Random matrices and history dependent stochastic processes. - PowerPoint PPT Presentation

Random matrices and history dependent stochastic processes. Margherita DISERTORI Bonn University & Hausdorff Center for Mathematics

History dependent stochastic processes memory effects, self-learning. . . ex: ants looking for the best route nest-food Lattice random Schr¨ odinger operators quantum diffusion for disordered materials These subjects are connected!

History dependent stochastic processes Example: linearly edge-reinforced random walk ( ERRW ) (Diaconis 1986) discrete time process ( X n ) n ≥ 0 , X n ∈ Z d or Λ ⊂⊂ Z d

History dependent stochastic processes Example: linearly edge-reinforced random walk ( ERRW ) (Diaconis 1986) discrete time process ( X n ) n ≥ 0 , X n ∈ Z d or Λ ⊂⊂ Z d Construction: jump only to nearest neighbors • set X 0 = i 0 starting point ω ij (0) = a > 0 ∀| i − j | = 1 initial weights 1 a P ( X 1 = i 1 | X 0 = i 0 ) = 2 da = 2 d ∀| i 0 − i 1 | = 1

History dependent stochastic processes Example: linearly edge-reinforced random walk ( ERRW ) (Diaconis 1986) discrete time process ( X n ) n ≥ 0 , X n ∈ Z d or Λ ⊂⊂ Z d Construction: jump only to nearest neighbors • set X 0 = i 0 starting point ω ij (0) = a > 0 ∀| i − j | = 1 initial weights 1 a P ( X 1 = i 1 | X 0 = i 0 ) = 2 da = 2 d ∀| i 0 − i 1 | = 1 � a +1 i 0 i 1 • update the weights ω ij (1) = a oth .

History dependent stochastic processes Example: linearly edge-reinforced random walk ( ERRW ) (Diaconis 1986) discrete time process ( X n ) n ≥ 0 , X n ∈ Z d or Λ ⊂⊂ Z d Construction: jump only to nearest neighbors • set X 0 = i 0 starting point ω ij (0) = a > 0 ∀| i − j | = 1 initial weights 1 a P ( X 1 = i 1 | X 0 = i 0 ) = 2 da = 2 d ∀| i 0 − i 1 | = 1 � a +1 i 0 i 1 • update the weights ω ij (1) = a oth . • set X 0 = i 0 , X 1 = i 1 , a +1 a P ( X 2 = i 0 | X 0 , X 1 ) = 2 da +1 > 2 da +1 = P ( X 2 = i 2 | X 0 , X 1 ) ∀| i 2 − i 1 | = 1 , i 2 � = i 1

History dependent stochastic processes Example: linearly edge-reinforced random walk ( ERRW ) (Diaconis 1986) discrete time process ( X n ) n ≥ 0 , X n ∈ Z d or Λ ⊂⊂ Z d Construction: jump only to nearest neighbors • set X 0 = i 0 starting point ω ij (0) = a > 0 ∀| i − j | = 1 initial weights 1 a P ( X 1 = i 1 | X 0 = i 0 ) = 2 da = 2 d ∀| i 0 − i 1 | = 1 � a +1 i 0 i 1 • update the weights ω ij (1) = a oth . • set X 0 = i 0 , X 1 = i 1 , a +1 a P ( X 2 = i 0 | X 0 , X 1 ) = 2 da +1 > 2 da +1 = P ( X 2 = i 2 | X 0 , X 1 ) ∀| i 2 − i 1 | = 1 , i 2 � = i 1 prefers to come back!

after n steps ω ij ( n ) P ( X n +1 = j | X n = i , ( X m ) m ≤ n ) = 1 | i − j | =1 � k , | k − j | =1 ω ik ( n ) ω e ( n ) = a + #crossings of e up to time n

after n steps ω ij ( n ) P ( X n +1 = j | X n = i , ( X m ) m ≤ n ) = 1 | i − j | =1 � k , | k − j | =1 ω ik ( n ) ω e ( n ) = a + #crossings of e up to time n a reinforcement parameter the first time e is crossed a → a + 1 ≫ a if a ≪ 1 strong reinforcement ≃ a if a ≫ 1 weak reinforcement

after n steps ω ij ( n ) P ( X n +1 = j | X n = i , ( X m ) m ≤ n ) = 1 | i − j | =1 � k , | k − j | =1 ω ik ( n ) ω e ( n ) = a + #crossings of e up to time n a reinforcement parameter the first time e is crossed a → a + 1 ≫ a if a ≪ 1 strong reinforcement ≃ a if a ≫ 1 weak reinforcement Generalizations Λ any locally finite graph variable initial weights a e

