Two-parameter Noncommutative Central Limit Theorem

Transcription

1 Two-parameter Noncommutative Central Limit Theorem Natasha Blitvić Vanderbilt University January 11, 2013 N. Blit. 11/1/ / 39

2 (Classical) Central Limit Theorem CLT (Classical) Probability space: (Ω, B, P). Theorem (Classical CLT) Let X 1, X 2,... be a sequence of independent and identically distributed random variables with E(X i ) = 0 and E(Xi 2 ) = 1 for all i N. Let N i=1 S N = X i. N Then, as N, S N dµ 1 (x) = 1 2π exp( x 2 /2) N. Blit. 11/1/ / 39

3 Free Central Limit Theorem CLT -probability space: (A, ϕ) Theorem (Free CLT) Let a 1, a 2,... be a sequence of freely independent and identically distributed self-adjoint elements of A with ϕ(a i ) = 0 and ϕ(ai 2 ) = 1 for all i N. Let S N = N i=1 a i N. Then, as N, S N dµ 0 (x) = 1 2π 4 x 2, x [ 2, 2] N. Blit. 11/1/ / 39

4 Speicher s Non-commutative CLT Condition (Speicher 1992) Consider a -algebra A, state ϕ : A C and {a i } i N A satisfying: 1. (vanishing means) for all i N, ϕ(a i ) = ϕ(a i ) = (normalized second moments) for all for all i, j N with i < j and ɛ, ɛ {1, }, ϕ(a i a i ) = 1 and ϕ(a ɛ i a ɛ i ) = 0 for (ɛ, ɛ ) (1, ). 3. (uniform moment bounds) for all n N and all n j(1),..., j(n) N, ɛ(1),..., ɛ(n) {1, }, ϕ( a ɛ(i) j(i) ) α n. N. Blit. 11/1/ / 39 i=1

5 4. ( independence ) If A i = -alg(a i ) (i = 1, 2,...), then ϕ(x 1 x 2... x n ) = ϕ(x 1 )... ϕ(x n ) whenever x 1 A i1,..., x n A in for i 1 <... < i n. Assume additionally that for all i j and all ɛ, ɛ {1, }, ai ɛ and satisfy the commutation relation a ɛ j a ɛ i a ɛ j = s(j, i) a ɛ j a ɛ i, s(j, i) { 1, 1}. N. Blit. 11/1/ / 39

6 Theorem (Non-commutative CLT, Speicher 1992) A non-commutative probability space (A, ϕ) and {a i } i N in A satisfying Condition 1. Fix q [ 1, 1], and let {s(i, j)} 1 i<j be independent, identically distributed random variables taking values in { 1, 1} with E(s(i, j)) = q. Let S N = 1 N N i=1 (a i + a i ). Then, for a.e. sign sequence {s(i, j)} 1 i<j, as N, S N q-gaussian. N. Blit. 11/1/ / 39

7 q-gaussian N. Blit. 11/1/ / 39

8 q-gaussian N. Blit. 11/1/ / 39

9 q-gaussian N. Blit. 11/1/ / 39

10 q-gaussian N. Blit. 11/1/ / 39

11 q-gaussian N. Blit. 11/1/ / 39

12 q-gaussian N. Blit. 11/1/ / 39

13 q-gaussian N. Blit. 11/1/ / 39

14 q-gaussian N. Blit. 11/1/ / 39

15 q-gaussian N. Blit. 11/1/ / 39

16 q-gaussian N. Blit. 11/1/ / 39

17 q-gaussian N. Blit. 11/1/ / 39

18 q-gaussian N. Blit. 11/1/ / 39

19 q-gaussian N. Blit. 11/1/ / 39

20 Combinatorial View Definition Let P 2 (2n) = {all pair-partitions of {1,..., 2n}}. For V = {(w 1, z 1 ),..., (w n, z n )} P 2 (2n), pairs (w i, z i ) and (w j, z j ) are said to cross if w i < w j < z i < z j. Then, 2n 1 2n lim ϕ(s N N ) = 0, lim ϕ(sn ) = N V P 2 (2n) q cross(v ), N. Blit. 11/1/ / 39

21 More general commutation structure? Why s(i, j) { 1, 1}? i µ(i,j) j For µ(i, j) R, Limit exists? q-gaussian? N. Blit. 11/1/ / 39

22 More general combinatorics? Definition Let P 2 (2n) = {all pair-partitions of {1,..., 2n}}. For V = {(w 1, z 1 ),..., (w n, z n )} P 2 (2n), pairs (w i, z i ) and (w j, z j ) are said to nest if w i < w j < z j < z i e i e j z i z j 2n 1... e i e j z j z i 2n The number of nestings of V is: nest(v ) = #{(w i, z i ), (w j, z j ) V i < j, w i < w j < z j < z i }. N. Blit. 11/1/ / 39

