Free Convolution with A Free Multiplicative Analogue of The Normal Distribution

Transcription

1 Free Convolution with A Free Multiplicative Analogue of The Normal Distribution Ping Zhong Indiana University Bloomington July 23, 2013 Fields Institute, Toronto Ping Zhong free multiplicative analogue of the normal distribution 1/17

2 Outline 1 Density of the measures µ γ t, where γ t is the semicircular distribution with variance t. Voiculescu transform -infinitely divisible distributions Density formula 2 Density of the measures µ λ t and ν σ t, where λ t and σ t are free multiplicative analogue of the normal distributions. Σ-transform -infinitely divisible distributions Density formula Free multiplicative analogue of the normal distributions Ping Zhong free multiplicative analogue of the normal distribution 2/17

3 Voiculescu transform M R : the set of probability measures on R For a measure µ M R, we define the Cauchy transform G µ : C + C by G µ (z) = + 1 z t dµ(t), z C+. We set F µ (z) = 1/G µ (z), z C +, so that F µ : C + C + is analytic. Voiculescu transform of µ: ϕ µ (z) = F 1 µ (z) z. (Voiculescu 86, Maassen 92, Bercovici-Voiculescu 93) ϕ µ ν (z) = ϕ µ (z)+ϕ ν (z). Ping Zhong free multiplicative analogue of the normal distribution 3/17

4 Infinite divisibility w.r.t. Given ν M R, we say that ν is -infinitely divisible if for every positive integer n, there exists a ν 1/n M R such that ν = ν 1/n ν 1/n ν 1/n. }{{} n times (Voiculescu 86, Bercovici-Voiculescu 93) ν is -infinitely divisible if and only if + 1+tz ϕ ν (z) = α+ z t dσ(t). where α R and σ is a finite positive measure on R. Ping Zhong free multiplicative analogue of the normal distribution 4/17

5 Boundary set F ν is a conformal map and its inverse map is H : z z +ϕ ν (z). (Biane 97, Chistyakov-Götze 11) The boundary of the set U = {F ν (z) : z C + } is the graph of a function defined on R. If x +iy U and y > 0, then 1 = R 1+t 2 (x t) 2 +y2dσ(t). (1) Ping Zhong free multiplicative analogue of the normal distribution 5/17

6 Biane s formula µ M R, γ t : semicircular distribution. Subordination function: F t : F µ γt (z) = F µ (F t (z)). The function F t can be regarded as the F-transform of a probability measure on R, denoted by γ t µ. Ping Zhong free multiplicative analogue of the normal distribution 6/17

7 Biane s formula µ M R, γ t : semicircular distribution. Subordination function: F t : F µ γt (z) = F µ (F t (z)). The function F t can be regarded as the F-transform of a probability measure on R, denoted by γ t µ. γ t ν is -infinitely divisible and ϕ γt ν (z) = z +tg µ(z). Set Ω t := F t (C + ), then Ω t is the graph of a function. Ping Zhong free multiplicative analogue of the normal distribution 6/17

8 Biane s formula µ M R, γ t : semicircular distribution. Subordination function: F t : F µ γt (z) = F µ (F t (z)). The function F t can be regarded as the F-transform of a probability measure on R, denoted by γ t µ. γ t ν is -infinitely divisible and ϕ γt ν (z) = z +tg µ(z). Set Ω t := F t (C + ), then Ω t is the graph of a function. Given x R, let y t (x) R such that x +iy t (x) Ω t, and f t (x) = H t (x +iy t (x)). Theorem (Biane 97) The density of µ γ t is given by p t (f t (x)) = yt(x) πt. Ping Zhong free multiplicative analogue of the normal distribution 6/17

9 Biane s formula µ M R, γ t : semicircular distribution. Subordination function: F t : F µ γt (z) = F µ (F t (z)). The function F t can be regarded as the F-transform of a probability measure on R, denoted by γ t µ. γ t ν is -infinitely divisible and ϕ γt ν (z) = z +tg µ(z). Set Ω t := F t (C + ), then Ω t is the graph of a function. Given x R, let y t (x) R such that x +iy t (x) Ω t, and f t (x) = H t (x +iy t (x)). Theorem (Biane 97) The density of µ γ t is given by p t (f t (x)) = yt(x) πt. The density p t is analytic whenever it is positive. Biane also proved that the number of connected components of the interior of the support is a non-increasing function of t. Ping Zhong free multiplicative analogue of the normal distribution 6/17

