Convergence in High Probability of the Quantum Diffusion in a Random - - PowerPoint PPT Presentation

▶
convergence in high probability of the quantum diffusion
SMART_READER_LITE
LIVE PREVIEW

Convergence in High Probability of the Quantum Diffusion in a Random - - PowerPoint PPT Presentation

Oct 16, 2022 357 likes •1.01k views

Convergence in High Probability of the Quantum Diffusion in a Random Band Matrix Model Project supervised by Prof. Antti Knowles- ETH Z urich Vlad Margarint Dept. of Mathematics, University of Oxford vlad.margarint@maths.ox.ac.uk 11 July

slide-1
SLIDE 1

Convergence in High Probability of the Quantum Diffusion in a Random Band Matrix Model

Project supervised by Prof. Antti Knowles- ETH Z¨ urich Vlad Margarint

  • Dept. of Mathematics, University of Oxford

vlad.margarint@maths.ox.ac.uk

11 July 2016

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 1 / 27

slide-2
SLIDE 2

Overview

1

Introduction of the Model

2

Main Result

3

Sketch of the Proof Graphical Representation Estimates and Combinatorics Truncation

4

References

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 2 / 27

slide-3
SLIDE 3

Introduction of the model

Preliminaries

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 3 / 27

slide-4
SLIDE 4

Introduction of the model

Preliminaries

Let Zd be the infinite lattice with the Euclidean norm | · |Zd and let M the number of points situated at distance at most W (W ≥ 2) from the origin, i.e. M = M(W ) = |{x ∈ Zd : 1 ≤ | · |Zd ≤ W } .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 3 / 27

slide-5
SLIDE 5

Introduction of the model

Preliminaries

Let Zd be the infinite lattice with the Euclidean norm | · |Zd and let M the number of points situated at distance at most W (W ≥ 2) from the origin, i.e. M = M(W ) = |{x ∈ Zd : 1 ≤ | · |Zd ≤ W } . For simplicity we consider throught the proof a d-dimensional finite periodic lattice ΛN ⊂ Zd (d ≥ 1) of linear size N equipped with the Euclidean norm | · |Zd. Specifically, we take ΛN to be a cube centered around the origin with side length N, i.e. ΛN := ([−N/2, N/2) ∩ Z)d .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 3 / 27

slide-6
SLIDE 6

Introduction of the Model

Preliminaries

In order to define the random matrices H with band width W in our model, let us first consider Sxy := 1(1 ≤ |x − y| ≤ W ) (M − 1) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 4 / 27

slide-7
SLIDE 7

Introduction of the Model

Preliminaries

In order to define the random matrices H with band width W in our model, let us first consider Sxy := 1(1 ≤ |x − y| ≤ W ) (M − 1) . We define the random band matrix (Hxy) through Hxy :=

  • SxyAxy ,

where (Axy) is Hermitian random matrix whose upper triangular entries (Axy : x ≤ y) are independent random variables uniformly distributed on the unit circle S1 ⊂ C .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 4 / 27

slide-8
SLIDE 8

Introduction of the Model

Preliminaries

In order to define the random matrices H with band width W in our model, let us first consider Sxy := 1(1 ≤ |x − y| ≤ W ) (M − 1) . We define the random band matrix (Hxy) through Hxy :=

  • SxyAxy ,

where (Axy) is Hermitian random matrix whose upper triangular entries (Axy : x ≤ y) are independent random variables uniformly distributed on the unit circle S1 ⊂ C . Note that the entries Hxy in the random band matrix H are indexed by x and y which are indices of points in ΛN .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 4 / 27

slide-9
SLIDE 9

Introduction of the Model

Let us consider also the function P(t, x) = |(e−itH/2)0x|2 , that describes the quantum transition probability of a particle starting in 0 and ending in position x after time t .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 5 / 27

slide-10
SLIDE 10

Introduction of the Model

Let us consider also the function P(t, x) = |(e−itH/2)0x|2 , that describes the quantum transition probability of a particle starting in 0 and ending in position x after time t . For κ > 0, we introduce the macroscopic time and space coordinates T and X, which are independent of W , and consider the microscopic time and space coordinates t = W dκT , x = W 1+dκ/2X .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 5 / 27

slide-11
SLIDE 11

Introduction of the Model

Let us consider also the function P(t, x) = |(e−itH/2)0x|2 , that describes the quantum transition probability of a particle starting in 0 and ending in position x after time t . For κ > 0, we introduce the macroscopic time and space coordinates T and X, which are independent of W , and consider the microscopic time and space coordinates t = W dκT , x = W 1+dκ/2X . Given φ ∈ Cb(Rd) , we define the main quantity that we investigate by YT,κ,W (φ) ≡ YT(φ) :=