Vertex-reinforced jump process ( VRJP ) (Werner 2000, Volkov, Davis)

Vertex-reinforced jump process ( VRJP ) (Werner 2000, Volkov, Davis) • continuous time jump process ( Y t ) t ≥ 0 , Y t ∈ Z d or Λ ⊂⊂ Z d

Vertex-reinforced jump process ( VRJP ) (Werner 2000, Volkov, Davis) • continuous time jump process ( Y t ) t ≥ 0 , Y t ∈ Z d or Λ ⊂⊂ Z d • conditioned on ( Y s ) s ≤ t jump from Y t = i to | j − i | = 1 with rate � initial weight W > 0 ω jk ( t ) = W (1+ L j ( t )) local time at j L j ( t )

Vertex-reinforced jump process ( VRJP ) (Werner 2000, Volkov, Davis) • continuous time jump process ( Y t ) t ≥ 0 , Y t ∈ Z d or Λ ⊂⊂ Z d • conditioned on ( Y s ) s ≤ t jump from Y t = i to | j − i | = 1 with rate � initial weight W > 0 ω jk ( t ) = W (1+ L j ( t )) local time at j L j ( t ) • process prefers to come back, W plays the same role as a

Vertex-reinforced jump process ( VRJP ) (Werner 2000, Volkov, Davis) • continuous time jump process ( Y t ) t ≥ 0 , Y t ∈ Z d or Λ ⊂⊂ Z d • conditioned on ( Y s ) s ≤ t jump from Y t = i to | j − i | = 1 with rate � initial weight W > 0 ω jk ( t ) = W (1+ L j ( t )) local time at j L j ( t ) • process prefers to come back, W plays the same role as a • generalization to variable initial rates W e and random initial rates

Vertex-reinforced jump process ( VRJP ) (Werner 2000, Volkov, Davis) • continuous time jump process ( Y t ) t ≥ 0 , Y t ∈ Z d or Λ ⊂⊂ Z d • conditioned on ( Y s ) s ≤ t jump from Y t = i to | j − i | = 1 with rate � initial weight W > 0 ω jk ( t ) = W (1+ L j ( t )) local time at j L j ( t ) • process prefers to come back, W plays the same role as a • generalization to variable initial rates W e and random initial rates Connections with ERRW [Sabot-Tarr` es 2013] hitting times for interacting Brownian motions nonlinear sigma models and statistical mechanics random matrices [Sabot-Tarr` es-Zeng 2015] [Sabot-Zeng 2015]

transience/recurrence for VRJP and ERRW as Λ → Z d positive recurrence at strong reinforcement: ERRW and VRJP for any d ≥ 1 [Merkl-Rolles 2009], [D.-Spencer 2010] [Sabot-Tarr` es 2013] [Angel-Crawford-Kozma.Angel 2014] for any reinforcement: ERRW and VRJP in d = 1 and strips [Merkl-Rolles 2009], [D.-Spencer 2010] [Sabot-Tarr` es 2013] [D.-Merkl-Rolles 2014] recurrence in d = 2 ERRW for any reinforcement, partial results for VRJP [Merkl-Rolles 2009], [Sabot-Zeng 2015] [Bauerschmidt-Helmuth-Swan 2018] transience in d ≥ 3 at weak reinforcement: ERRW and VRJP [D.-Spencer-Zirnbauer 2010], [D.-Sabot-Tarr` es 2015] ⇒ phase transition in d ≥ 3

Random matrices

Random matrices • set up: Λ ⊂ Z d finite , H Λ ∈ C Λ × Λ H ∗ Λ = H Λ H Λ random with some probability d P Λ ( H ) Question: lim Λ → Z d d P Λ ( H ) =? spectral properties of the limit operator?

