When J. Ginibre met E. Schr odinger Thomas J. Bothner Department - - PowerPoint PPT Presentation

When J. Ginibre met E. Schr¨

• dinger

Thomas J. Bothner

Department of Mathematics

King’s College London

Joint with Jinho Baik, arXiv:1808.02419

CIRM - Integrability and Randomness April 11th, 2019

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 1 / 37

Did they actually meet?

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

F K G

Lf gingin z Lthis i Lg niz i Figure 1: E.S.: 1887 - 1961 and J.G.: 1938 - ???

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 2 / 37

GOE to the comparison

Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices

X = 1 2(Y + YT) ∈ Rn×n : Yjk

iid

∼ N(0, 1). (Mehta 1960)

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 3 / 37

GOE to the comparison

Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices

X = 1 2(Y + YT) ∈ Rn×n : Yjk

iid

∼ N(0, 1). (Mehta 1960)

Equivalently think of this setup as a log-gas system

λ1 < λ2 < . . . < λn, λj ∈ R,

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 3 / 37

GOE to the comparison

Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices

X = 1 2(Y + YT) ∈ Rn×n : Yjk

iid

∼ N(0, 1). (Mehta 1960)

Equivalently think of this setup as a log-gas system

λ1 < λ2 < . . . < λn, λj ∈ R,

with joint pdf for the particles’ locations equal to (Hsu 1939)

f (λ1, . . . , λn) = 1 Zn

• 1≤j<k≤n

|λk − λj| exp

• − 1

2

n

• j=1

λ2

j

• .

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 3 / 37

GOE to the comparison

Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices

X = 1 2(Y + YT) ∈ Rn×n : Yjk

iid

∼ N(0, 1). (Mehta 1960)

Equivalently think of this setup as a log-gas system

λ1 < λ2 < . . . < λn, λj ∈ R,

with joint pdf for the particles’ locations equal to (Hsu 1939)

f (λ1, . . . , λn) = 1 Zn

• 1≤j<k≤n

|λk − λj| exp

• − 1

2

n

• j=1

λ2

j

• .

Objective: What can we say about the underlying limit laws?

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 3 / 37

The eigenvalues {λj}n

j=1 form a Pfaffian point process (Dyson 1970),

Rk(λ1, . . . , λk) := n! (n − k)!

• Rn−k f (λ1, . . . , λn)

n

• j=k+1

dλj = Pf

• Kn(λi, λj)

k

i,j=1,

with a Hilbert-Schmidt class 2 × 2 matrix-valued kernel Kn. Now analyze Rk asymptotically in different scaling regimes.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 4 / 37

The eigenvalues {λj}n

j=1 form a Pfaffian point process (Dyson 1970),

Rk(λ1, . . . , λk) := n! (n − k)!

• Rn−k f (λ1, . . . , λn)

n

• j=k+1

dλj = Pf

• Kn(λi, λj)

k

i,j=1,

with a Hilbert-Schmidt class 2 × 2 matrix-valued kernel Kn. Now analyze Rk asymptotically in different scaling regimes. (A) The global eigenvalue regime: define the ESD

µX(s) = 1 n# {1 ≤ j ≤ n, λj ≤ s}, s ∈ R,

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 4 / 37

The eigenvalues {λj}n

j=1 form a Pfaffian point process (Dyson 1970),

Rk(λ1, . . . , λk) := n! (n − k)!

• Rn−k f (λ1, . . . , λn)

n

• j=k+1

dλj = Pf

• Kn(λi, λj)

k

i,j=1,

with a Hilbert-Schmidt class 2 × 2 matrix-valued kernel Kn. Now analyze Rk asymptotically in different scaling regimes. (A) The global eigenvalue regime: define the ESD

µX(s) = 1 n# {1 ≤ j ≤ n, λj ≤ s}, s ∈ R,

then, as n → ∞, the random measure µX/√n converges almost surely to the Wigner semi-circular distribution (Wigner 1955)

ρsc(λ) = 1 π

• 2 − λ2+ dλ.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 4 / 37

• 1.5
• 1
• 0.5

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7

• 1.5
• 1
• 0.5

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 2: Wigner’s law for one (rescaled) 2000 × 2000 GOE matrix on the left,

plotted is the rescaled histogram of the 2000 eigenvalues and the semicircular density ρsc(λ). On the right we compare Wigner’s law to the exact eigenvalue density for n = 4 and the associated eigenvalue histogram (sampled 4000 times).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 5 / 37