  • x

P(W dκT, x)φ

  • x

W 1+dκ/2

  • .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 5 / 27

slide-12
SLIDE 12

Known and new result

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 6 / 27

slide-13
SLIDE 13

Known and new result

Theorem (¨ Erdos L. , Knowles A. 2011)

Let 0 < κ < 1/3 be fixed. Then for any φ ∈ Cb(Rd) and for any T0 > 0 we have that lim

W →∞ EYT(φ) =

  • RddX L(T, X)φ(X) ,

uniformly in N ≥ W 1+d/6 and 0 ≤ T ≤ T0 . Here L(T, X) := 1 dλ 4 π λ2 √ 1 − λ2 G(λT, X) , and G is the heat kernel G(T, X) := d + 2 2πT d/2 e− d+2

2T |X|2 . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 6 / 27

slide-14
SLIDE 14

Known and new result

Theorem (Knowles A., M. 2015)

Fix T0 > 0 and κ such that 0 < κ < 1/3 . Choose a real number β satisfying 0 < β < 2/3 − 2κ . Then there exists C ≥ 0 and W0 ≥ 0 depending only on T0, κ and β such that for all T ∈ [0, T0] , W ≥ W0 and N ≥ W 1+ d

6 we have

Var(YT(φ)) ≤ C||φ||2

∞

W dβ .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 7 / 27

slide-15
SLIDE 15

Sketch of the Proof

Expanding in non-backtracking powers

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 8 / 27

slide-16
SLIDE 16

Sketch of the Proof

Expanding in non-backtracking powers

Let us introduce H(1) = H , H(n)

x0,xn : =

  • x1,...,xn−1

n−2

  • i=0

1(xi = xi+2)

  • Hx0x1, . . . , Hxn−1xn

(n ≥ 2) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 8 / 27

slide-17
SLIDE 17

Sketch of the Proof

Expanding in non-backtracking powers

Let us introduce H(1) = H , H(n)

x0,xn : =

  • x1,...,xn−1

n−2

  • i=0

1(xi = xi+2)

  • Hx0x1, . . . , Hxn−1xn

(n ≥ 2) . Let Uk be the k-th Cebyshev polynomial of the second kind and let αk(t) := 2 π

1

  • −1
  • 1 − ζ2e−itζUk(ζ)dζ .

We define the quantity am(t) =

k≥0 αm+2k(t) (M−1)k . Then we have that

e−itH/2 =

  • m≥0

am(t)H(m) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 8 / 27

slide-18
SLIDE 18

Expansion in non-backtracking powers

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 9 / 27

slide-19
SLIDE 19

Expansion in non-backtracking powers

Plugging in the definition of YT(φ) we have Var(YT(φ)) =

  • y1,y2

φ

  • y1

W 1+dκ/2

  • φ
  • y2

W 1+dκ/2

  • P(t, y1); P(t, y2)

≤ ||φ||2

∞

  • y1
  • y2

|P(t, y1); P(t, y2)| .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 9 / 27

slide-20
SLIDE 20

Expansion in non-backtracking powers

Plugging in the definition of YT(φ) we have Var(YT(φ)) =

  • y1,y2

φ

  • y1

W 1+dκ/2

  • φ
  • y2

W 1+dκ/2

  • P(t, y1); P(t, y2)

≤ ||φ||2

∞

  • y1
  • y2

|P(t, y1); P(t, y2)| . Moreover, P(t, y1); P(t, y2) = =

  • n11,n12≥0
  • n21,n22≥0

an11(t)an12(t)an21(t)an22(t)H(n11)

0y1 H(n12) y10 ; H(n21) 0y2 H(n22) y20 .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 9 / 27

slide-21
SLIDE 21

Graphical Representation

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 10 / 27

slide-22
SLIDE 22

Graphical Representation

r(L1) s(L1) r(L2) s(L2) i a(i) b(i) L1 L2

Figure: The graphical representation of a path of vertices.