10 Multiplicative free convolution on M T M T : the set of probability measures on T; M : the set of probability measures on C with nonzero mean. Given µ M T, we define ψ µ (z) = and set η µ (z) = ψ µ (z)/(1+ψ µ (z)). η µ : D D is analytic. T tz 1 tz d µ(t) Ping Zhong free multiplicative analogue of the normal distribution 7/17

11 Multiplicative free convolution on M T M T : the set of probability measures on T; M : the set of probability measures on C with nonzero mean. Given µ M T, we define ψ µ (z) = T tz 1 tz d µ(t) and set η µ (z) = ψ µ (z)/(1+ψ µ (z)). η µ : D D is analytic. For µ M T M, set Σ µ (z) = η 1 µ (z)/z. Theorem (Voiculescu 87) Given µ,ν M T M, then we have Σ µ ν (z) = Σ µ (z)σ ν (z). Ping Zhong free multiplicative analogue of the normal distribution 7/17

12 Subordination distributions λ t : the free multiplicative analogue ) of the normal distribution on T such that Σ λt (z) = exp( t 1+z 2 1 z. Subordination function η t : D D: η µ λt (z) = η µ (η t (z)). Subordination distribution ρ t M T : η ρt (z) = η t (z), for t > 0. Lemma The measure ρ t is -infinitely divisible and its Σ-transform is ( ) t 1+ξz Σ ρt (z) = Σ λt (η µ (z)) = exp 2 1 ξz dµ(ξ). (2) Let Φ t,µ (z) := zσ ρt (z), then Φ t,µ (η ρt (z)) = z for all z T. T Ping Zhong free multiplicative analogue of the normal distribution 8/17

13 Subordination distributions λ t : the free multiplicative analogue ) of the normal distribution on T such that Σ λt (z) = exp( t 1+z 2 1 z. Subordination function η t : D D: η µ λt (z) = η µ (η t (z)). Subordination distribution ρ t M T : η ρt (z) = η t (z), for t > 0. Lemma The measure ρ t is -infinitely divisible and its Σ-transform is ( ) t 1+ξz Σ ρt (z) = Σ λt (η µ (z)) = exp 2 1 ξz dµ(ξ). (2) Let Φ t,µ (z) := zσ ρt (z), then Φ t,µ (η ρt (z)) = z for all z T. The function η ρt is a conformal map. (Belinschi-Bercovici 05) η ρt extends to be a one-to-one and continuous function on D. T Ping Zhong free multiplicative analogue of the normal distribution 8/17

14 A formula for the density of µ λ t µ M T ; µ t := µ λ t. Subordination distribution: η µt = η µ (η ρt (z)), ρ t ID(,T). Φ t,µ (z) := zσ ρt (z). Ω t,µ := {z D : Φ t,µ (z) < 1} = η ρt (D). Lemma Ω t,µ is the graph of a function of θ. Ping Zhong free multiplicative analogue of the normal distribution 9/17

15 A formula for the density of µ λ t µ M T ; µ t := µ λ t. Subordination distribution: η µt = η µ (η ρt (z)), ρ t ID(,T). Φ t,µ (z) := zσ ρt (z). Ω t,µ := {z D : Φ t,µ (z) < 1} = η ρt (D). Lemma Ω t,µ is the graph of a function of θ. Given θ [ π,π], let v t (θ) (0,1] such that v t (θ)e iθ Ω t,µ. Theorem (Z. 12) The density of µ t is given by p t (Ψ t,µ (e iθ )) = ln(v t (θ))/πt, where Ψ t,µ = Φ t,µ (v t (θ)e iθ ). The density p t is analytic whenever it is positive. The number of connected components of the interior of the support of µ t is a non-increasing function of t. Ping Zhong free multiplicative analogue of the normal distribution 9/17

