random walks on some noncommutative spaces philippe biane
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RANDOM WALKS ON SOME NONCOMMUTATIVE SPACES Philippe Biane Vietri - PowerPoint PPT Presentation

RANDOM WALKS ON SOME NONCOMMUTATIVE SPACES Philippe Biane Vietri Sul Mare, 01/09/2009 Classical random walk . 0.0 7.5 15.0 22.5 30.0 S n = X 1 + . . . + X n X k = 1 Brownian motion Scale by in time and in space.


  1. RANDOM WALKS ON SOME NONCOMMUTATIVE SPACES Philippe Biane Vietri Sul Mare, 01/09/2009

  2. Classical random walk . 0.0 7.5 15.0 22.5 30.0 S n = X 1 + . . . + X n X k = ± 1

  3. Brownian motion Scale by ε in time and √ ε in space. √ ε ↑ → ε 3 . −3 0.00 0.25 0.50 0.75 1.00 Y

  4. PITMAN THEOREM (1975) B t ; t ≥ 0 Brownian motion; I t = inf 0 ≤ s ≤ t B s R t = B t − 2 I t ; t ≥ 0 is distributed as the norm of a three dimensional Brownian motion(=Bessel 3 process) 3 . −3 0.00 0.25 0.50 0.75 1.00 I(t)=inf(Y(s);0<s<t) Y −I R(t)=Y(t)−2I(t) Explained by considering a random walk in a non-commutative space.

  5. DISCRETE VERSION X i = ± 1; S n = X 1 + X 2 + . . . + X n 1/2 1/2 � � � � � � � � � � � � � � � � � � � � � � � � � � R n = S n − 2 min 0 ≤ k ≤ n S k is a Markov chain(=discrete Bessel 3 process)

  6. P ( R n +1 = k + 1 | R n = k ) = k + 1 2 k P ( R n +1 = k − 1 | R n = k ) = k − 1 2 k (k−1)/2k (k+1)/2k � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � k−1 k k+1 when n → ∞ S [ nt ] / √ n → n →∞ Brownian motion R [ nt ] / √ n → n →∞ norm of 3D-Brownian motion

  7. PROOF OF PITMAN’S THEOREM ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� � � ���� ���� � � ���� ���� ���� ���� ���� ���� S ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� �� �� ���� ���� ���� ���� �� �� � � ���� ���� ���� ���� � � �� �� ��� ��� ��� ��� � � ���� ���� � � ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ���� ���� ��� ��� ��� ��� ���� ���� � � � � � � ��� ��� ���� ���� � � ��� ��� � � � � ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� �� �� ��� ��� ��� ��� ��� ��� ��� � � � � ��� �� �� ����� ����� � � � � ��� ��� ��� ��� �� �� ����� ����� ��� ��� ��� ��� ����� ����� ��� ��� ��� ��� ����� ����� ��� ��� ��� ��� S−2I ����� ����� ��� ��� ��� ��� ����� ����� ��� ��� ��� ��� ����� ����� ��� ��� ��� ��� ��� ��� ����� ����� ��� ��� ����� ����� ��� ��� ��� ��� ����� ����� ��� ��� � � � � � � � � ����� ����� �� �� ����� ����� �� �� ����� ����� �� �� ����� ����� �� �� ����� ����� ��� ��� ����� ����� ��� ��� ��� ��� ����� ����� ��� ��� ����� ����� ��� ��� ����� ����� � � � � ����� ����� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� � � ����� ����� � � ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������

  8. Quantum Bernoulli random walks We ”quantize” the set of increments of the random walk {± 1 } to obtain M 2 ( C ). The subset of hermitian operators in M 2 ( C ) is a four dimensional real subspace, generated by the identity matrix I as well as the three matrices � 0 � � 0 � � 1 � 1 − i 0 σ x = σ y = σ z = − 1 1 0 i 0 0 The matrices σ x , σ y , σ z are the Pauli matrices. They satisfy the commutation relations [ σ x , σ y ] = 2 i σ z ; [ σ y , σ z ] = 2 i σ x ; [ σ z , σ x ] = 2 i σ y (1)