Universality I Wigner’s law is a universal limiting law (Arnold 1967, ...), it holds true for any (properly centered and scaled) symmetric or Hermitian Wigner matrix X = (Xjk)n

j,k=1 with E|Xjk|2 < ∞ where Xjk, j < k are

iid real or complex variables and Xjj iid real variables independent of the upper triangular ones.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 6 / 37

Universality I Wigner’s law is a universal limiting law (Arnold 1967, ...), it holds true for any (properly centered and scaled) symmetric or Hermitian Wigner matrix X = (Xjk)n

j,k=1 with E|Xjk|2 < ∞ where Xjk, j < k are

iid real or complex variables and Xjj iid real variables independent of the upper triangular ones. (B) The local eigenvalue regime: We shall zoom in on the right edge point λ0 = √ 2n and let n be even (Forrester, Nagao, Honner 1999),

1 √ 2n

1 6

Kn √ 2n + x √ 2n

1 6

, √ 2n + y √ 2n

1 6

• → QAi(x, y),

as n → ∞ uniformly in x, y ∈ R chosen from compact subsets.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 6 / 37

Here, QAi is Hilbert-Schmidt on L2(s0, ∞) with kernel entries

Q11(x, y) = Q22(y, x) = KAi(x, y) + 1 2 Ai(x) y

−∞

Ai(t) dt Q12(x, y) = − ∂ ∂y KAi(x, y) − 1 2 Ai(x)Ai(y),

and

Q21(x, y) = − ∞

x

KAi(t, y) dt − 1 2 y

x

Ai(t) dt + 1 2 ∞

x

Ai(t) dt · ∞

y

Ai(t) dt − 1 2 sgn(x − y),

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 7 / 37

Here, QAi is Hilbert-Schmidt on L2(s0, ∞) with kernel entries

Q11(x, y) = Q22(y, x) = KAi(x, y) + 1 2 Ai(x) y

−∞

Ai(t) dt Q12(x, y) = − ∂ ∂y KAi(x, y) − 1 2 Ai(x)Ai(y),

and

Q21(x, y) = − ∞

x

KAi(t, y) dt − 1 2 y

x

Ai(t) dt + 1 2 ∞

x

Ai(t) dt · ∞

y

Ai(t) dt − 1 2 sgn(x − y),

where we use the trace-class kernel (on L2(s0, ∞))

KAi(x, y) = ∞ Ai(x + s)Ai(y + s) ds.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 7 / 37

Universality II The limiting kernel QAi(x, y) is once more universal (Soshnikov 1999), it governs the soft edge scaling limits of the k-point correlation functions for any (properly centered and scaled) real Wigner matrix X (modulo some decay constraints).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 8 / 37

Universality II The limiting kernel QAi(x, y) is once more universal (Soshnikov 1999), it governs the soft edge scaling limits of the k-point correlation functions for any (properly centered and scaled) real Wigner matrix X (modulo some decay constraints). The above kernel allows us to formulate the following central limit theorem for the largest eigenvalue λmax in the GOE, as n → ∞,

λmax(X) ⇒ √ 2n + 1 √ 2n

1 6 F1

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 8 / 37

Universality II The limiting kernel QAi(x, y) is once more universal (Soshnikov 1999), it governs the soft edge scaling limits of the k-point correlation functions for any (properly centered and scaled) real Wigner matrix X (modulo some decay constraints). The above kernel allows us to formulate the following central limit theorem for the largest eigenvalue λmax in the GOE, as n → ∞,

λmax(X) ⇒ √ 2n + 1 √ 2n

1 6 F1

where the cdf of F1 equals (Tracy, Widom 2005)

• P(F1 ≤ s)

2 = det

2 (1 − GQAiG−1 ↾L2(s,∞)⊕L2(s,∞)).

with G = diag(g, g −1) and g(x) = √ 1 + x2.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 8 / 37

• 6
• 4
• 2

2 4 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

• 6
• 4
• 2

2 4 6 0.2 0.4 0.6 0.8 1

Figure 3: Tracy-Widom distribution F1 (blue) versus N(0, 1) (red).

mean variance skewness kurtosis N(0, 1) 1 F1

• 1.20653

1.60778 0.29346 0.16524

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 9 / 37

There are other explicit formulæ for the cdf of F1:

• 1. Airy determinant and resolvent formula (Forrester 2006)
• P(F1 ≤ s)

2 = det(1 − (KAi + U ⊗ V ) ↾L2(s,∞))

where (U ⊗ V )(x, y) = Ai(x)A(y) and A(x) = x

−∞ Ai(y) dy.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 10 / 37

There are other explicit formulæ for the cdf of F1:

• 1. Airy determinant and resolvent formula (Forrester 2006)
• P(F1 ≤ s)

2 = det(1 − (KAi + U ⊗ V ) ↾L2(s,∞))

where (U ⊗ V )(x, y) = Ai(x)A(y) and A(x) = x

−∞ Ai(y) dy.

