110 in OEIS). Recurrence relation: B 0 = 1, n � n � � B n +1 = B k . k k =0 The exponential generating function: ∞ B n n ! z n = exp( e z − 1) . � B ( z ) = n =0 The number of partitions in P ( n ) consisting on k blocks: Stirling � n � numbers of the second kind : . k WM () Free probability 03.03.2014 2 / 19

Definition. A partition π ∈ P ( n ) is called noncrossing if for every 1 ≤ k 1 < k 2 < k 3 < k 4 ≤ n we have implication: k 1 , k 3 ∈ U ∈ π, k 2 , k 4 ∈ V ∈ π = ⇒ U = V . NC ( n )-the class of noncrossing partitions of the set { 1 , 2 , . . . , n } . � 2 n +1 1 � The number of elements in NC ( n ): the Catalan numbers : 2 n +1 : n 1 , 1 , 2 , 5 , 14 , 42 , 132 , 429 , 1430 , 4862 , 16796 , . . . (sequence A000108 in OEIS). They satisfy recurrence: C 0 = 1 and n � for n ≥ 0 . C n +1 = C i C n − i i =0 WM () Free probability 03.03.2014 3 / 19

The generating function: ∞ 2 C n z n = � 1 + √ 1 − 4 z . C ( z ) = n =0 The number of π ∈ NC ( n ) having k blocks: the Narayana numbers : � n �� � 1 n . k − 1 n k For π ∈ NC ( n ) define sequence Λ( π ) = ( x 1 , x 2 , . . . , x n ) as follows: � | U | − 1 if k is the first element of a block U ∈ π , x k = − 1 otherwise. Note that the sequence Λ( π ) has the following properties: 1. x k ∈ {− 1 , 0 , 1 , 2 , 3 , . . . } , 2. x 1 + x 2 + . . . + x k ≥ 0 for 1 ≤ k ≤ n , 3. x 1 + x 2 + . . . + x n = 0. Proposition. The map Λ is a bijection of NC ( n ) onto the class of sequences satisfying (1-2-3). WM () Free probability 03.03.2014 4 / 19

Classical cumulants Let X be a random variable, µ its distribution, a probability measure on R . We assume that X is bounded. Moments of X , µ : � t n d µ ( t ) . m n ( X ) = m n ( µ ) := E ( X n ) = R Cumulants κ n ( µ ) = κ n of X and µ are defined as ∞ t n log(E( e tX )) = � κ n n ! n =1 Then for independent random variables X ∼ µ , Y ∼ ν we have κ n ( X + Y ) = κ n ( X ) + κ n ( Y ) (1) or κ n ( µ ∗ ν ) = κ n ( µ ) + κ n ( ν ) . WM () Free probability 03.03.2014 5 / 19

Relation between moments and cumulants: � � m n ( µ ) = κ | V | ( µ ) . (2) π ∈ P ( n ) V ∈ π Examples: The normal distribution N ( a , σ 2 ), with density − ( x − a ) 2 1 � � √ exp 2 σ 2 σ 2 π we have κ 1 = a , κ 2 = σ 2 and κ n = 0 for n ≥ 3. The Poisson distribution ∞ λ k exp( − λ ) � δ k , k ! k =0 so that Pr( X = k ) = λ k exp( − λ ) , we have κ n = λ for all n ≥ 1. k ! WM () Free probability 03.03.2014 6 / 19

