Dynamical Freezing in a Spin Glass with Logarithmic Correlations - PowerPoint PPT Presentation

Setup Motivation Results Proof Idea Dynamical Freezing in a Spin Glass with Logarithmic Correlations Based on joint work with A. Cortines, J. Gold and A. Svejda Weizmann Institute of Science, 18/1/2018

Setup Motivation Results Proof Idea Outline Setup Motivation Results Proof Idea

Setup Motivation Results Proof Idea Setup Potential: � � , V N = ( 0 , N ) 2 ∩ Z 2 , where h N ≡ h N ( x ) : x ∈ V N h N ∼ Discrete Gaussian Free Field on V N (with 0 b.c.). � � ∞ � = ⇒ h N ∼ N ( 0 , G N ) , where G N ( x , y ) = E x n = 0 1 { S n = y , ∀ k ≤ n : S k ∈ V N } . • G N is the Green Function on V N : • C ov = − g log � y − x � � � h N ( X ) , h N ( y ) + O ( 1 ) . N � 2 = 2 g log � y − x � + O ( 1 ) . • E � h N ( x ) − h N ( y ) (In the bulk, g = 2 /π .) = ⇒ Logarithmic correlations.

Setup Motivation Results Proof Idea Setup - Cont’d Dynamics: � � X N = X N ( t ) : t ≥ 0 , where: X N is (conditionally on h N ) a Markov jump process on V N with transition rates: � e β [ a h N ( y ) − ( 1 − a ) h N ( x )] if x ∼ y , q N ( x , y ) := 0 otherwise . x ∼ y stands for nearest neighbors in the torus. β ≥ 0 is the interaction strength parameter (inverse temperature). a ∈ [ 0 , 1 ] is a symmetry parameter. Question Long and short time behavior of X N for large β ?

Setup Motivation Results Proof Idea Outline Setup Motivation Results Proof Idea

Setup Motivation Results Proof Idea Motivation: Glauber Dynamics for a Spin Glass Easy to check that for all a and β : e β h N ( x ) q N ( x , y ) = e β h N ( y ) q N ( y , x ) = e β a [ h N ( x )+ h N ( y )] = ⇒ X N is reversible with respect to � M N ,β � e β h N ( x ) δ x . M N ,β = M N ,β ( V N ) , M N ,β = x ∈ V N � M N ,β is a spin-glass Gibbs distribution with energy states: � � − h N ( x ) : x ∈ V N . = ⇒ X N is its Glauber dynamics.

Setup Motivation Results Proof Idea Limiting Gibbs Distribution There exists β c ∈ ( 0 , ∞ ) such that � M β is asymptotically discrete if and only if β > β c . Equivalently, iff β > β c , for large N most of the mass of � M N ,β is concentrated on (essentially) finitely many vertices (energy levels). “Known”: � ⇒ � M N ,β ( N · ) = M β as N → ∞ , where (formally): e β h ( x ) d x � � M β ( d x ) = V e β h ( y ) d y , h ∼ continous GFF on V . � M β is a normalized version of the Liouville Quantum Gravity Measure with respect to the continuous Gaussian free field on V := [ 0 , 1 ] 2 at parameter β . Only really proved for β > β c (Zindy-Arguin, Biskup, L.).

Setup Motivation Results Proof Idea Limiting Gibbs Distribution - Supercritical case What is actually proved is � M N ,β = ⇒ M β as N → ∞ , where  N 2 √ g β (log N ) − 3 √ g β/ 4 M N ,β ( d x ) β > β c ,   � E e β h N ( x ) � − 1 , � M N ,β ( d x ) = M N ,β ( d x ) β < β c ,  � E e β h N ( x ) � − 1 ,  (log N ) M N ,β ( d x ) β = β c . M β is the (“real”) LQGM on V . Indeed a.s. discrete iff β > β c . | M β | and M β = � When β > β c we have � M β M β = k τ k δ ξ k where � � �� � Z ( d x ) ⊗ κ β | Z | t − 1 − β c /β d t ( ξ k , τ k ) : k ≥ 1 atoms of χ β ∼ PPP . ! Z = | Z | � Z is some random measure on V , κ β ∈ ( 0 , ∞ ) . M β = � Equivalently � k ≥ 1 � τ k δ ξ k , where ( � τ k : k ≥ 1 ) are Poisson Dirichlet with parameter β/β c and ξ k ∼ � Z are independently i.i.d.

Setup Motivation Results Proof Idea Glassy Phase The regime β > β c is called the glassy (or frozen) phase. Conjectured to be a universal feature of (mean-field) spin-glass-type distributions at low temperature. “Static” features : Few and isolated dominant energy levels. Poisson Dirichlet statistics in the limit. Overlap distribution supported on { 0 , 1 } (1 RSB). · · · ”Dynamical” features - Dynamical Freezing: Metastability. Tunneling. Aging.