• 2. Expression in terms of Painlev´

e-II (Tracy, Widom 1996)

• P(F1 ≤ s)

2 = exp

• −

∞

s

(x − s)q2(x)dx − ∞

s

q(x)dx

• where q solves d2q

dx2 = xq + 2q3 with q(x) ∼ Ai(x), x → +∞.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 10 / 37

There are other explicit formulæ for the cdf of F1:

• 1. Airy determinant and resolvent formula (Forrester 2006)
• P(F1 ≤ s)

2 = det(1 − (KAi + U ⊗ V ) ↾L2(s,∞))

where (U ⊗ V )(x, y) = Ai(x)A(y) and A(x) = x

−∞ Ai(y) dy.

• 2. Expression in terms of Painlev´

e-II (Tracy, Widom 1996)

• P(F1 ≤ s)

2 = exp

• −

∞

s

(x − s)q2(x)dx − ∞

s

q(x)dx

• where q solves d2q

dx2 = xq + 2q3 with q(x) ∼ Ai(x), x → +∞.

• 3. Single determinantal formula (Ferrari, Spohn 2005)

P(F1 ≤ s) = det(1 − F ↾L2(s,∞)), KAi = F 2.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 10 / 37

Somewhat GOE but non-symmetric

We now consider the Real Ginibre ensemble (GinOE), i.e. matrices

X ∈ Rn×n : Xjk

iid

∼ N(0, 1). (Ginibre 1965)

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 11 / 37

Somewhat GOE but non-symmetric

We now consider the Real Ginibre ensemble (GinOE), i.e. matrices

X ∈ Rn×n : Xjk

iid

∼ N(0, 1). (Ginibre 1965)

Equivalently think of this setup as a log-gas system

λ1 < λ2 < . . . < λL

• →

α=(λ1,...,λL)

; x1 < . . . < xM; y1, . . . , yM > 0

• →

β=(x1+iy1,...,xM+iyM)

: L + 2M = n

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 11 / 37

Somewhat GOE but non-symmetric

We now consider the Real Ginibre ensemble (GinOE), i.e. matrices

X ∈ Rn×n : Xjk

iid

∼ N(0, 1). (Ginibre 1965)

Equivalently think of this setup as a log-gas system

λ1 < λ2 < . . . < λL

• →

α=(λ1,...,λL)

; x1 < . . . < xM; y1, . . . , yM > 0

• →

β=(x1+iy1,...,xM+iyM)

: L + 2M = n

with (L, M)-partial joint pdf (Lehmann, Sommers 1991)

fL,M( α, β) = 1 Zn,L,M

• 1≤j<k≤n

|zk − zj| exp  −1 2

n

• j=1

z2

j

  ×

n

• j=1
• erfc(

√ 2 |ℑzj|),

• z ≡ (

α, β, β∗) ∈ Cn.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 11 / 37

The saturn effect The above pdf is not absolutely continuous, in particular P(z1, . . . , zn ∈ R) = 2− n

4 (n−1)

(Edelman 1997)

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 12 / 37

The saturn effect The above pdf is not absolutely continuous, in particular P(z1, . . . , zn ∈ R) = 2− n

4 (n−1)

(Edelman 1997) Objective: What can we say about the underlying limit laws?

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 12 / 37

The saturn effect The above pdf is not absolutely continuous, in particular P(z1, . . . , zn ∈ R) = 2− n

4 (n−1)

(Edelman 1997) Objective: What can we say about the underlying limit laws? The eigenvalues {zj}n

j=1 form a Pfaffian point field (Borodin, Sinclair

2009), with µ ∈ Rℓ, ν ∈ Cm, and µ ∨ α = (µ1, . . . , µℓ, α1, . . . , αL−ℓ),

Rℓ,m( µ, ν) :=

• (L,M)

L≥ℓ,M≥m

1 (L − ℓ)!(M − m)!