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 10 / 27

slide-23
SLIDE 23

Each vertex i ∈ V (L) carries a label xi ∈ ΛN .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 11 / 27

slide-24
SLIDE 24

Each vertex i ∈ V (L) carries a label xi ∈ ΛN . For each configuration of labels x we assign a lumping Γ = Γ(x) of the set of edges E(L). The lumping Γ = Γ(x) associated with the labels x is given by the equivalence relation e ∼ e′ ⇔ {xa(e), xb(e)} = {xa(e′), xb(e′)} .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 11 / 27

slide-25
SLIDE 25

Graphical Representation

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 12 / 27

slide-26
SLIDE 26

Graphical Representation

Using the graph L we may now write the covariance as H(n11)

0y1 H(n12) y10 ; H(n21) 0y2 H(n22) y20 =

  • x∈ΛV (L)

N

Qy1,y2(x)A(x) ,

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 12 / 27

slide-27
SLIDE 27

Graphical Representation

Using the graph L we may now write the covariance as H(n11)

0y1 H(n12) y10 ; H(n21) 0y2 H(n22) y20 =

  • x∈ΛV (L)

N

Qy1,y2(x)A(x) , where Qy1,y2(x) = 1(xr(L1) = 0)1(xr(L2) = 0)1(xs(L1) = y1)1(xs(L2) = y2)

  • i∈Vb(L)

1(xa(i) = xb(i)) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 12 / 27

slide-28
SLIDE 28

Graphical Representation

Using the graph L we may now write the covariance as H(n11)

0y1 H(n12) y10 ; H(n21) 0y2 H(n22) y20 =

  • x∈ΛV (L)

N

Qy1,y2(x)A(x) , where Qy1,y2(x) = 1(xr(L1) = 0)1(xr(L2) = 0)1(xs(L1) = y1)1(xs(L2) = y2)

  • i∈Vb(L)

1(xa(i) = xb(i)) . and A(x) = E

  • e∈E(L)

Hxe − E

  • e∈E(L1)

HxeE

  • e∈E(L2)

Hxe .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 12 / 27

slide-29
SLIDE 29

Graphical Representation

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 13 / 27

slide-30
SLIDE 30

Graphical Representation

Let Pc(E(L)) be the set of connected even lumpings, i.e. the set of all lumpings Γ for which each lump γ ∈ Γ has even size and there exists γ ∈ Γ such that γ ∩ E(Lk) = ∅ , for k ∈ {1, 2} .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 13 / 27

slide-31
SLIDE 31

Graphical Representation

Let Pc(E(L)) be the set of connected even lumpings, i.e. the set of all lumpings Γ for which each lump γ ∈ Γ has even size and there exists γ ∈ Γ such that γ ∩ E(Lk) = ∅ , for k ∈ {1, 2} .

Lemma

We have that H(n11)

0y1 H(n12) y10 ; H(n21) 0y2 H(n22) y20 =

  • Γ∈Pc(E(L))

Vy1,y2(Γ) , where Vy1,y2(Γ) =

x

1(Γ(x) = Γ)Qy1,y2(x)A(x) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 13 / 27

slide-32
SLIDE 32

Graphical Representation

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 14 / 27

slide-33
SLIDE 33

Graphical Representation

We define Mc the set of all connected pairings

  • n11,n12,n21,n22

{Π ∈ Pc(E(L(n11, n12, n21, n22))) : |π| = 2 , ∀π ∈ Π} .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 14 / 27

slide-34
SLIDE 34

Graphical Representation

We define Mc the set of all connected pairings

  • n11,n12,n21,n22

{Π ∈ Pc(E(L(n11, n12, n21, n22))) : |π| = 2 , ∀π ∈ Π} . We call the lumps π ∈ Π of a pairing Π bridges.

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 14 / 27

slide-35
SLIDE 35

First estimate

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 15 / 27

slide-36
SLIDE 36

First estimate

Consider J{e,e′}(x) = 1(xa(e) = xb(e′))1(xa′(e) = xb(e)) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 15 / 27

slide-37
SLIDE 37

First estimate

Consider J{e,e′}(x) = 1(xa(e) = xb(e′))1(xa′(e) = xb(e)) .

