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
Overview Introduction of the Model 1 Main Result 2 Sketch of the Proof 3 Graphical Representation Estimates and Combinatorics Truncation References 4 Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 2 / 27
Introduction of the model Preliminaries Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 3 / 27
Introduction of the model Preliminaries Let Z d be the infinite lattice with the Euclidean norm | · | Z d and let M the number of points situated at distance at most W ( W ≥ 2) from the origin, i.e. M = M ( W ) = |{ x ∈ Z d : 1 ≤ | · | Z d ≤ W } . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 3 / 27
Introduction of the model Preliminaries Let Z d be the infinite lattice with the Euclidean norm | · | Z d and let M the number of points situated at distance at most W ( W ≥ 2) from the origin, i.e. M = M ( W ) = |{ x ∈ Z d : 1 ≤ | · | Z d ≤ W } . For simplicity we consider throught the proof a d -dimensional finite periodic lattice Λ N ⊂ Z d ( d ≥ 1) of linear size N equipped with the Euclidean norm | · | Z d . Specifically, we take Λ N to be a cube centered around the origin with side length N , i.e. Λ N := ([ − N / 2 , N / 2) ∩ Z ) d . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 3 / 27
Introduction of the Model Preliminaries In order to define the random matrices H with band width W in our model, let us first consider S xy := 1 (1 ≤ | x − y | ≤ W ) . ( M − 1) Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 4 / 27
Introduction of the Model Preliminaries In order to define the random matrices H with band width W in our model, let us first consider S xy := 1 (1 ≤ | x − y | ≤ W ) . ( M − 1) We define the random band matrix ( H xy ) through � H xy := S xy A xy , where ( A xy ) is Hermitian random matrix whose upper triangular entries ( A xy : x ≤ y ) are independent random variables uniformly distributed on the unit circle S 1 ⊂ C . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 4 / 27
Introduction of the Model Preliminaries In order to define the random matrices H with band width W in our model, let us first consider S xy := 1 (1 ≤ | x − y | ≤ W ) . ( M − 1) We define the random band matrix ( H xy ) through � H xy := S xy A xy , where ( A xy ) is Hermitian random matrix whose upper triangular entries ( A xy : x ≤ y ) are independent random variables uniformly distributed on the unit circle S 1 ⊂ C . Note that the entries H xy in the random band matrix H are indexed by x and y which are indices of points in Λ N . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 4 / 27
Introduction of the Model Let us consider also the function P ( t , x ) = | ( e − itH / 2 ) 0 x | 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
Introduction of the Model Let us consider also the function P ( t , x ) = | ( e − itH / 2 ) 0 x | 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 d κ T , x = W 1+ d κ/ 2 X . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 5 / 27
Introduction of the Model Let us consider also the function P ( t , x ) = | ( e − itH / 2 ) 0 x | 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 d κ T , x = W 1+ d κ/ 2 X . Given φ ∈ C b ( R d ) , we define the main quantity that we investigate by x � � � P ( W d κ T , x ) φ Y T ,κ, W ( φ ) ≡ Y T ( φ ) := . W 1+ d κ/ 2 x Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 5 / 27
Known and new result Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 6 / 27