16 An example (Biane 97) where Q n (t) = n 1 tk k=0 ( 1)k k! nk 1( n k+1). T ξ n dλ t (ξ) = Q n (t)e n 2 t, (3) Let µ = δ 1, Ω t = Ω t,δ1, Φ t = Φ t,δ1 and U t = U t,δ1, we recover Biane s result. 1 The support of λ t is T if t 4. 2 Let θ t = arccos ( 1 t 2) 1 2 (4 t)t, then the support of λ t is { e iθ : θ t θ θ t } T for t < 4. Proposition The density of λ t is symmetric with respect to x-axis; and the density of λ t has a unique maximum at 1. Ping Zhong free multiplicative analogue of the normal distribution 10/17

17 the density of λ t t=1 t=4 t=5 p t arg(φ t (e iθ )) Ping Zhong free multiplicative analogue of the normal distribution 11/17

18 A formula for the density of ν σ t 1+z σ t : Σ σt (z) = e t 2 z 1 ; ν MR +: ν δ 0 ; ν t := ν σ t. τ t : η νt = η ν (η τt (z)), then τ t ID(,R + ), and ( t + ) 1+ξz Σ τt (z) = Σ λt (η ν (z)) = exp 2 0 ξz 1 dν(ξ). Let H t,ν (z) = zσ τt (z), and let Γ t,ν be the connected component of the set {z C + : I(H t,ν (z)) > 0} whose boundary contains the left half line (,0). Proposition (Belinschi-Bercovici 05) The function η τt extends to be a continuous function on C + R, and η τt is one to one on this set. If ξ R + satisfies I(η τt (ξ)) > 0, then η τt can be extended analytically to a neighborhood of ξ. Ping Zhong free multiplicative analogue of the normal distribution 12/17

19 A formula for the density of ν σ t Lemma Given r (0,+ ), then there exists only one θ [0,π), denoted by u t (r), such that re iθ (Γ t,ν ). Let Λ r,ν (r) = H t,ν (re iut(r) ). Theorem (Z. 12) The measure ν t has a density given by ( ) 1 q t = 1 Λ t,ν (r) π Λ t,ν(r) u t(r) t for r (0,+ ). Ping Zhong free multiplicative analogue of the normal distribution 13/17

20 The density of ν t Corollary 1 The number of connected components of the interior of the support of ν t is a non-increasing function of t. 2 ν t is absolutely continuous and q t is analytic whenever it is positive. 3 If ν is compactly supported on (0,+ ), then the support of ν t has only one connected component for large t. Ping Zhong free multiplicative analogue of the normal distribution 14/17

21 An example We then give a description of σ t, part of results were known by Biane. Let ν = δ 1, H t (z) = H t,δ1 = z exp( t z+1 2 z 1 ), and V t = {0 < r < + : u t (r) > 0}. Then ν t = σ t and V t = (x 1 (t),x 2 (t)), where x 1 (t) = (2+t t(t +4))/2 and x 2 (t) = (2+t + t(t +4))/2 = 1/x 1 (t). 1 Let x 3 (t) = Λ t (x 2 (t)) = 2+t ( ) t(t +4) t(t +4) exp 2 2 and x 4 (t) = 1/x 3 (t). Ping Zhong free multiplicative analogue of the normal distribution 15/17

22 Symmetry of σ t Proposition For t > 0, let s t (x) be the density of the measure σ t at x. (1) The support of the function s t is the interval [x 3 (t),x 4 (t)]. ([ ]) (2) For [α,β] (0,1], we have σ t ([α,β]) = σ 1 t β, 1 α. (3) s t (x)x and s t are strictly decreasing functions of x on the interval [1,x 4 (t)). Remark For any integer n, let a n = 1+ 2t/n, b n = 1/a n and µ n = (δ an +δ bn )/2, then µ n n σ t weakly as n +. Ping Zhong free multiplicative analogue of the normal distribution 16/17

23 the density of σ t t=0.4 t=0.6 t=1 q t /Λ t Ping Zhong free multiplicative analogue of the normal distribution 17/17