Maximal angle of a system of self-repelling particles on the circle Antoine Dahlqvist Technische Universität Berlin Berlin-Padova workshop October 25, 2014 1 / 16
Dyson Brownian motion on the circle 1 A random matrix model, a diffusion on U ( N ) 2 Moment method 3 Embed and compare 4 2 / 16
Dyson Brownian motion on the circle N ∈ N ∗ , ω t , 1 , . . . , ω t , N ∈ U , processes satisfying 1 i ω p , t dB t , p + 1 √ N ( ∇ log V ( ω t )) p dt , 1 ≤ p ≤ N , d ω t , p = (Unitary Dyson) N � | z p − z q | 2 , V ( z 1 , . . . , z N ) = p < q B t , 1 , . . . , B t , N N independant standard real Brownian motions. Proposition The SDE (Unitary Dyson) admits a unique strong solution ( ω t ) t ≥ 0 such that ω 0 , 1 = . . . = ω 0 , N = 1 and a.s. for all t > 0 , V ( ω t ) � = 0 . 3 / 16
Dyson Brownian motion on the circle F IGURE : − i log ( ω x / 20 ) 0 ≤ x ≤ 100 � N Questions : if µ t , N = 1 p = 1 δ ω p , µ t , N → µ t , with supp ( µ t ) = exp i [ − θ t , θ t ]? For N all t >0, a.s., ω t , max → exp [ i θ t ]? 4 / 16
Dyson Brownian motion on the circle F IGURE : − i log ( ω x / 20 ) 0 ≤ x ≤ 100 � N Questions : if µ t , N = 1 p = 1 δ ω p , µ t , N → µ t , with supp ( µ t ) = exp i [ − θ t , θ t ]? For N all t >0, a.s., ω t , max → exp [ i θ t ]? 4 / 16
A random matrix model, a diffusion on U ( N ) A scalar product �· , ·� on M N ( C ) : � X , Y � = N Tr ( X ∗ Y ) . [GUE Matrix] : ( H t , N ) t ≥ 0 gaussian process on H N = { H : ∈ M N ( C ) : H ∗ = H } such that ∀ X , Y ∈ H N , E [ � X , H t , N �� Y , H s , N � ] = t ∧ s � X , Y � . ... Y l , m , t 1 √ B p , t , N ... Y l , m , t 1 2 ( B 1 l , m , t + iB 2 Y l , m , t = l , m , t ) √ l , m , t ) l < m , ( B p , t ) p : N 2 independant standard real brownian ( B 1 l , m , t ) l < m , ( B 2 motions. 5 / 16
A random matrix model, a diffusion on U ( N ) A scalar product �· , ·� on M N ( C ) : � X , Y � = N Tr ( X ∗ Y ) . [GUE Matrix] : ( H t , N ) t ≥ 0 gaussian process on H N = { H : ∈ M N ( C ) : H ∗ = H } such that ∀ X , Y ∈ H N , E [ � X , H t , N �� Y , H s , N � ] = t ∧ s � X , Y � . ... Y l , m , t 1 √ B p , t , N ... Y l , m , t 1 2 ( B 1 l , m , t + iB 2 Y l , m , t = l , m , t ) √ l , m , t ) l < m , ( B p , t ) p : N 2 independant standard real brownian ( B 1 l , m , t ) l < m , ( B 2 motions. 5 / 16
A random matrix model, a diffusion on U ( N ) Set K t , N = iH t , N , ( U t , N ) t ≥ 0 the solution of the M N ( C ) - valued SDE, � = U t , N . dK t , N − 1 2 U t , N dt = U t , N ◦ dK t , N , dU t , N = Id . U 0 Proposition i) A.s. for all t ≥ 0 , U t , N ∈ U ( N ) = { U ∈ M N ( C ) : UU ∗ = Id } . ii) The eigenvalues ( ω t , 1 , . . . , ω t , N ) of U t , N satisfy the equation (Unitary Dyson). Remark Λ : unitary diagonal matrices, X = U ( N ) / Λ , Θ : X × Λ − → U ( N ) → UDU − 1 ( U . Λ , D ) �− Then, Jac (Θ)( U . Λ , D ) = V ( D 1 , 1 , . . . , D N , N ) . 6 / 16