• RL−ℓ
• CM−m fL,M(

µ ∨ α, ν ∨ β) ×

L−ℓ

• j=1

dαj

M−m

• k=1

d2βk = Pf

• [KR,R

n

(µj, µk)]ℓ×ℓ

j,k=1

[KR,C

n

(µj, νk)]ℓ×m

j,k=1

[KC,R

n

(νj, µk)]m×ℓ

j,k=1

[KC,C

n

(νj, νk)]m×m

j,k=1

• Thomas J. Bothner (KCL)

Ginibre and Schr¨

• dinger

April 11th, 2019 12 / 37

using four 2 × 2 matrix-valued Hilbert-Schmidt kernels K#,#

n

. Now analyze Rℓ,m asymptotically in different scaling regimes.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 13 / 37

using four 2 × 2 matrix-valued Hilbert-Schmidt kernels K#,#

n

. Now analyze Rℓ,m asymptotically in different scaling regimes. (A) The global eigenvalue regime: define the ESD

µX(s, t) = 1 n#{1 ≤ j ≤ n, ℜzj ≤ s, ℑzj ≤ t}, s, t ∈ R,

then, as n → ∞, the random measure µX/√n converges almost surely to the uniform distribution on the unit disk (Ginibre 1965)

ρc(z) = 1 πχ|z|<1(z) d2z

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 13 / 37

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

Figure 4: The circular law for 1000 real (rescaled) Ginibre matrices of varying

dimensions n × n in comparison with the unit circle boundary. We plot n = 4, 8, 16 from left to right. A saturn effect is clearly visible on the real line.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 14 / 37

• 1.5
• 1
• 0.5

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 n=5 n=10 n=20 n=50 n=100 2 0.05 0.1 0.15 1 2 0.2 0.25 1 0.3 0.35

• 1
• 1
• 2
• 2

Figure 5: Densities of normalized real (left) and complex (right) eigenvalues for

n = 5, 10, 20, 50, 100 (left) and n = 100 (right). The larger n, the better their approach to the uniform density on [−1, 1] (left) and x2 + y 2 ≤ 1 (right).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 15 / 37

Universality III The circular law is a universal limiting law (Tao, Vu 2010), it holds true for any (properly centered and scaled) non-Hermitian real or complex matrix X = (Xjk)n

j,k=1 with E|Xjk|2 < ∞ where Xjk are iid

real or complex variables.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 16 / 37

Universality III The circular law is a universal limiting law (Tao, Vu 2010), it holds true for any (properly centered and scaled) non-Hermitian real or complex matrix X = (Xjk)n

j,k=1 with E|Xjk|2 < ∞ where Xjk are iid

real or complex variables. (B) The local eigenvalue regime: We shall zoom in on √n for n even and discuss “only” the scaling limit of KR,R

n

(·, ·) (Borodin, Sinclair 2009; Poplavskyi, Tribe, Zaboronski 2016):

KR,R

n

(√n + x, √n + y) → Pe(x, y)

as n → ∞ uniformly in x, y ∈ R chosen from compact subsets.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 16 / 37

Here, Pe is Hilbert-Schmidt on L2(s0, ∞) with kernel entries

P11(x, y) = P22(y, x) = Ke(x, y) + 1 2e(x) y

−∞

e(t) dt P12(x, y) = − ∂ ∂y Ke(x, y) − 1 2e(x)e(y),

and

P21(x, y) = − ∞

x

Ke(t, y) dt − 1 2 y

x

e(t) dt + 1 2 ∞

x

e(t) dt · ∞

y

e(t) dt − 1 2 sgn(x − y)

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 17 / 37

Here, Pe is Hilbert-Schmidt on L2(s0, ∞) with kernel entries

P11(x, y) = P22(y, x) = Ke(x, y) + 1 2e(x) y

−∞

e(t) dt P12(x, y) = − ∂ ∂y Ke(x, y) − 1 2e(x)e(y),

and

P21(x, y) = − ∞

x

Ke(t, y) dt − 1 2 y

x

e(t) dt + 1 2 ∞

x

e(t) dt · ∞

y

e(t) dt − 1 2 sgn(x − y)

where we use the trace-class kernel (on L2(s0, ∞))

Ke(x, y) = ∞ e(x + s)e(y + s) ds; e(x) = 1 √π e−x2.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 17 / 37