  9. The random walk For ω a state on M 2 ( C ), in ( M 2 ( C ) , ω ) ⊗ N we put x n = I ⊗ ( n − 1) ⊗ σ x ⊗ I ⊗∞ , y n = I ⊗ ( n − 1) ⊗ σ y ⊗ I ⊗∞ , z n = I ⊗ ( n − 1) ⊗ σ z ⊗ I ⊗∞ x n is a commuting family of operators, a sequence of independent Bernoulli random variables. n n n � � � X n = x i ; Y n = y i Z n = z i i =1 i =1 i =1 are Bernoulli random walks. They do not commute but obey [ X n , Y m ] = 2 iZ n ∧ m (2) as well as the similar relations obtained by cyclic permutation of X , Y , Z . ( X n , Y n , Z n ); n ≥ 1 is a quantum Bernoulli random walk .

  10. The spin process � I + X 2 n + Y 2 n + Z 2 Let S n = n Proposition For all n , m one has [ S n , S m ] = 0 Thus we have a commutative process and we can try to compute its distribution.

  11. Theorem Let ω be the tracial state 1 2 Tr , then S n is distributed as a Markov chain on the positive integers, with probability transitions p ( k , k − 1) = k − 1 p ( k , k + 1) = k + 1 ; . 2 k 2 k

  12. Random walks on groups ˆ W = abelian group W = dual group ξ ∈ ˆ W = character of W A ( ˆ W )=group algebra of ˆ F ( W )=algebra of functions on W W ∆ : A ( ˆ W ) → A ( ˆ W ) ⊗ A ( ˆ F ( W ) → F ( W × W ) W ) f ( x ) → f ( x + y ) ∆( ξ ) = ξ ⊗ ξ φ =positive definite function on ˆ µ : F ( W ) → C W � =probability measure on W φ ( ξ ) = W ξ ( x ) d µ ( x ) state ω on A ( ˆ W )

  13. M = ⊗ ∞ ( A ( ˆ Ω = ( W , µ ) ∞ W ) , ω ) j n : A ( ˆ Y n = w 1 + . . . + w n W ) → M j n +1 = (∆ ⊗ I ⊗ ( n +1) ) ◦ I ⊗ j n f → f ( w 1 + . . . + w n ) Markov operator � Φ( f ) = ( I ⊗ ω ) ◦ ∆ Φ( f )( x ) = W f ( x + y ) d µ ( y )

  14. Random walks on duals of compact groups Replace ˆ W by a compact group G . φ =continuous positive definite functions on G , with φ ( e ) = 1. =state ν on A ( G ). ν = distribution of the increments. Φ ν : A ( G ) → A ( G ) Φ ν = ( I ⊗ ν ) ◦ ∆ is a completely positive map. It generates a semigroup Φ n ν ; n ≥ 1.

  15. ( N , ω ) = ( A ( G ) , ν ) ∞ j n : A ( G ) → N defined by j n ( λ g ) = λ ⊗ n ⊗ I g The morphisms ( j n ) n ≥ 0 , define a random walk on the noncommutative space dual to G , with Markov operator. Φ ν : A ( G ) → A ( G ) Φ ν = ( I ⊗ ν ) ◦ ∆ The quantum Bernoulli random walk is obtained for G = SU (2), and ν the tracial state associated with the 2-dimensional representation.

  16. The dual of SU (2) as a noncommutative space G = SU (2) = unitary 2 × 2 matrices with determinant 1. ˆ G = { 1 , 2 , 3 , . . . } A ( SU (2)) = ⊕ ∞ n =1 M n ( C ) is the noncommutative space dual to SU (2). The Pauli matrices belong to the Lie algebra su (2), they define unbounded operators X , Y , Z , on L 2 ( SU (2)). They generate oneparameter sugbroups isomorphic to U (1). This is true also of any linear combination xX + yY + zZ with x 2 + y 2 + z 2 = 1.

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