Activated Dynamics of the Cascading 2-GREM Scaling limits & Aging

Transcription

1 Activated Dynamics of the Cascading 2-GREM Scaling limits & Aging Véronique Gayrard CNRS, Université d Aix-Marseille Based on joint work with L.R. Fontes, arxiv: Spin Glasses and Related Topics, Banff September 30-October 5, 2018

2 V n = { 1,1} n. 2-level GREM Hamiltonian (H n (σ),σ V n ) centered gaussian process on (Ω,F,P) with where and E(H n (σ)h n (η)) = na(1 ρ n (σ,η)), ρ n (σ,η) = 1 n 1 (min{i σ(i) η(i)} 1) A(x) 1 a 0 p 1 x Alternatively, setting n 1 = pn, n = n 1 +n 2, σ = σ 1 σ 2 V n1 V n2, H n (σ) = H n (1) (σ 1 )+H n (2) (σ 1 σ 2 ) = ang σ (1) 1 + (1 a)ng σ (2) 1 σ 2, where {g (1) σ 1,g (2) σ 1 σ 2 ; σ V n } are i.i.d. N(0,1).

3 STATIC Ground states (extremes of H n ): H n (1) (σ1 1 ) H(1) n (σ1 2 ) H(1) n (σ x 1 1 )... and for each x 1 {1,...,2 n 1} H n (2) (σ x 1 1 σx 11 2 ) H(2) n (σ x 1 1 σx 12 2 ) H(2) n (σ x 1 2 )... 1 σx 1x 2 a > p a < p Free energy for a > p exhibits two discontinuities at the critical inverse temperatures β 1 = β p a < β = 2ln2 < β 2 = β 1 p 1 a

4 Which dynamics to choose? Glauber dynamics: Markov jump process (X n (t),t > 0) on V n with rates e βh n(σ) λn (σ,η) = e βh n(η) λn (η,σ) σ η, and λ n (σ,η) = 0 else. Random Hopping: Metropolis: λ RH n (σ,η) e βh n(σ), σ η. Because H n is hierarchical, i.e. λ Metro n (σ,η) e β[h n(η) H n (σ)] +, σ η. H n (σ) = H (1) n (σ 1 )+H (2) n (σ 1 σ 2 ), σ = σ 1 σ 2 V n1 V n2, so is Metropolis λ Metro n (σ,η) e β[h n(η) H n (σ)] +½σ 1 +e β[h(2) n (η) H n (2) (σ)] +½σ 2. η η We choose λ n (σ,η) e βh n(σ) ½σ 1 η +e βh(2) n (σ) ½σ 2 η.

5 AGING... t 0 t w t w +t time unit = c n µ n : initial distribution c n : time scale C n (t w,t), t w,t > 0: time-time correlation function, e.g., C n (t w,t) = P µn [n 1( ) X n (c n t w ),X n (c n (t w +t)) 1 δ ], 0 < δ < 1...occurs if C (ρ) 1 or lim n C n(t w,t) = C (t/t w ), lim tw,t 0 t/tw=ρ lim n C n(t w,t) = C (ρ). Asymptotically, C n (t w,t) is not time translation invariant ρ

6 Random hopping dynamics of the REM static critical temperature: β = 2log2. time-scale: given β eff > 0, c n (β eff ) = exp { nββ eff β 2β eff ( log ( β 2 eff n/2 ) +log4π )}. start = uniform distribution on V n if β eff β, any distribution else. Stationary (II) Ergodic (IV) Stationary (I) Aging (III)

7 Metropolis dynamics of the REM Stationary (II) Stationary (I) Aging (III) Phase transition lines are not understood: aging is proved for that is, on time scales β > β eff, β eff < β (1 cst. ) logn n, (V.G., PTRF 18) c n (β eff ) < e cst nlogn c n (β ).

8 K-processes Markov processes in random environment on countably infinite state space, N { }, with a single unstable state { }. Simplest example: the uniform K-process (as in the RHD of the REM). when in a stable state x, waits an exponential time of mean value γ(x), x N γ(x) < then jumps to the unstable state, & from there, enters any finite set of stable states with uniform distribution. More general re-entrance mechanisms (not uniform but weighted) or state spaces (hierarchical) needed to describe more involved models or dynamics. Ergodic (infinite volume) dynamics of mean-field spin glasses are modeled in physics through so-called (finite volume) trap models. Two GREM-like trap models were proposed (Bouchaud & Dean, Sasaki & Nemoto). Is one of them the right one?

9 Back to the cascading 2-GREM (a > p) Set λ n (σ,η) e β(h(1) n (σ 1 )+H n (2) (σ 1 σ 2 )) ½σ 1 +e βh(2) n (σ) ½σ 2 η η c i n max σ i V ni e βh(i) n (σ) e nββ ap..., i = 1, e nββ (1 a)(1 p)..., i = 2. Two sources of dynamical phase transitions on extreme scales Static: β 1 = β p a β = 2ln2 β 2 = β 1 p 1 a Fine tuning: Given ζ n < b n n 2β 2 2 let β n (a,p,ζ n ) be the solution in β of In explicit form: We say that β 1 = β 1 n (a,p,ζ n ) is 2 n 2/c 1 n = e ζn. β n (a,p,ζ n ) β FT( 1 ζ n bn ), β FT β 2 1 p pa. above at below fine tuning temperature if ζ n + ζ n ζ ζ n.

10 Assume that a > p,β > β 2 and β FT > β 2. β 1 β β 2 β FT Theorem (Scaling limits).the rescaled process X n (c n t) has three possible scaling limits, Y = Y 1 Y 2, above (c n = c 2 n), below (c n = 2 n 2c 1 nc 2 n) and at fine tuning (c n = e ζ c 2 n), all of them ergodic processes expressed through K-processes. Convergence is in law w.r.t. the environment. Choose as time-time correlation function C(t w,t) = 1P(A 1 A 2 )+pp(a 1 A c 2 )+(1 p)p(ac 1 A 2) where for i = 1,2, A i = A i (t w,t) = {Y i does not jump between times t w and t w +t}. Theorem (Aging). ( ) Asl 1 α2 1+ρ, above FT, ( ) ( ) lim C(t w,t) = pasl 1 α1 α tw,t ρ +(1 p)aslα2 1 1+ρ, at FT, ( ) t/tw ρ pasl 1 α1, below FT, 1+ρ where α i = β i /β (0,1) and Asl is the p.d.f. of the arcsine law.

11 γn(x 1 1 ) = e βh(1) n (σ x 1 1 ) /c 1 n x 1, γn(x 2 1 x 2 ) = e βh(2) n (σ x 1 1 σx 2 2 ) /c 2 n Set S = S 1 S 2, S i = {1,...,M i }, i = 1,2. For x = x 1 x 2,y = y 1 y 2 S, [ x (1 λ 1 n )+λ x 1 n νn(x 1 1 ) ] 1 Proba(x y) = (1+o(1)) M2, if x 1 = y 1, λ x 1 n νn 1(y 1) M 1, else, 2 where λ x 1 n λ x 1 = 1, above FT, 1 1+e ζ M 2 γ 1 (x 1 ) and ν 1 n is the random measure on S 1 ν 1 n (x 1), at FT, 0, below FT, γ1 (x 1 ) z S γ 1 (z), 1 γ1 (x 1 )λ x 1 z S γ 1 (z)λ 1 1 M, 1 above FT, z, at FT, below FT.

12 None of the GREM-like trap models of theoretical physics correctly predicted the dynamics of the GREM at extreme (i.e. ergodic) time-scales.

13 THANK YOU FOR YOUR ATTENTION!