Universality IV It is an open question whether Pe(x, y) is universal, i.e. whether it governs the scaling limits of the real-real (ℓ, m)-point correlation functions for any (properly centered and scaled) non-Hermitian real matrix X with iid entries (modulo some decay constraints).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 18 / 37

Universality IV It is an open question whether Pe(x, y) is universal, i.e. whether it governs the scaling limits of the real-real (ℓ, m)-point correlation functions for any (properly centered and scaled) non-Hermitian real matrix X with iid entries (modulo some decay constraints). The above kernel allows us to formulate the following central limit theorem for the largest real eigenvalue λmax ∈ R in the GinOE, as n → ∞,

λmax(X) ⇒ √n + G

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 18 / 37

Universality IV It is an open question whether Pe(x, y) is universal, i.e. whether it governs the scaling limits of the real-real (ℓ, m)-point correlation functions for any (properly centered and scaled) non-Hermitian real matrix X with iid entries (modulo some decay constraints). The above kernel allows us to formulate the following central limit theorem for the largest real eigenvalue λmax ∈ R in the GinOE, as n → ∞,

λmax(X) ⇒ √n + G

where the cdf of G equals (Rider, Sinclair 2014; Poplavskyi, Tribe, Zaboronski 2017)

• P(G ≤ s)

2 = det

2 (1 − GPeG−1 ↾L2(s,∞)⊕L2(s,∞))

with G = diag(g, g−1) and g(x) = √ 1 + x2.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 18 / 37

• 10
• 8
• 6
• 4
• 2

2 4 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 10
• 8
• 6
• 4
• 2

2 4 6 0.2 0.4 0.6 0.8 1 1.2

Figure 6: Tracy-Widom distribution F1 (blue) versus G (red).

mean variance skewness kurtosis G

• 1.30319

3.97536

• 1.76969

5.14560 F1

• 1.20653

1.60778 0.29346 0.16524

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 19 / 37

What other explicit formulæ for the cdf of G are available?

• 1. Exponential determinant and resolvent formula (Rider, Sinclair

2014; Poplavskyi, Tribe, Zaboronski 2017)

• P(G ≤ s)

2 = det(1 − (Ke + Ue ⊗ Ve) ↾L2(s,∞))

where (Ue ⊗ Ve)(x, y) = e(x)E(y) and E(y) = x

−∞ e(y) dy.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 20 / 37

What other explicit formulæ for the cdf of G are available?

• 1. Exponential determinant and resolvent formula (Rider, Sinclair

2014; Poplavskyi, Tribe, Zaboronski 2017)

• P(G ≤ s)

2 = det(1 − (Ke + Ue ⊗ Ve) ↾L2(s,∞))

where (Ue ⊗ Ve)(x, y) = e(x)E(y) and E(y) = x

−∞ e(y) dy.

• 2. Any Tracy-Widom type closed form expression?

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 20 / 37

What other explicit formulæ for the cdf of G are available?

• 1. Exponential determinant and resolvent formula (Rider, Sinclair

2014; Poplavskyi, Tribe, Zaboronski 2017)

• P(G ≤ s)

2 = det(1 − (Ke + Ue ⊗ Ve) ↾L2(s,∞))

where (Ue ⊗ Ve)(x, y) = e(x)E(y) and E(y) = x

−∞ e(y) dy.

• 2. Any Tracy-Widom type closed form expression?
• 3. Any Ferrari-Spohn type determinantal formula?

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 20 / 37

What other explicit formulæ for the cdf of G are available?

• 1. Exponential determinant and resolvent formula (Rider, Sinclair

2014; Poplavskyi, Tribe, Zaboronski 2017)

• P(G ≤ s)

2 = det(1 − (Ke + Ue ⊗ Ve) ↾L2(s,∞))

where (Ue ⊗ Ve)(x, y) = e(x)E(y) and E(y) = x

−∞ e(y) dy.

• 2. Any Tracy-Widom type closed form expression?
• 3. Any Ferrari-Spohn type determinantal formula?

Roadblock: KAi(x, y) has a Christoffel-Darboux type structure, i.e.

KAi(x, y) = φ(x)ψ(y) − ψ(x)φ(y) x − y , φ(z) = Ai(z), ψ(z) = Ai′(z),

but this is not true for Ke(x, y)!

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 20 / 37

What other explicit formulæ for the cdf of G are available?