A random matrix model, a diffusion on U ( N ) Set K t , N = iH t , N , ( U t , N ) t ≥ 0 the solution of the M N ( C ) - valued SDE, � = U t , N . dK t , N − 1 2 U t , N dt = U t , N ◦ dK t , N , dU t , N = Id . U 0 Proposition i) A.s. for all t ≥ 0 , U t , N ∈ U ( N ) = { U ∈ M N ( C ) : UU ∗ = Id } . ii) The eigenvalues ( ω t , 1 , . . . , ω t , N ) of U t , N satisfy the equation (Unitary Dyson). Remark Λ : unitary diagonal matrices, X = U ( N ) / Λ , Θ : X × Λ − → U ( N ) → UDU − 1 ( U . Λ , D ) �− Then, Jac (Θ)( U . Λ , D ) = V ( D 1 , 1 , . . . , D N , N ) . 6 / 16
Moment method µ t , N = 1 � δ λ . N λ ∈ Spec ( U t , N ) U w n µ t , N ( dw ) = 1 � N Tr ( U n Moments : t , N ) , n ∈ Z . Theorem (P . Biane, F . Xu, T. Lévy) For all t ≥ 0 , f ∈ C ( U , R ) , a.s., µ t , N ( f ) − → µ t ( f ) . µ t unique measure on U , such that ∀ n ∈ Z ∗ , a.s., � | n | | n |− 1 ( −| n | t ) k 1 ω n µ t ( d ω ) = 1 � � −| n | t N Tr ( U n � 2 . t , N ) − → e | n | k ! k + 1 U k = 0 7 / 16
Moment method µ t , N = 1 � δ λ . N λ ∈ Spec ( U t , N ) U w n µ t , N ( dw ) = 1 � N Tr ( U n Moments : t , N ) , n ∈ Z . Theorem (P . Biane, F . Xu, T. Lévy) For all t ≥ 0 , f ∈ C ( U , R ) , a.s., µ t , N ( f ) − → µ t ( f ) . µ t unique measure on U , such that ∀ n ∈ Z ∗ , a.s., � | n | | n |− 1 ( −| n | t ) k 1 ω n µ t ( d ω ) = 1 � � −| n | t N Tr ( U n � 2 . t , N ) − → e | n | k ! k + 1 U k = 0 7 / 16
Moment method d µ t ( x ) = ρ t ( x ) dx , F IGURE : Plot of ρ t ( e 2 i πθ ) , Axes : X : angle : θ ∈ ] − 1 2 , 1 2 [ , Y : time t ∈ [ 0 , 6 ] . supp ( ρ t ) = [ − θ t , θ t ] θ 4 = π. 8 / 16
Moment method ∀ t ≥ 0 , S N t = Spec ( U t , N ) , S t = supp ( µ t ) . Theorem ( D., B. Collins,T. Kemp) ∀ t ≥ 0 , d Haus ( S N t , S t ) → 0 , as N → ∞ , in probability. i log ( U t , N ) , i U t , N − Id Problems : i) Back to H n : 1 U t , N + Id ∈ H n sign moments, no hope to study easily asymptotics of moments. ii) Contrary to the GUE case no independance between coefficients of U ( N ) , no compatibility between the measures of U t , N + 1 and U t , N , a priori, no recursion formula. iii) Stieltjes transform approach : leads to study the stability of a complex Burger equation ∂ t H t , N ( z ) = zH t , N ( z ) ∂ z H t , N ( z ) + 1 N 2 R t , N ( z ) , H 0 , N ( z ) = 1 + z 1 − z , H t , N : { z : | z | < 1 } → − i H . 9 / 16
