A nonlinear sigma model connected with stochastic processes and - PowerPoint PPT Presentation

Quantissima, Venice 2017 A nonlinear sigma model connected with stochastic processes and quantum diffusion. Margherita DISERTORI joint work with T.Spencer and M.Zirnbauer C. Sabot and P. Tarr` es F. Merkl and S. Rolles Bonn University & Hausdorff Center for Mathematics

I. A supersymmetric nonlinear sigma model H 2 | 2 (Zirnbauer 1996) originally introduced as a toymodel for quantum diffusion ”spin” model: i ∼ j ∈ Λ e Wij � Si,Sj � � j ∈ Λ e − ǫj � e,Sj � δ ( � S j ,S j � +1) dS Λ dµ ( S )= � Z Λ ◮ Λ ⊂ Z d finite cube, i ∼ j ≡| i − j | =1 ◮ spin S is a (super)vector S =( x,y,z,ξ,η ) , x,y,z even, ξ,η odd elements in a real Grassmann algebra ◮ � S,S ′ � = xx ′ + yy ′ − zz ′ + ξη ′ − ηξ ′ ◮ nonlinear constraint � S,S � = − 1 ⇒ z = √ 1+ x 2 + y 2 +2 ξη ◮ W ij > 0 , � S,S ′ �≤− 1 ⇒ ferromagnetic interaction ◮ ǫ j ≥ 0 mass term, e =(0 , 0 , 1 , 0 , 0) � e,S j � =( z j − 1) ≥ 0 Question: are the spins aligned as Λ → Z d ?

Horospherical coordinates: ( x,y,ξ,η ) → ( u,s, ¯ ψ,ψ ) dµ ( S ) → dµ ( u,s, ¯ ψ,ψ )= j ∈ Λ e − ǫj (cosh uj − 1) e − 1 i ∼ j ∈ Λ e − Wij (cosh( ui − uj ) − 1) � 2 ( s,Ms ) e − ( ¯ ψ,Mψ ) d [ u,s, ¯ � ψ,ψ ] ψ j dψ j e − uj , d [ u,s, ¯ j ∈ Λ du j ds j d ¯ ψ,ψ ]= � i ∼ j W ij e ui + uj ( s i − s j ) 2 + � j ǫ j e uj s 2 ( s,Ms )= � j main features • zero mass: non-compact symmetry group z 2 − x 2 − y 2 − 2 ξη = const • positive mass: compact symmetry subgroup (SUSY) z 2 =1+ x 2 + y 2 +2 ξη = const ⇒ normalized measure Z =1 • for this model a phase transition has been proved in d ≥ 3 [D.-Spencer-Zirnbauer 2010]

II. H 2 | 2 as a random walk in a random environment The setting • finite volume Λ ⊂ Z d , undirected edges E Λ = { e =( j ∼ k ) | j,k ∈ Λ , | j − k | =1 } • discrete time process: ( X n ) n ≥ 0 , X n ∈ Λ • time evolution: n → n +1 : conditional probability ωij ( n ) P ( X n +1 = j | X n = i, ( X k ) k ≤ n )= 1 i ∼ j � k,k ∼ i ωik ( n ) ω ij ( n ) ≥ 0 local conductance at time n ◮ if ω independent of n : Markov chain in the environment ω ◮ if ω time dependent: memory effect Question: is the process recurrent or transient as Λ → Z d ?

( X n ) n ≥ 0 is a random walk in a random environment (RWRE) if P ω � P 0 , Λ [ · ] = 0Λ [ · ] dρ 0 , Λ ( ω ) P 0 , Λ [ · ] prob. for the process on Λ starting at 0 0 , Λ [ · ] Markov chain in a frozen environment ω = { ω e } e ∈ E Λ P ω dρ 0 , Λ ( ω ) mixing measure H 2 | 2 maps to the mixing measure of two RWRE models ◮ linearly edge-reinforced random walk ERRW (Diaconis 1986) ◮ vertex-reinforced jump process VRJP (Werner 2000, Volkov, Davis) both processes tend to come back to sites already visited in the past (attractive interaction) and are RWRE