Definition. A (noncommutative) probability space is a pair ( A , φ ), where A is a complex unital ∗ -algebra and φ is a state on A , i.e. a linear map A → C such that φ ( 1 ) = 1 and φ ( a ∗ a ) ≥ 0 for all a ∈ A . Definition: A family {A i } i ∈ I of unital (i.e. 1 ∈ A i ) subalgebras is called free if φ ( a 1 a 2 . . . a m ) = 0 whenever m ≥ 1, a 1 ∈ A i 1 , . . . , a m ∈ A i m , i 1 , . . . , i m ∈ I , i 1 � = i 2 � = . . . � = i m and φ ( a 1 ) = . . . = φ ( a m ) = 0. Main example: Unital free product. Let ( A i , φ i ), i ∈ I , noncommutative probability spaces. Put A 0 i := Ker φ i . Then the unital free product A = ∗ i ∈ I A i can be represented as � A 0 i 1 ⊗ A 0 i 2 ⊗ . . . ⊗ A 0 i m = C 1 ⊕ A 0 . A := C 1 ⊕ (3) m ≥ 1 i 1 ,..., im ∈ I i 1 � = i 2 � = ... � = im with the state defined by φ ( 1 ) = 1 and φ ( a ) = 0 for a ∈ A 0 . Then {A i } i ∈ I is a free family in ( A , φ ) WM () Free probability 03.03.2014 7 / 19

Suppose a k ∈ A 1 , b k ∈ A 2 . We write a k = α k 1 + a 0 k , where α k = φ ( a k ), φ ( a 0 k ) = 0, b k = β k 1 + b 0 k where β k = φ ( b k ), φ ( b 0 k ) = 0, Then ( α 1 1 + a 0 1 )( β k 1 + b 0 � � φ ( a 1 b 1 ) = φ k ) = α 1 β 1 + α 1 φ ( b 0 1 ) + β 1 φ ( a 0 a 0 1 b 0 � � 1 ) + φ = α 1 β 1 = φ ( a 1 ) φ ( b 1 ) . k In a similar way: φ ( a 1 b 1 a 2 ) = φ ( a 1 a 2 ) φ ( b 1 ) and φ ( a 1 b 1 a 2 b 2 ) = φ ( a 1 a 2 ) φ ( b 1 ) φ ( b 2 ) + φ ( a 1 ) φ ( a 2 ) φ ( b 1 b 2 ) − φ ( a 1 ) φ ( a 2 ) φ ( b 1 ) φ ( b 2 ) . WM () Free probability 03.03.2014 8 / 19

Proposition. Assume, that a ∈ A 1 , b ∈ A 2 , and A 1 , A 2 are free. Then the moments φ (( a + b ) n ) of a + b depend only on the moments φ ( a n ) of a and the moments φ ( b n ) of b . Distribution of a self-adjoint element a = a ∗ ∈ A is the probability measure µ on R satisfying: � t n d µ ( t ) , φ ( a n ) = n = 1 , 2 , . . . , R so that φ ( a n ) are moments of µ . If a , b are free and the distribution of a , b is µ, ν respectively then the distribution of a + b will be denoted µ ⊞ ν - the additive free convolution. We want to compute the moments φ (( a + b ) n ) from φ ( a n ) and φ ( b n ). WM () Free probability 03.03.2014 9 / 19

For a ∈ A we define its free cumulants r n ( a ) by the relation: � � φ ( a n ) = r | V | ( a ) . (4) V ∈ π π ∈ NC ( n ) In particular φ ( a ) = r 1 ( a ) , φ ( a 2 ) = r 1 ( a ) 2 + r 2 ( a ) , φ ( a 3 ) = r 1 ( a ) 3 + 3 r 1 ( a ) r 2 ( a ) + r 3 ( a ) , The moment sequence φ ( a n ) and the cumulant sequence r n ( a ) determine each other. We are going to prove Theorem. If a , b ∈ A are free (i.e. belong to free subalgebras) then r n ( a + b ) = r n ( a ) + r n ( b ) . (5) WM () Free probability 03.03.2014 10 / 19

Examples. 1. Catalan numbers: if � 2 n + 1 � 1 φ ( a n ) = then r n ( a ) = 1 for all n ≥ 1. (6) n 2 n + 1 2. More generally: Fuss/Raney numbers: if � pn + r � r φ ( a n ) = then (7) n pn + r � ( p − r ) n + r � r r n ( a ) = (8) ( p − r ) n + r . n W. M� lotkowski, Fuss-Catalan numbers in noncommutative probability, Documenta Mathematica 15 (2010). WM () Free probability 03.03.2014 11 / 19