23 Generalized Non-commutative CLT Condition Consider a -algebra A, state ϕ : A C and {a i } i N A satisfying: 1. (vanishing means) for all i N, ϕ(a i ) = ϕ(a i ) = (normalized second moments) for all for all i, j N with i < j and ɛ, ɛ {1, }, ϕ(a i a i ) = 1 and ϕ(a ɛ i a ɛ i ) = 0 for (ɛ, ɛ ) (1, ). 3. (uniform moment bounds) for all n N and all n j(1),..., j(n) N, ɛ(1),..., ɛ(n) {1, }, ϕ( a ɛ(i) j(i) ) α n. i=1 N. Blit. 11/1/ / 39

24 4. ( independence ) If A i = -alg(a i ) (i = 1, 2,...), then ϕ(x 1 x 2... x n ) = ϕ(x 1 )... ϕ(x n ) whenever x 1 A i1,..., x n A in for i 1 <... < i n. Assume additionally that for all i j and all ɛ, ɛ {1, }, ai ɛ and satisfy the commutation relations a ɛ j a i a j = µ 1,1 (j, i) a j a i, a i a j = µ, (j, i) a j a i a i a j = µ,1 (j, i) a j a i, a i a j = µ 1, (j, i) a j a i for µ 1,1 (j, i), µ, (j, i), µ,1 (j, i), µ 1, (j, i) R. N. Blit. 11/1/ / 39

25 a ɛ i a ɛ j = µ ɛ,ɛ(j, i) a ɛ j a ɛ i, µ ɛ,ɛ(j, i) R Lemma Coefficients {µ ɛ,ɛ(i, j)} satisfy the consistency relations: 1 µ ɛ,ɛ(j, i) = µ ɛ,ɛ (i, j) µ 1,1 (i, j) = for all i j, ɛ, ɛ {1, }, (1) 1 µ, (i, j), µ 1, (i, j) = 1 µ,1 (i, j) (2) In addition, when ϕ is assumed to be positive, µ,1 (i, j) µ,1 (i, j) = µ, (i, j) µ, (i, j). (3) N. Blit. 11/1/ / 39

26 Theorem (Generalized non-commutative CLT, B. 2012) Consider a -probability space (A, ϕ) and {a i } i N in A satisfying Condition 2. In addition, suppose that {µ, (i, j)} 1 i<j satisfy for some t > 0. Let S N = 1 N N i=1 µ,1 (i, j) = t µ, (i, j) (3 ) (a i + a i ). N. Blit. 11/1/ / 39

27 For fixed q R, let {µ, (i, j)} 1 i<j be independent, identically distributed random variables with E(µ, (i, j)) = q/t and E(µ, (i, j) 2 ) = 1, and populate the remaining µ ɛ,ɛ (i, j), for ɛ, ɛ {1, } and i j (i, j N), by consistency conditions (1)-(3 ). N. Blit. 11/1/ / 39

28 For fixed q R, let {µ, (i, j)} 1 i<j be independent, identically distributed random variables with E(µ, (i, j)) = q/t and E(µ, (i, j) 2 ) = 1, and populate the remaining µ ɛ,ɛ (i, j), for ɛ, ɛ {1, } and i j (i, j N), by consistency conditions (1)-(3 ). Then, for a.e. coefficient sequence {µ, (i, j)} 1 i<j, S n (q, t)-gaussian. 2n 1 2n lim ϕ(s N N ) = 0, lim ϕ(sn ) = N V P 2 (2n) q cross(v ) t nest(v ) N. Blit. 11/1/ / 39

29 Proof sketch Show that Z N = a a N N converges in mixed moments. Fix k N and a -pattern ɛ(1),..., ɛ(k) {1, }, and look at the mixed moment ϕ(z ɛ(1) N... Z ɛ(k) N ) = 1 ( ) N k/2 ϕ a ɛ(1) i(1)... aɛ(k) i(k). i(1),...,i(k) [N] ( ) Repetition patterns of indices in ϕ a ɛ(1) i(1)... aɛ(k) i(k) induce set partitions. N. Blit. 11/1/ / 39