• 1. Exponential determinant and resolvent formula (Rider, Sinclair

2014; Poplavskyi, Tribe, Zaboronski 2017)

• P(G ≤ s)

2 = det(1 − (Ke + Ue ⊗ Ve) ↾L2(s,∞))

where (Ue ⊗ Ve)(x, y) = e(x)E(y) and E(y) = x

−∞ e(y) dy.

• 2. Any Tracy-Widom type closed form expression?
• 3. Any Ferrari-Spohn type determinantal formula?

Roadblock: KAi(x, y) has a Christoffel-Darboux type structure, i.e.

KAi(x, y) = φ(x)ψ(y) − ψ(x)φ(y) x − y , φ(z) = Ai(z), ψ(z) = Ai′(z),

but this is not true for Ke(x, y)! − → Do remember N. Wiener and F!

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 20 / 37

And finally they meet

Define F(t; γ) :=

• det(1 − γ(Ke + Ue ⊗ Ve) ↾L2(t,∞)),

t ∈ R, γ ∈ [0, 1]. and consider the following Riemann-Hilbert problem (RHP):

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 21 / 37

And finally they meet

Define F(t; γ) :=

• det(1 − γ(Ke + Ue ⊗ Ve) ↾L2(t,∞)),

t ∈ R, γ ∈ [0, 1]. and consider the following Riemann-Hilbert problem (RHP): Zakharov-Shabat (ZS) RHP For any (x, γ) ∈ R × [0, 1] determine X(z) = X(z; x, γ) ∈ C2×2 such that (1) X(z) is analytic for z ∈ C \ R and continuous on C±.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 21 / 37

And finally they meet

Define F(t; γ) :=

• det(1 − γ(Ke + Ue ⊗ Ve) ↾L2(t,∞)),

t ∈ R, γ ∈ [0, 1]. and consider the following Riemann-Hilbert problem (RHP): Zakharov-Shabat (ZS) RHP For any (x, γ) ∈ R × [0, 1] determine X(z) = X(z; x, γ) ∈ C2×2 such that (1) X(z) is analytic for z ∈ C \ R and continuous on C±. (2) The limits X±(z) := limǫ↓0 X(z ± iǫ), z ∈ R satisfy X+(z) = X−(z)

• 1 − |r(z)|2

−¯ r(z)e−2ixz r(z)e2ixz 1

• ;

r(z) = r(z; γ) = −i√γe− 1

4 z2

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 21 / 37

And finally they meet

Define F(t; γ) :=

• det(1 − γ(Ke + Ue ⊗ Ve) ↾L2(t,∞)),

t ∈ R, γ ∈ [0, 1]. and consider the following Riemann-Hilbert problem (RHP): Zakharov-Shabat (ZS) RHP For any (x, γ) ∈ R × [0, 1] determine X(z) = X(z; x, γ) ∈ C2×2 such that (1) X(z) is analytic for z ∈ C \ R and continuous on C±. (2) The limits X±(z) := limǫ↓0 X(z ± iǫ), z ∈ R satisfy X+(z) = X−(z)

• 1 − |r(z)|2

−¯ r(z)e−2ixz r(z)e2ixz 1

• ;

r(z) = r(z; γ) = −i√γe− 1

4 z2

(3) As z → ∞, we require X(z) = I + X1z−1 + O

• z−2

, Xi = Xi(x, γ).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 21 / 37

Baik-B 2018 The ZS-RHP is uniquely solvable for any (x, γ) ∈ R × [0, 1]. Also, X 12

1 (·, γ) ∈ R is continuous for any γ ∈ [0, 1]

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 22 / 37

Baik-B 2018 The ZS-RHP is uniquely solvable for any (x, γ) ∈ R × [0, 1]. Also, X 12

1 (·, γ) ∈ R is continuous for any γ ∈ [0, 1] and

• F(t; γ)

2 = exp

• − 1

4 ∞

t

(x − t)

• y

x 2; γ

• 2

dx

• (1)

×

• cosh µ(t; γ) − √γ sinh µ(t; γ)
• ,

using the abbreviations

µ(t; γ) := − i 2 ∞

t

y x 2; γ

• dx,

and y(x; γ) := 2iX 12

1 (x, γ).

Identity (1) mirrors a corresponding γ-deformed TW identity in the superimposed GOE (Forrester 2006).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 22 / 37

But where is Prof. Schr¨

• dinger?

Set

Ψ(z) := X(z)e−ixzσ3, z ∈ C \ R,

then ∂Ψ

∂x Ψ−1 is entire,

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 23 / 37

But where is Prof. Schr¨

• dinger?