3. Aerated Catalan numbers: if � 1 � � 2 k +1 1 � if n = 2 k , if n = 2, φ ( a n ) = k 2 k +1 then r n ( a ) = (9) 0 if n � = 2. 0 if n odd, 4. More generally, aerated Fuss/Raney numbers, if � � pk + r r � if n = 2 k , φ ( a n ) = k pk + r (10) 0 if n is odd, then � � ( p − 2 r ) k + r r � if n = 2 k , ( p − 2 r ) k + r r n ( a ) = k (11) 0 if n is odd. WM () Free probability 03.03.2014 12 / 19

Free Gaussian law γ a , r : 1 � 4 r 2 − ( x − a ) 2 χ [ a − 2 r , a +2 r ] ( x ) dx , (12) 2 π r 2 then r 1 ( γ a , r ) = a , r 2 ( γ a , r ) = r 2 and r n ( γ a , r ) = 0 for r ≥ 3. Free Poisson law ̟ t : � 4 t − ( x − 1 − t ) 2 χ [(1 −√ t ) 2 , (1+ √ t ) 2 ] ( x ) dx max { 1 − t , 0 } δ 0 + (13) 2 π x then r n ( ̟ t ) = t for all n ≥ 1. WM () Free probability 03.03.2014 13 / 19

Cuntz algebra Let H be a Hilbert space and define the full Fock space of H : ∞ � H ⊗ m . F ( H ) := C Ω ⊕ m =1 Fix an orthonormal basis e i , i ∈ I . Then the vectors e i 1 ⊗ e i 2 ⊗ . . . ⊗ e i m , m ≥ 0, i 1 , i 2 , . . . , i m ∈ I , form an orthonormal basis of F ( H ). The vector corresponding to the empty word ( m = 0) will be denoted by Ω. For i ∈ I define operator ℓ i : ℓ i e i 1 ⊗ . . . ⊗ e i m = e i ⊗ e i 1 ⊗ . . . ⊗ e i m in particular ℓ i Ω = e i , and its adjoint: � e i 2 ⊗ . . . ⊗ e i m if m ≥ 1 and i 1 = i ℓ ∗ i e i 1 ⊗ e i 2 ⊗ . . . ⊗ e i m = 0 otherwise. WM () Free probability 03.03.2014 14 / 19

Note the relation � 1 if i = j ℓ ∗ i ℓ j = (14) 0 otherwise. Define: A - the unital algebra generated by all ℓ i , ℓ ∗ i , i ∈ I , A i - the unital subalgebra generated by ℓ i , ℓ ∗ i . For a ∈ A we put φ ( a ) := � a Ω , Ω � . By (14), A i is the linear span of i ) n : m , n ≥ 0 } , { ℓ m i ( ℓ ∗ while A is the linear span of { ℓ i 1 ℓ i 2 . . . ℓ i m ℓ ∗ j 1 ℓ ∗ j 2 . . . ℓ ∗ j n : m , n ≥ 0 } . Proposition. 1. If m + n > 0 then ℓ i 1 ℓ i 2 . . . ℓ i m ℓ ∗ j 1 ℓ ∗ j 2 . . . ℓ ∗ � � φ = 0 j n 2. The family {A i } i ∈ I is free in ( A , φ ). WM () Free probability 03.03.2014 15 / 19

Lemma. Suppose that x 1 , x 2 , . . . , x n ∈ {− 1 , 0 , 2 , 3 , . . . } and denote ℓ − 1 := ℓ ∗ i . Then i φ ( ℓ x n 1 . . . ℓ x 2 1 ℓ x 1 1 ) = 1 iff the sequence ( x 1 , x 2 , . . . , x n ) satisfies conditions (1-2-3) from page 4 and 1 . . . ℓ x 2 1 ℓ x 1 φ ( ℓ x n 1 ) = 0 otherwise. Proposition. Let ∞ � T 1 = ℓ ∗ α k ℓ k − 1 1 + 1 k =1 for some α n ∈ C . Then α k are free cumulants of T 1 : � � φ ( T n 1 ) = α | V | . π ∈ NC ( n ) V ∈ π WM () Free probability 03.03.2014 16 / 19