Known and new result Theorem (¨ Erdos L. , Knowles A. 2011) Let 0 < κ < 1 / 3 be fixed. Then for any φ ∈ C b ( R d ) and for any T 0 > 0 we have that � W →∞ E Y T ( φ ) = lim R d dX L ( T , X ) φ ( X ) , uniformly in N ≥ W 1+ d / 6 and 0 ≤ T ≤ T 0 . Here � 1 λ 2 d λ 4 L ( T , X ) := √ 1 − λ 2 G ( λ T , X ) , π 0 and G is the heat kernel � d / 2 � d + 2 2 T | X | 2 . e − d +2 G ( T , X ) := 2 π T Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 6 / 27
Known and new result Theorem (Knowles A., M. 2015) Fix T 0 > 0 and κ such that 0 < κ < 1 / 3 . Choose a real number β satisfying 0 < β < 2 / 3 − 2 κ . Then there exists C ≥ 0 and W 0 ≥ 0 depending only on T 0 , κ and β such that for all T ∈ [0 , T 0 ] , W ≥ W 0 and N ≥ W 1+ d 6 we have Var ( Y T ( φ )) ≤ C || φ || 2 ∞ . W d β Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 7 / 27
Sketch of the Proof Expanding in non-backtracking powers Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 8 / 27
Sketch of the Proof Expanding in non-backtracking powers Let us introduce H (1) = H , � n − 2 � H ( n ) � � 1 ( x i � = x i +2 ) ( n ≥ 2) . x 0 , x n : = H x 0 x 1 , . . . , H x n − 1 x n x 1 ,..., x n − 1 i =0 Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 8 / 27
Sketch of the Proof Expanding in non-backtracking powers Let us introduce H (1) = H , � n − 2 � H ( n ) � � 1 ( x i � = x i +2 ) ( n ≥ 2) . x 0 , x n : = H x 0 x 1 , . . . , H x n − 1 x n x 1 ,..., x n − 1 i =0 Let U k be the k -th Cebyshev polynomial of the second kind and let 1 α k ( t ) := 2 � � 1 − ζ 2 e − it ζ U k ( ζ ) d ζ . π − 1 α m +2 k ( t ) We define the quantity a m ( t ) = � ( M − 1) k . Then we have that k ≥ 0 e − itH / 2 = a m ( t ) H ( m ) . � m ≥ 0 Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 8 / 27
Expansion in non-backtracking powers Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 9 / 27
Expansion in non-backtracking powers Plugging in the definition of Y T ( φ ) we have y 1 y 2 � � � � � Var ( Y T ( φ )) = φ φ � P ( t , y 1 ); P ( t , y 2 ) � W 1+ d κ/ 2 W 1+ d κ/ 2 y 1 , y 2 � � ≤ || φ || 2 |� P ( t , y 1 ); P ( t , y 2 ) �| . ∞ y 1 y 2 Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 9 / 27
Expansion in non-backtracking powers Plugging in the definition of Y T ( φ ) we have y 1 y 2 � � � � � Var ( Y T ( φ )) = φ φ � P ( t , y 1 ); P ( t , y 2 ) � W 1+ d κ/ 2 W 1+ d κ/ 2 y 1 , y 2 � � ≤ || φ || 2 |� P ( t , y 1 ); P ( t , y 2 ) �| . ∞ y 1 y 2 Moreover, � P ( t , y 1 ); P ( t , y 2 ) � = a n 11 ( t ) a n 12 ( t ) a n 21 ( t ) a n 22 ( t ) � H ( n 11 ) 0 y 1 H ( n 12 ) y 1 0 ; H ( n 21 ) 0 y 2 H ( n 22 ) � � = y 2 0 � . n 11 , n 12 ≥ 0 n 21 , n 22 ≥ 0 Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 9 / 27
Graphical Representation Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 10 / 27
Graphical Representation i b(i) a(i) L 1 r(L 1 ) s(L 1 ) L 2 s(L 2 ) r(L 2 ) Figure: The graphical representation of a path of vertices. Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 10 / 27
Each vertex i ∈ V ( L ) carries a label x i ∈ Λ N . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 11 / 27
Each vertex i ∈ V ( L ) carries a label x i ∈ Λ N . 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 ′ ⇔ { x a ( e ) , x b ( e ) } = { x a ( e ′ ) , x b ( e ′ ) } . Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 11 / 27
Graphical Representation Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 12 / 27
Graphical Representation Using the graph L we may now write the covariance as � H ( n 11 ) 0 y 1 H ( n 12 ) y 1 0 ; H ( n 21 ) 0 y 2 H ( n 22 ) � y 2 0 � = Q y 1 , y 2 ( x ) A ( x ) , x ∈ Λ V ( L ) N Vlad Margarint (University of Oxford) Quantum Dynamics and Random Matrices 11 July 2016 12 / 27
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