Set

Ψ(z) := X(z)e−ixzσ3, z ∈ C \ R,

then ∂Ψ

∂x Ψ−1 is entire, in fact by (3) above and Liouville’s theorem

∂Ψ ∂x =

• −izσ3 + 2i

y ¯ y

• Ψ,

(2)

which is the famous Zakharov-Shabat system of 1972.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 23 / 37

But where is Prof. Schr¨

• dinger?

Set

Ψ(z) := X(z)e−ixzσ3, z ∈ C \ R,

then ∂Ψ

∂x Ψ−1 is entire, in fact by (3) above and Liouville’s theorem

∂Ψ ∂x =

• −izσ3 + 2i

y ¯ y

• Ψ,

(2)

which is the famous Zakharov-Shabat system of 1972. It is directly related to several of the most interesting nonlinear evolution equations in 1 + 1 dimensions which are solvable by the IST method.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 23 / 37

For instance, in order to solve the Cauchy problem for the defocusing nonlinear Schr¨

• dinger equation,

iyt + yxx − 2|y|2y = 0, y(x, 0) = y0(x) ∈ S(R); y = y(x, t) : R2 → C,

• ne first computes the reflection coefficient r(z) ∈ S(R) associated

to y0 through the direct scattering transform. Note that y0 → r is a bijection from S(R) onto S(R) ∩ {r : r∞ < 1} (Beals, Coifman 1984).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 24 / 37

For instance, in order to solve the Cauchy problem for the defocusing nonlinear Schr¨

• dinger equation,

iyt + yxx − 2|y|2y = 0, y(x, 0) = y0(x) ∈ S(R); y = y(x, t) : R2 → C,

• ne first computes the reflection coefficient r(z) ∈ S(R) associated

to y0 through the direct scattering transform. Note that y0 → r is a bijection from S(R) onto S(R) ∩ {r : r∞ < 1} (Beals, Coifman 1984). Second, one considers the ZS-RHP above subject to the replacement e2ixz → e2i(2tz2+xz), t ∈ R, and provided this problem is solvable, its (unique) solution solves dNLS with y(x, 0) = y0(x) via y(x, t) = 2iX 12

1 (x, t).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 24 / 37

Some corollaries

First, tail estimates for G. Baik-B 2018 Let γ ∈ [0, 1], then as t → +∞,

F(t; γ) = 1 − γ 4erfc(t) + O

• γ

3 2 t−1e−2t2

.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 25 / 37

Some corollaries

First, tail estimates for G. Baik-B 2018 Let γ ∈ [0, 1], then as t → +∞,

F(t; γ) = 1 − γ 4erfc(t) + O

• γ

3 2 t−1e−2t2

.

On the other hand, as t → −∞,

F(t; γ) = eη1(γ)tη0(γ)

• 1 + o(1)
• ,

η1(γ) = 1 2 √ 2π Li 3

2 (γ)

in terms of the polylogarithm Lis(z) and with a t-independent positive factor η0(γ). Also, η0(1) = 0.75277069.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 25 / 37

• 2
• 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 GinOE Right asy.

• 18
• 16
• 14
• 12
• 10
• 8
• 6
• 4
• 2

2 10-5 10-4 10-3 10-2 10-1 100 GinOE Left asy.

Figure 7: We doublecheck our tail estimates (blue) against the numerically

computed values of F(t; 1) (red).

Second, the analogue of the Ferrari-Spohn formula for G:

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 26 / 37

Baik-B 2018 We have

F(t; 1) = det(1 − S ↾L2(t,∞)) (3)

where S : L2(R) → L2(R) is the integral operator with kernel

S(x, y) = 1 2√πe− 1

4 (x+y)2 Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 27 / 37

Baik-B 2018 We have

F(t; 1) = det(1 − S ↾L2(t,∞)) (3)

where S : L2(R) → L2(R) is the integral operator with kernel

S(x, y) = 1 2√πe− 1

4 (x+y)2

Next objectives: What is the probabilistic interpretation of F(t; γ) for all γ ∈ [0, 1]? What is the exact value of η0(γ)? Can we generalize (3) for F(t; γ), γ ∈ [0, 1]?

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 27 / 37

Some proof ideas

Recall that

Ke(x, y) = ∞ e(x + s)e(y + s) ds; e(x) = 1 √πe−x2 (4)

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 28 / 37

Some proof ideas

Recall that

Ke(x, y) = ∞ e(x + s)e(y + s) ds; e(x) = 1 √πe−x2 (4)

and use

e−x2 = 1 2√π

• Γ

e− 1

4 λ2±ixλ dλ,

x ∈ R.