Lemma

We have |P(t, y1); P(t, y2)| ≤

  • Π∈Mc

|an11(Π)(t)an12(Π)(t)an21(Π)(t)an22(Π)(t)|

  • x

Qy1,y2(x)

  • {e,e′}∈Π

Sxe

  • {e,e′}∈Π

J{e,e′}(x) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 15 / 27

slide-38
SLIDE 38

Collapsing of parallel bridges

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 16 / 27

slide-39
SLIDE 39

Collapsing of parallel bridges

To each Π ∈ Mc we associate a couple (Σ, lΣ), where Σ ∈ Mc has no parallel bridges and lΣ := (lσ)σ∈Σ ∈ NΣ . The integer lσ denotes the number of parallel bridges of Π that were collapsed into the bridge σ

  • f Σ . Inverting the procedure we obtain a bijection Π ←

→ (Σ, lΣ) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 16 / 27

slide-40
SLIDE 40

Collapsing of parallel bridges

To each Π ∈ Mc we associate a couple (Σ, lΣ), where Σ ∈ Mc has no parallel bridges and lΣ := (lσ)σ∈Σ ∈ NΣ . The integer lσ denotes the number of parallel bridges of Π that were collapsed into the bridge σ

  • f Σ . Inverting the procedure we obtain a bijection Π ←

→ (Σ, lΣ) . We further define the set of admissible skeletons as G = {S(Π) : Π ∈ Mc} .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 16 / 27

slide-41
SLIDE 41

Collapsing of parallel bridges

To each Π ∈ Mc we associate a couple (Σ, lΣ), where Σ ∈ Mc has no parallel bridges and lΣ := (lσ)σ∈Σ ∈ NΣ . The integer lσ denotes the number of parallel bridges of Π that were collapsed into the bridge σ

  • f Σ . Inverting the procedure we obtain a bijection Π ←

→ (Σ, lΣ) . We further define the set of admissible skeletons as G = {S(Π) : Π ∈ Mc} . We further define |lΣ| =

σ∈Σ lσ for Σ ∈ G .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 16 / 27

slide-42
SLIDE 42

Collapsing of parallel bridges

L1 L2

s(L1) r(L2) s(L2) r(L1)

Figure: Graphical representation of the skeleton for a given configuration

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 17 / 27

slide-43
SLIDE 43

Collapsing of parallel bridges

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 18 / 27

slide-44
SLIDE 44

Collapsing of parallel bridges

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 18 / 27

slide-45
SLIDE 45

Collapsing of parallel bridges

Lemma

We have that

  • y1
  • y2

P(t, y1); P(t, y2) ≤

  • Σ∈G
  • lΣ

|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)|R(Σ) , where R(Σ) =

  • x∈ΛV (Σ)

N

1(xr(L1(Σ)) = 0)1(xr(L2(Σ)) = 0)

  • {e,e′}∈Σ
  • Sl{e,e′}
  • xe
  • σ∈Σ

Jσ(x) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 18 / 27

slide-46
SLIDE 46

Orbits of vertices

τ τ

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 19 / 27

slide-47
SLIDE 47

Orbits of vertices

For fixed Σ ∈ G we define τ : V (Σ) → V (Σ) as follows. Let i ∈ V (Σ) and let e be the unique edge such that {{i, b(i)}, e} ∈ Σ . Then, for any vertex i of Σ ∈ G we define τi = b(e). We denote the

  • rbit of the vertex i ∈ Σ by [i] := {τ ni : n ∈ N}

i τi τ2i s(L1) s(L2) r(L2) r(L1)

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 19 / 27

slide-48
SLIDE 48

Orbits of vertices

Let Z(Σ) := {[i] : i ∈ V (Σ)} be the set of orbits of Σ and |Σ| be the number of bridges of the skeleton Σ and let L(Σ) = |Z ∗(Σ)| with Z ∗(Σ) := Z(Σ) \ {[r(L1)], [r(L2)]} .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 20 / 27

slide-49
SLIDE 49

Orbits of vertices

Let Z(Σ) := {[i] : i ∈ V (Σ)} be the set of orbits of Σ and |Σ| be the number of bridges of the skeleton Σ and let L(Σ) = |Z ∗(Σ)| with Z ∗(Σ) := Z(Σ) \ {[r(L1)], [r(L2)]} .

Lemma

We have the inequality L(Σ) ≤ 2|Σ| 3 + 2 3 .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 20 / 27

slide-50
SLIDE 50

Orbits of vertices

Let Z(Σ) := {[i] : i ∈ V (Σ)} be the set of orbits of Σ and |Σ| be the number of bridges of the skeleton Σ and let L(Σ) = |Z ∗(Σ)| with Z ∗(Σ) := Z(Σ) \ {[r(L1)], [r(L2)]} .

Lemma

We have the inequality L(Σ) ≤ 2|Σ| 3 + 2 3 .

