Edge-Reinforced Random Walk, Vertex-Reinforced Jump Process and the - PowerPoint PPT Presentation

Edge-Reinforced Random Walk, Vertex-Reinforced Jump Process and the second generalised Ray-Knight theorem Pierre Tarrès, University of Oxford (joint work with Christophe Sabot) Bath, 23 June 2014

CONTENTS I) DEFINITION of Vertex-Reinforced Jump Process (VRJP) II) LINK with Edge-Reinforced Random Walk (ERRW) III) LINK with supersymmetric (SuSy) hyperbolic sigma model IV) Ray-Knight and the reversed VRJP V) The magnetized reversed VRJP: inverting Ray-Knight

I) Definition of Vertex Reinforced Jump Process (VRJP) ◮ G = ( V , E ) non-oriented locally finite graph ◮ ( W e ) e ∈ E be positive conductances on edges ◮ ϕ = ( ϕ i ) i ∈ V , ϕ i > 0. ◮ VRJP ( Y s ) s ≥ 0 continuous-time process: Y 0 = i 0 and, if Y s = i , then Y (conditionally) jumps to j ∼ i at rate W i , j L j ( s ) , where � t L j ( s ) = ϕ j + ✶ Y u = j du . 0 ◮ Proposed by Werner (’00) and studied by Davis, Volkov (’02,’04), Collevechio (’09), Basdevant and Singh (’10) on trees.

I) Definition of Vertex Reinforced Jump Process (VRJP) ◮ Change of time ℓ i := 1 � 2 ( L 2 i − ϕ 2 i ) , t := ℓ i i ∈ V defines time-changed VRJP ( Z t ) t ≥ 0 . ◮ Between times t and t + dt , X (conditionally) jumps to j ∼ i with probability � ϕ 2 j + 2 ℓ j ( t ) W i , j i + 2 ℓ i ( t ) dt , ϕ 2 where � t ℓ i ( t ) = ✶ Z u = i du . 0

II) LINK with ERRW ( ∀ i , ϕ i = 1) ◮ G = ( V , E ) non-oriented locally finite graph ◮ ( a e ) e ∈ E weights on edges, a e > 0. ◮ ERRW on V with initial weights ( a e ) , starting from i 0 ∈ V , is the discrete-time process ( X n ) defined by X 0 = i 0 and Z n ( { X n , j } ) P ( X n + 1 = j | X k , k ≤ n ) = ✶ j ∼ X n � i ∼ X n Z n ( { X n , i } ) where Z n ( e ) = a e + � n − 1 k = 0 ✶ { X k − 1 , X k } = e . : ◮ Coppersmith-Diaconis (’86), Pemantle (’88), Merkl-Rolles (’05-09).

II) LINK with ERRW ( ∀ i , ϕ i = 1) Theorem (Sabot, T. ’11) ◮ W e ∼ Gamma ( a e ) independent, e ∈ E ◮ Y VRJP with Gamma ( a e , 1 ) independent conductances. Then ( Y t ) t ≥ 0 (at jump times) "law" ( X n ) . = Two ingredients : ◮ Rubin construction (Davis ’90, Sellke ’94) : (˜ X t ) continuous-time version of ( X n ) through independent sequences of exponential random variables for each edge. ◮ Kendall transform (’66): Representation of the timeline at each edge as a Poisson Point Process with Gamma random parameter after change of time.