ℜλ ℑλ

π 4

− π

4 3π 4 5π 4

Γ Figure 8: An admissible choice for the contour Γ in (??).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 28 / 37

After substitution into (4),

Ke(x, y) = 1 (2π)2

• Γλ
• Γw

e− 1

4 (λ2+w 2)e−i(xλ−yw)

∞ e−iu(λ−w) du

• dw dλ

we choose (λ, w) ∈ Γλ × Γw such that ℑw > ℑλ and obtain

Ke(x, y) = 1 (2π)2

• Γλ
• Γw

e− 1

4 (λ2+w 2)−i(xλ−yw)

i(λ − w) dw dλ.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 29 / 37

After substitution into (4),

Ke(x, y) = 1 (2π)2

• Γλ
• Γw

e− 1

4 (λ2+w 2)e−i(xλ−yw)

∞ e−iu(λ−w) du

• dw dλ

we choose (λ, w) ∈ Γλ × Γw such that ℑw > ℑλ and obtain

Ke(x, y) = 1 (2π)2

• Γλ
• Γw

e− 1

4 (λ2+w 2)−i(xλ−yw)

i(λ − w) dw dλ.

Use residue theorem Suppose w ∈ Γw satisfies ℑw > 0. Then for any y, t ∈ R : y = t,

1 2πi ∞

−∞

eiµ(y−t) dµ µ − w = eiw(y−t)χ(t,∞)(y).

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 29 / 37

Now combine (Γλ = R, Γw ≡ Γ),

Ke(x, y)χ(t,∞)(y) =

• R2

e−ixλ √ 2π

• 1

(2π)2

• Γ

e− 1

4 (λ2+w 2)−it(µ−w)

(λ − w)(w − µ) dw

• =:E(λ,µ)

eiyµ √ 2π dµ dλ

Thus, Keχ(t,∞) on L2(R) is simply the operator composition FEF−1. After properly modifying the function spaces, E is trace-class and can be massaged into the aforementioned Christoffel-Darboux structure.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 30 / 37

Thank you very much for your attention!!!

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 31 / 37

References I

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random matrices, J. Math. Anal. Appl. 20 (1967), 262-268.

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Ginibre ensemble and its relation to the Zakharov-Shabat system, preprint arXiv:1808.02419

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matrices and its scaling limits, Comm. Math. Phys. 291 (2009),

• no. 1, 177-224.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 32 / 37

References II

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Erratum to: The Ginibre ensemble of real random matrices and its scaling limits, Commun. Math. Phys. 346 (2016), 1051-1055.

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Tracy-Widom distribution, J. Phys. A: Math. Gen. 38 (2005), L557.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 33 / 37

References III

• P. Forrester, T. Nagao, G. Honner, Correlations for the
• rthogonal-unitary and symplectic-unitary transitions at the hard

and soft edges, Nuclear Physics B 553 (1999), 601-643.

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matrix ensembles with orthogonal and symplectic symmetry, Nonlinearity 19 (2006), 2989-3002.

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matrices, Journal of Math. Phys. 6, 440 (1965).

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equations, Ann. Eugen. 9 (1939), 250-258.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 34 / 37

References IV

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matrices, Physical Review Letters 67, no. 8 (1991).

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nuclear spectra, Nuclear Physics 18 (1960), 395-419.

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the largest real eigenvalue for the real Ginibre ensemble, Ann.

• Appl. Probab. 27 (2017), no. 3, 1395-1413.
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random matrices, Commun. Math. Phys. 207 (1999), 697-733.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 35 / 37

References V

• T. Tao, V. Vu, Random matrices: universality of ESDs and the

circular law, Ann. Probab. 38 (2010), no. 5, 2023-2065.

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ensembles, Commun. Math. Phys. 177 (1996), 727-754.

• C. Tracy, H. Widom, Matrix kernels for the Gaussian orthogonal

and symplectic ensembles, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 2197-2207.

• E. Wigner, Characteristic vectors of bordered matrices with

infinite dimensions, Ann. of Math. (2) 62 (1955), 548-564.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

April 11th, 2019 36 / 37

References VI

• V. Zakharov, A. Shabat, Exact theory of two-dimensional

self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69.

Thomas J. Bothner (KCL) Ginibre and Schr¨

• dinger

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