P ω ( u,W ) VRJP as RWRE: P VRJP � 0 ,W, Λ [ · ] = [ · ] dρ 0 ,W, Λ ( u ) 0Λ ◮ u ∈ R Λ random vector with prob. law dρ 0 ,W, Λ ( u ) Wij 2 e uj − ui , W ij > 0 ◮ ω ij ( u,W )= connection with H 2 | 2 : dρ 0 ,W, Λ ( u ) = u − marginal of H 2 | 2 with mass at 0: ǫ j = δ j 0 ǫ i ∼ j e − Wij (cosh( ui − uj ) − 1) e − ǫ (cosh u 0 − 1) √ = � det M ( u ) du j du j e − uj (2 π ) − 1 / 2 , du = � ERRW as a RWRE: P ERRW � P VRJP 0 ,a, Λ [ · ] = 0 ,W, Λ [ · ] dγ 0 ,a, Λ ( W ) { W e } e ∈ E independent gamma distributed r.v.: e e − We W ae − 1 dγ 0 ,a, Λ ( W ) ∝ � dW e a e > 0 e

from mixing measure to localization/transience remember: P ω P 0 , Λ [ · ] = � 0Λ [ · ] dρ 0 , Λ ( ω ) Two possible criterions: e 0 a fixed arbitrary edge attached to 0 ◮ positive recurrence: s ] ≤ K e − c | e − e 0 | dρ 0 , Λ ( ω ) [( ω e /ω e 0 ) � unif. in Λ for some 0 <s ≤ 1 , c,K> 0 . s ] ≤ e − c | j − k | in quantum diffusion analog to E H [ | ( E − H ) − 1 jk | ◮ transience ( d ≥ 3) : � dρ 0 , Λ ( ω ) [ ( ω e 0 /ω e ) ] ≤ K unif. in Λ for some K> 0 . ≤ K ǫ = | Λ | − 1 in quantum diffusion � 2 � analog to E H | ( E + iǫ − H ) − 1 jj |

Some results positive recurrence ◮ strong reinf.: ERRW and VRJP for any d ≥ 1 [Merkl-Rolles 2009], [D.-Spencer 2010] [Sabot-Tarr` es 2013] [Angel-Crawford-Kozma 2014] ◮ any reinf.: ERRW and VRJP in d = 1 and strips [Merkl-Rolles 2009], [D.-Spencer 2010] [Sabot-Tarr` es 2013] [D.-Merkl-Rolles 2014] transience ◮ weak reinf.: ERRW and VRJP in d ≥ 3 [D.-Spencer-Zirnbauer 2010], [D.-Sabot-Tarr` es 2015] ⇒ phase transition in d ≥ 3 key tool: Ward identities inherited by the supersymmetric structure of the H 2 | 2 measure

III. H 2 | 2 as a Random Schr¨ odinger operator M ( u ) as a RS matrix: e − u M ( u ) e − u = H β ( u ) =2 β ( u ) − P W Laplacian: P W ij = 1 i ∼ j W ij , diagonal disorder: β ij = 1 i = j β i β j ( u ):= e − 2 uj M jj = ǫ j e − uj + � k ∼ j e uk − uj W jk law of the random potential: [Sabot-Tarr` es-Zeng 2015] 2 ( ǫ,H − 1 − 1 ǫ ) e − � j βj � dν ǫ,W, Λ ( β )= 1 1 β Z 1 Hβ> 0 (det Hβ )1 / 2 e j dβ j Z ǫ,W, Λ =( π 2 ) | V | / 2 e − � e ∈ E We e − � j ǫj

properties of dν ǫ,W, Λ ( β ) : explicit formula for the Laplace transform eǫj (1 − √ 1+ λj ) √ i ∼ j e Wij (1 − √ 1+ λj e − � j λjβj dν ǫ,W, Λ ( β )= � 1+ λk ) � √ 1+ λj � j λ j ≥− 1 Some consequences: [Sabot-Tarr` es-Zeng 2015] [D.,Merkl-Rolles 2016] • the r.v. β j ,β k are independent if | j − k |≥ 2 . • wired boundary conditions ǫ Λ i = � j ∈ Λ c,j ∼ i W ij i.e. zero boundary conditions for u Λ variables: u Λ j =0 ∀ j ∈ Λ c ◮ ∃ a unique prob. measure on R Z d dν ∞ ǫ,W ( β ) with marginals dν ǫ,W, Λ ( β ) (Kolmogorov ext. th.) j = e u Λ ◮ martingale: set ψ Λ E [ ψ Λ ′ |F Λ ] = ψ Λ . j , j ∈ Z d : ∀ Λ ⊂ Λ ′

Some other results • ERRW in d = 2 recurrent for any reinforcement [Merkl-Rolles 2009], [Sabot-Zeng 2015] • infinite hierarchy of martingales with generating function: M Λ ( θ )= e ( u Λ ,θ ) e − 1 2 ( θ,H − 1 θ ∈ ( −∞ , 0] Z d θ ) , Λ [D.,Merkl-Rolles 2016]

Some open problems ◮ d = 2 positive recurrence (exponential localization) is expected for both VRJP and ERRW ◮ d ≥ 3 complete phase diagram? At the moment results only for very small/large reinf ◮ spectral properties for the RS H β

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