30 Note: By standard arguments, only pair-partitions can contribute to the limit. Set k = 2n. If lim N E(ϕ(Z ɛ(1) N... Z ɛ(n) N )) exists, Markov inequality + more careful estimates yield convergence a.e. Remains to factor ( ) ϕ a ɛ(1) i(1)... aɛ(2n) i(2n) If i(1) = i(2) < i(3) = i(4) <... < i(2n 1) = i(2n), then ( ) ( ) ( ) ϕ a ɛ(1) i(1)... aɛ(2n) i(2n) = ϕ a ɛ(1) i(1) aɛ(2) i(2)... ϕ a ɛ(2n 1) i(2n 1) aɛ(2n) i(2n) e.g. ( ) ( ) ( ) ϕ a ɛ j i a ɛ j i a ɛ i j aɛ i j = ϕ a ɛ j i a ɛ j i ϕ a ɛ i j aɛ i j i < j N. Blit. 11/1/ / 39

31 Otherwise, use the commutation relation. For i < j, need to factor ( ) ϕ a ɛ j j a ɛ j j a ɛ i i a ɛ i i ( ) ϕ a ɛ i i a ɛ j j a ɛ i i a ɛ j j ( ) and ϕ a ɛ j j a ɛ i i a ɛ j j a ɛ i i ( ) ϕ a ɛ i i a ɛ j j a ɛ j j a ɛ i i ( ) and ϕ a ɛ j j a ɛ i i a ɛ i i a ɛ j j N. Blit. 11/1/ / 39

32 Coefficients incurred: µ ɛi,ɛ (j, i)µ ɛ j i,ɛ j (j, i)µ ɛ i,ɛ (j, i)µ j ɛ i,ɛ (j, i) j µ ɛ i,ɛ j (i, j) and µ ɛ j,ɛ i (j, i) µ ɛ i,ɛ j (i, j)µ ɛ i,ɛ (i, j) and µ ɛ j j,ɛ i (j, i)µ ɛj,ɛ (j, i) i N. Blit. 11/1/ / 39

33 If ϕ(a i a i ) = 1 and ϕ(aɛ i aɛ i ) = 0 for (ɛ, ɛ ) (1, ), µ 1, (j, i)µ 1,1 (j, i)µ, (j, i)µ,1 (j, i) µ,1 (i, j) and µ,1 (j, i) µ,1 (i, j)µ, (i, j) and µ 1,1 (i, j)µ,1 (i, j) N. Blit. 11/1/ / 39

34 Recall: a ɛ i a ɛ j = µ ɛ,ɛ(j, i) a ɛ j a ɛ i, µ ɛ,ɛ(j, i) R And coefficients {µ ɛ,ɛ(i, j)} satisfy the consistency relations: 1 µ ɛ,ɛ(j, i) = µ ɛ,ɛ (i, j) µ 1,1 (i, j) = for all i j, ɛ, ɛ {1, }, (1) 1 µ, (i, j), µ 1, (i, j) = 1 µ,1 (i, j) (2) µ,1 (i, j) = t µ, (i, j) (3 ) N. Blit. 11/1/ / 39

35 Applying consistency relations (1)-(3 ) and taking E 1 q and q t and t N. Blit. 11/1/ / 39

36 Theorem (Extended Jordan-Wigner Transform, Biane 1997) Non-commutative probability space (A, ϕ) with A = M 2 (C) n and ϕ(a) = ae0 n, e n 0. Fix q [ 1, 1] and s(i, j) { 1, 1} for all 1 i < j n. Let [ ] [ ] [ ] σ 1 =, σ =, γ = and, for i = 1,..., n, let the element a i M 2 (C) n be given by a i = σ s(1,i) σ s(2,i)... σ s(i 1,i) γ σ 1... σ }{{} 1. =σ (n i) 1 Then, for every n N, a 1,..., a n satisfy Condition 1. N. Blit. 11/1/ / 39

37 Theorem (Two-parameter Jordan-Wigner Transform, B. 2012) Non-commutative probability space (A, ϕ) with A = M 2 (C) n and ϕ(a) = ae0 n, e n 0. Fix q t and µ ɛ,ɛ (i, j) R for all 1 i < j n, ɛ, ɛ {1, }. Let [ ] [ ] 1 0 σ x = 0 0 1, γ = t x 0 0 and, for i = 1,..., n, let the element a i M 2 (C) n be given by a i = σ µ(1,i) σ µ(2,i)... σ µ(i 1,i) γ σ 1... σ }{{} 1. =σ (n i) 1 Then, for every n N, a 1,..., a n satisfy Condition 2. N. Blit. 11/1/ / 39

38 Connections. Looking ahead q = -1 anti-symmetric free q = 0 q = 1 symmetric q = 0 < t < 1 More general commutation structure. Beyond i.i.d. Practical applications? N. Blit. 11/1/ / 39

39 Taknh yuo N. Blit. 11/1/ / 39