III) LINK with SuSy hyperbolic sigma model ( ∀ i , ϕ i = 1) VRJP Mixture of MJPs ( G finite, | V | = N )) Let P i 0 := law of ( X t ) starting at i 0 ∈ V . Theorem (Sabot, T. ’11) i) ∀ i ∈ V , ∃ U i := lim [( log ℓ i ( t )) / 2 − � i ∈ V ( log ℓ i ( t )) / 2 N ] s.t., conditionally on U = ( U i ) i ∈ V , X is a Markov jump process starting from i 0 with jump rate from i to j W i , j e U j − U i . In particular VRJP (jump times) MC with conductances W U ij := W ij e U i + U j . =

III) LINK with SuSy hyperbolic sigma model ( ∀ i , ϕ i = 1) Mixing law of VRJP ( G finite, | V | = N ) ii) Under P i 0 , ( U i ) has density on H 0 := { ( u i ) , � u i = 0 } N ( 2 π ) ( N − 1 ) / 2 e u i 0 e − H ( W , u ) � D ( W , u ) , where, if T is the set of (non-oriented) spanning trees of G , � H ( W , u ) := 2 W i , j ( cosh ( u i − u j ) − 1 ) , { i , j }∈ E � � W { i , j } e u i + u j . D ( W , u ) := T ∈T { i , j }∈ T Introduced by Zirnbauer (1991) in quantum field theory as the SuSy hyperbolic sigma model.

III) LINK with SuSy hyperbolic sigma model ( ∀ i , ϕ i = 1) Recurrence/transience of ERRW/VRJP Theorem (Recurrence: Sabot-T.’11-’12 (using Disertori and Spencer ’10), Angel, Crawford and Kozma ’12 ) For any graph of bounded degree there exists β c > 0 such that, if W e < β c (resp. a e < β c ) for all e ∈ E, the VRJP (resp. ERRW) is recurrent. Theorem (Transience: Sabot and T. ’12 (using Disertori, Spencer and Zirnbauer ’10), Disertori, Sabot and T. ’13) On Z d , d ≥ 3 , there exists β c > 0 such that, if W e > β c (resp. a e > β c ) for all e ∈ E, then VRJP (resp. ERRW) is transient a.s.

IV) Ray-Knight and the VRJP ◮ P i 0 law of MJP X = ( X t ) t ≥ 0 starting at i 0 , with local time ℓ . ◮ U = E \ { i 0 } , ◮ P G , U = C exp {−E ( ϕ, ϕ ) / 2 } δ 0 ( ϕ i 0 ) � x ∈ U d ϕ x . ◮ σ u = inf { t ≥ 0 ; ℓ i 0 t > u } , u ≥ 0. x , y ∈ V W x , y ( f ( x ) − f ( y )) 2 Dirichlet form at ◮ E ( f , f ) = 1 � 2 f : V → R . Theorem (Generalized second Ray-Knight theorem) For any u > 0 , � σ u + 1 � ℓ x under P i 0 ⊗ P G , U , has the same law as 2 ϕ 2 x x ∈ V √ � 1 � under P G , U . 2 u ) 2 2 ( ϕ x + x ∈ V

IV) Ray-Knight and the VRJP Let � ϕ 2 Φ i = i + 2 ℓ i ( t ) . Let P i 0 , t (resp. P VRJP ) be the laws of VRJP (resp. MJP) ( X t ) t ≥ 0 i 0 , t starting at i 0 , with conductances ( W e ) e ∈ E , up to time t . An elementary calculation yields d P VRJP � � j � = i 0 ϕ j � 1 i 0 , t = exp 2 ( E (Φ , Φ) − E ( ϕ, ϕ )) d P i 0 , t � j � = X t Φ j ◮ Exponential part is the holding probability, where the fraction is the product of the jump probabilities. ◮ Implies partial exchangeability easily ◮ Yields a martingale of the MJP ◮ Can be used to obtain large deviation estimates

IV) Ray-Knight and the VRJP ◮ Given Φ = (Φ i ) i ∈ V , Φ i > 0, time-reversed VRJP defined as follows: ˜ Y 0 = i 0 and, if ˜ Y s = i , then ˜ Y (conditionally) jumps to j ∼ i at rate � t W i , j ˜ L j ( s ) , where ˜ L j ( s ) = Φ j − Y u = j du . ✶ ˜ 0 ◮ Change of time ℓ i := 1 ˜ i − ˜ ˜ 2 (Φ 2 L 2 � i ) , t := ℓ i i ∈ V defines time-changed VRJP (˜ Z t ) t ≥ 0 . ◮ Between times t and t + dt , X (conditionally) jumps to j ∼ i with probability � � t j − 2 ˜ � Φ 2 � ℓ j ( t ) � dt , where ˜ W i , j ℓ i ( t ) = Z u = i du . ✶ ˜ i − 2 ˜ Φ 2 ℓ i ( t ) 0