Lemma

Let Σ ∈ G and lΣ ∈ ΛV (Σ)

N

. We have that R(Σ) ≤ C

  • M

M − 1 |lΣ| M−|Σ|/3+2/3 .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 20 / 27

slide-51
SLIDE 51

Truncation

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 21 / 27

slide-52
SLIDE 52

Truncation

We introduce a cut-off at |lΣ| < Mµ for µ < 1/3 . We define E ≤ =

  • Σ∈G
  • |lΣ|≤Mµ

|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)|R(Σ) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 21 / 27

slide-53
SLIDE 53

Truncation

We introduce a cut-off at |lΣ| < Mµ for µ < 1/3 . We define E ≤ =

  • Σ∈G
  • |lΣ|≤Mµ

|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)|R(Σ) .

Lemma

1

For any time t and for any n ∈ N we have |an(t)| ≤ Ctn

n! , with C universal

constant.

2

We have

n≥0

|an(t)|2 = 1 + O(M−1) , uniformly in t ∈ R .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 21 / 27

slide-54
SLIDE 54

Main Lemma

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 22 / 27

slide-55
SLIDE 55

Main Lemma

Lemma

For any Σ ∈ G with |Σ| ≥ 3 we have

  • lΣ

1(|lΣ| ≤ Mµ)|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)| ≤ CMµ(|Σ|−2) (|Σ| − 3)! .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 22 / 27

slide-56
SLIDE 56

End of the proof

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 23 / 27

slide-57
SLIDE 57

End of the proof

In order to finish the argument

1 We prove using Stirling approximation and some arguments involving

geometric series that the remaining terms are tiny. E > =

  • Σ∈G
  • |lΣ|≥Mµ

|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)|R(Σ) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 23 / 27

slide-58
SLIDE 58

End of the proof

In order to finish the argument

1 We prove using Stirling approximation and some arguments involving

geometric series that the remaining terms are tiny. E > =

  • Σ∈G
  • |lΣ|≥Mµ

|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)|R(Σ) .

2 We find bounds by direct computation for the cases |Σ| = 0, |Σ| = 1

and |Σ| = 2 , and we conclude the proof.

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 23 / 27

slide-59
SLIDE 59

Summary

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 24 / 27

slide-60
SLIDE 60

Summary

We started with Var(YT(φ)) ≤ ||φ||2

∞

  • y1
  • y2

|P(t, y1); P(t, y2)| .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 24 / 27

slide-61
SLIDE 61

Summary

We started with Var(YT(φ)) ≤ ||φ||2

∞

  • y1
  • y2

|P(t, y1); P(t, y2)| . Via graphical and combinatorial arguments we arrived at

  • y1
  • y2

P(t, y1); P(t, y2) ≤

  • Σ∈G
  • lΣ

|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)|R(Σ) .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 24 / 27

slide-62
SLIDE 62

Summary

We started with Var(YT(φ)) ≤ ||φ||2

∞

  • y1
  • y2

|P(t, y1); P(t, y2)| . Via graphical and combinatorial arguments we arrived at

  • y1
  • y2

P(t, y1); P(t, y2) ≤

  • Σ∈G
  • lΣ

|an11(Σ,lΣ)(t)an12(Σ,lΣ)(t)an21(Σ,lΣ)(t)an22(Σ,lΣ)(t)|R(Σ) . Truncating the expression in |lΣ| and summing the two terms under the specific conditions on the parameters imposed by the setting (i.e. 0 < κ ≤ 1/3, . 0 < β < 2/3 − 2κ . ), we obtain a bound for the variance of the form Var(YT(φ)) ≤ C||φ||2

∞

W dβ .

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 24 / 27

slide-63
SLIDE 63

Thank you for your attention!

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 25 / 27

slide-64
SLIDE 64

References

Erd¨

  • s L., Knowles A. Quantum Diffusion and Eigenfunction

Delocalization in a Random Band Matrix Model Comm. Math. Phys.,303 509-554, 2011. Bai, Z.D., Yin, Y.Q Limit of the smallest eigenvalue of a large dimensional sample covariance matrix Ann. Probab., 21 12751294, 1993. Erd¨

  • s L., Knowles A. The Altshuler-Shklovskii formulas for random

band matrices II: the general case. Preprint arXiv:1309.5107, 76 pages, to appear in Ann. H. Poincar. Anderson P., Absences of diffusion in certain random lattices Phys. Revue., 109 14921505, 1958. Spencer T., Random banded and sparse matrices (Chapter 23) Oxford handbook of Random Matrix Theory¨ edited by G.Akemann, J.Baik and

  • P. Di Francesco.

Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 26 / 27