Moment method ∀ t ≥ 0 , S N t = Spec ( U t , N ) , S t = supp ( µ t ) . Theorem ( D., B. Collins,T. Kemp) ∀ t ≥ 0 , d Haus ( S N t , S t ) → 0 , as N → ∞ , in probability. i log ( U t , N ) , i U t , N − Id Problems : i) Back to H n : 1 U t , N + Id ∈ H n sign moments, no hope to study easily asymptotics of moments. ii) Contrary to the GUE case no independance between coefficients of U ( N ) , no compatibility between the measures of U t , N + 1 and U t , N , a priori, no recursion formula. iii) Stieltjes transform approach : leads to study the stability of a complex Burger equation ∂ t H t , N ( z ) = zH t , N ( z ) ∂ z H t , N ( z ) + 1 N 2 R t , N ( z ) , H 0 , N ( z ) = 1 + z 1 − z , H t , N : { z : | z | < 1 } → − i H . 9 / 16
Moment method ∀ t ≥ 0 , S N t = Spec ( U t , N ) , S t = supp ( µ t ) . Theorem ( D., B. Collins,T. Kemp) ∀ t ≥ 0 , d Haus ( S N t , S t ) → 0 , as N → ∞ , in probability. i log ( U t , N ) , i U t , N − Id Problems : i) Back to H n : 1 U t , N + Id ∈ H n sign moments, no hope to study easily asymptotics of moments. ii) Contrary to the GUE case no independance between coefficients of U ( N ) , no compatibility between the measures of U t , N + 1 and U t , N , a priori, no recursion formula. iii) Stieltjes transform approach : leads to study the stability of a complex Burger equation ∂ t H t , N ( z ) = zH t , N ( z ) ∂ z H t , N ( z ) + 1 N 2 R t , N ( z ) , H 0 , N ( z ) = 1 + z 1 − z , H t , N : { z : | z | < 1 } → − i H . 9 / 16
Moment method Strategy for the upperbound : ∀ ε > 0 , t ≥ 0 to prove that a.s., lim sup 1 i log ω t , max ≤ θ t + ε, it is sufficient to prove that ∀ f ∈ C ∞ ( U ) , ∃ δ, K > 0 , with | E [ µ t , N ( f )] − µ t ( f ) | ≤ KN − 1 − δ , (Speed order 1) and if supp ( f ) ∩ supp ( µ t ) = ∅ , Var ( µ t , N ( f )) ≤ KN − 3 − δ . (Speed order 2) Choose f ∈ C ∞ bump function with P N ,ε = P ( i − 1 log ( ω t , max ) > θ t + ε ) ≤ P ( N µ t , N ( f ) ≥ 1 ) . Then, P N ,ε ≤ ( N − 1 − E [ µ t , N ( f )]) − 2 Var ( µ t , N ( f )) ≤ K ′ N − 1 − δ . 10 / 16
Moment method Strategy for the upperbound : ∀ ε > 0 , t ≥ 0 to prove that a.s., lim sup 1 i log ω t , max ≤ θ t + ε, it is sufficient to prove that ∀ f ∈ C ∞ ( U ) , ∃ δ, K > 0 , with | E [ µ t , N ( f )] − µ t ( f ) | ≤ KN − 1 − δ , (Speed order 1) and if supp ( f ) ∩ supp ( µ t ) = ∅ , Var ( µ t , N ( f )) ≤ KN − 3 − δ . (Speed order 2) Choose f ∈ C ∞ bump function with P N ,ε = P ( i − 1 log ( ω t , max ) > θ t + ε ) ≤ P ( N µ t , N ( f ) ≥ 1 ) . Then, P N ,ε ≤ ( N − 1 − E [ µ t , N ( f )]) − 2 Var ( µ t , N ( f )) ≤ K ′ N − 1 − δ . 10 / 16
Embed and compare To get (Speed order 1), it is sufficient to prove that ∃ m ∈ N ∗ , δ, K > 0, such that � ≤ K n m � � � � � ω n µ t , N ( d ω )] − ω n µ t ( d ω ) � � E [ N 1 + δ . � � U U N , d ∈ N ∗ , on the same probability space, consider ( U i t , N ) 1 ≤ i ≤ d , t ≥ 0 i.i.d. sequence of law ( U t , N ) t ≥ 0 , ( U t , dN ) t ≥ 0 independant of it and set U 1 0 t , N ... U t , d , N = ∈ U ( dN ) U d 0 t , N For any P ∈ C [ X ] , ( dN ) − 1 E [ Tr ( P ( U t , d , N ))] = E [ µ t , N ( P )] 11 / 16
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