Random matrices • set up: Λ ⊂ Z d finite , H Λ ∈ C Λ × Λ H ∗ Λ = H Λ H Λ random with some probability d P Λ ( H ) Question: lim Λ → Z d d P Λ ( H ) =? spectral properties of the limit operator? odinger H Λ = − ∆ Λ + λ ˆ • special case: random Schr¨ V [Anderson 1958 ] − ∆ Λ lattice Laplacian, λ > 0 parameter V = diag ( { V x } x ∈ Λ ) , V ∈ R Λ random vector d P Λ ( V ) ˆ motivation: quantum mechanics, disordered conductors

odinger H Λ = − ∆ Λ + λ ˆ random Schr¨ V two limit cases: λ = 0 : H = − ∆ : l 2 ( Z d ) → l 2 ( Z d ) extended states: H has only generalized eigenfunctions ψ λ ( k ) ( x ) = e ik · x �∈ l 2 ( Z d ) conductor λ ≫ 1 : H ≃ ˆ V diagonal matrix ⇒ localized eigenfunctions insulator

odinger H Λ = − ∆ Λ + λ ˆ random Schr¨ V two limit cases: λ = 0 : H = − ∆ : l 2 ( Z d ) → l 2 ( Z d ) extended states: H has only generalized eigenfunctions ψ λ ( k ) ( x ) = e ik · x �∈ l 2 ( Z d ) conductor λ ≫ 1 : H ≃ ˆ V diagonal matrix ⇒ localized eigenfunctions insulator results/conjectures in d ≥ 2 Assume V independent or short range correlated: large disorder λ ≫ 1 : exponentially localized eigenfunctions ∀ d ≥ 2 [Fr¨ ohlich-Spencer 1983], [Aizenman-Molchanov 1993 ] and many other results later. . . d = 2 exponentially localized eigenfunctions ∀ λ (conjecture) d ≥ 3 phase transition at weak disorder (conjecture)

odinger operator: H W ( β ) := 2ˆ A special example of random Schr¨ β − WP − P = − ∆ − 2 d Id (off-set Laplacian) P ij = 1 | i − j | =1 β ∈ R Λ random vector with distribution � 2 � | Λ | / 2 e W 2 d | Λ | e − � j ∈ Λ β j 1 1 H ( β ) > 0 2 d β Λ 1 π (det H W ( β ))

odinger operator: H W ( β ) := 2ˆ A special example of random Schr¨ β − WP − P = − ∆ − 2 d Id (off-set Laplacian) P ij = 1 | i − j | =1 β ∈ R Λ random vector with distribution � 2 � | Λ | / 2 e W 2 d | Λ | e − � j ∈ Λ β j 1 1 H ( β ) > 0 2 d β Λ 1 π (det H W ( β )) features β x > 0 ∀ x a.s

odinger operator: H W ( β ) := 2ˆ A special example of random Schr¨ β − WP − P = − ∆ − 2 d Id (off-set Laplacian) P ij = 1 | i − j | =1 β ∈ R Λ random vector with distribution � 2 � | Λ | / 2 e W 2 d | Λ | e − � j ∈ Λ β j 1 1 H ( β ) > 0 2 d β Λ 1 π (det H W ( β )) features β x > 0 ∀ x a.s short range correlations! | i − j | =1 ( √ 1+ λ i √ E [ e − � j λ j β j ] = e − W � 1+ λ j − 1) � 1 + λ j ) − 1 � j (

odinger operator: H W ( β ) := 2ˆ A special example of random Schr¨ β − WP − P = − ∆ − 2 d Id (off-set Laplacian) P ij = 1 | i − j | =1 β ∈ R Λ random vector with distribution � 2 � | Λ | / 2 e W 2 d | Λ | e − � j ∈ Λ β j 1 1 H ( β ) > 0 2 d β Λ 1 π (det H W ( β )) features β x > 0 ∀ x a.s short range correlations! | i − j | =1 ( √ 1+ λ i √ E [ e − � j λ j β j ] = e − W � 1+ λ j − 1) � 1 + λ j ) − 1 � j ( for wired boundary conditions lim Λ → Z d d P Λ ( β ) exists