IV) Ray-Knight and the VRJP Let � i − 2 ˜ Φ 2 ϕ i = ℓ i ( t ) , and assume it is well-defined at time t for all i ∈ V . Let ˜ P VRJP be the law of time-reversed VRJP. Then, similarly, i 0 , t d ˜ P VRJP � � j � = i 0 Φ j � 1 i 0 , t = exp 2 ( E (Φ , Φ) − E ( ϕ, ϕ )) � d P i 0 , t j � = X t ϕ j

IV) Ray-Knight and the VRJP Let t = σ u and � ϕ 2 Φ i = i + 2 ℓ i ( σ u ) , σ ( ϕ ) = ( sign ( ϕ i )) i ∈ V , σ = + ⇐ ⇒ ∀ i ∈ V , σ i = 1 . Let us explain why, heuristically, L ( ℓ | σ ( ϕ ) = +) =˜ ℓ, where ˜ ℓ is distributed under ˜ P VRJP i 0 ,σ u .

IV) Ray-Knight and the VRJP Indeed, by Ray-Knight Theorem above, and a change of variables, P i 0 ⊗ P G , U (Φ + d Φ) = exp ( − 1 2 E (Φ , Φ)) d Φ (1) = exp ( − 1 ϕ j � 2 E (Φ , Φ)) d ϕ Φ j j � = i 0 Therefore d ( P i 0 ⊗ P G , U ) � � j � = i 0 Φ j � 1 P i 0 ⊗ P G , U (Φ + d Φ) = exp 2 ( E (Φ , Φ) − E ( ϕ, ϕ )) d P i 0 , t � j � = X t ϕ j d ˜ � P VRJP � i 0 , t d P i 0 , t = d ˜ P VRJP = . i 0 , t d P i 0 , t Problem By Ray-Knight Theorem, Φ is the modulus a variable √ ϕ ) = P G , U , but ϕ + ˜ 2 u with L ( ˜ √ σ = + does not imply ˜ ϕ + 2 u = + , i.e. (1) does not hold.

V) The magnetized time-reversed VRJP ◮ Magnetized time-reversed VRJP ( ˇ Y s ) s ≥ 0 , ˇ Y 0 = i 0 . ◮ (Φ x ) x ∈ V positive reals. � s ◮ ˇ L i ( s ) = Φ i − Y u = i du . 0 ✶ ˇ ◮ < · > s (resp. F ( s ) ) the expectation (resp. partition function) of the Ising model with interaction J i , j ( s ) = W i , j ˇ L i ( s )ˇ L j ( s ) , and with boundary condition σ i 0 = + 1. ◮ Conditioned on the past at time s , if ˇ Y s = i , jumps from i to j with a rate L j ( s ) < σ j > s W i , j ˇ . < σ i > s ◮ ˇ Y well-defined up to time S = sup { s ≥ 0 , ˇ L i ( s ) > 0 for all i } .

V) The magnetized time-reversed VRJP Lemma ˇ Y S = i 0 . Let ˇ be the law of ( ˇ P VRJP Y s ) s ≥ 0 up to the time S , and set i 0 � ϕ 2 Φ i = i + 2 ℓ i ( σ u ) . Theorem (Sabot, T. ’14) � 1 � 2 (Φ 2 − ˇ L 2 ( S )) , σ ˇ L (( ℓ, ϕ ) | Φ) = L ( S ) , where ◮ ˇ L ( S ) distributed under ˇ P VRJP Φ , i 0 ◮ conditionally on ˇ L ( S ) , σ has law < · > S .