Setup Motivation Results Proof Idea Dynamical Freezing in other Spin-Glass models Previously studied models: The Random Energy Model. Underlying graph is H n = {− 1 , + 1 } n with N = 2 n . h N ( x ) are i.i.d. centered Gaussians with variance n . The Bouchaud Trap Model. Underlying graph is V N (or Z d for d ≥ 2). e h N ( x ) are i.i.d. stable random variables with index 1. The p-spin model. Underlying graph is H n (or the sphere S n − 1 in R n ). � � � p , p ≥ 2. � 1 h N ( x ) is Gaussian, with E h N ( x ) h N ( y ) = n n � x , y � (In our case correlations are logarithmic).

Setup Motivation Results Proof Idea Outline Setup Motivation Results Proof Idea

Setup Motivation Results Proof Idea In-equlibirium timescales From now on we take a = 0 (random hopping times dynamics). Let τ N = τ N ( β ) := N 2 √ g β (log N ) 1 − 3 √ g β/ 4 . ! Evolution rate of X N in equilibrium. Let V = V ∪{∞} with ∞ at finite distance from V = [ 0 , 1 ] 2 . Theorem (Cortines, Gold, L. ’17) � � Let β > β c . There exists a process Y β ≡ Y β ( t ) : t ≥ 0 taking values in V such that for any t > 0 , � � � � 1 N X N ( τ N t ) : t ∈ [ 0 , t ] = ⇒ Y β ( t ) : t ∈ [ 0 , t ] as N → ∞ , as random functions in L 1 � � [ 0 , t ] → V . Q: What is Y β ?

Setup Motivation Results Proof Idea The limiting process Ingredient 1 - The K process: Fix ( τ k ) k ≥ 1 be a deterministic decreasing sequence of “trap depths”. � � Let A k = ( A k ( u ) : u ≥ 0 ) k ≥ 1 be indep. rate 1 Poisson processes. The jumps of A k are denoted by σ k = ( σ k ( j ) : j = 0 , . . . ) . � Let E k ( j ) : k , j = 1 , . . . ) be i.i.d Exp ( 1 ) (indep of A k -s). The clock process is T : [ 0 , ∞ ) → [ 0 , ∞ ) , where: A k ( u ) ∞ � � T ( u ) := τ k E k ( j ) , k = 1 j = 1 The K -process is K : [ 0 , ∞ ) → N := N ∪{∞} , where: � � �� � � � k if t ∈ T σ k ( j ) − , T σ k ( j ) for some k , j , K ( t ) := ∞ otherwise.

Setup Motivation Results Proof Idea The limiting process Ingredient 2 - The trapping landscape: � � Recall that ( ξ k , τ k ) : k = 1 , . . . enumerate the atoms of �� � Z ( d z ) ⊗ κ β | Z | t − 1 − β c /β d t χ β ∼ PPP , M β = � k τ k δ ξ k / � (Limiting Gibbs dist. is � k τ k for β > β c ). Order ( ξ k , τ k ) k ≥ 1 so that τ 1 > τ 2 > . . . . Think of ( ξ k , τ k ) ∈ V × R + are position and depth of k -th deepest trap. Construction of Y β : � � Draw the trapping landscape: ( ξ k , τ k ) : k ≥ 1 . � � Draw a K -process K = K ( t ) : t ≥ 0 with trap depths ( τ k ) k ≥ 1 . The limiting process Y β : [ 0 , ∞ ) → V is � ξ K ( t ) if K ( t ) � = ∞ , Y β ( t ) := ∞ if K ( t ) = ∞ .

Setup Motivation Results Proof Idea Properties of the Limiting Process The K -process was introduced in this context by Fontes and Mathieu (2008) who studied dynamics for BTM on the complete graph. Named after Kolmogorov (1951) who studied a variant of K ( t ) . The evolution of Y β can be described as follows: When at trap k the process spends ∼ Exp ( τ − 1 ) time at position k ξ k . Then jumps to ∞ . At ∞ the process spends zero time (that is the exit time is a.s. 0). Then jumps to a new trap. = ⇒ ∞ is called an unstable state.. Traps are chosen such that following a visit to ∞ : - The hitting time of any finite subset of traps is finite a.s. - The entrance distribution to any such subset is uniform.

Setup Motivation Results Proof Idea Properties of the Limiting Process - Cont’d The set of times when the process is at ∞ has Leb. measure zero. = ⇒ ∞ is called a fictitious state.. On a (compactifed version of) N ∪ {∞} the process K is strongly Markovian and has a version with càdlàg paths. Y β can be seen as super-critical Liouville Brownian motion. Formally, � u � � T − 1 ( t ) e β h X ( s ) d s , Y β ( t ) = X , T ( u ) = s = 0 where ( X ( t ) : t ≥ 0 ) is a SBM on V and h is the CGFF on V . This has been constructed recently for β ≤ β c .

Setup Motivation Results Proof Idea The Asymptotic Picture For large N , the process X N : - Spends most ( ≍ τ N ) time trapped in clusters of diameter O ( 1 ) of dominant or meta-stable energy states of � M N ,β . - Spends negligible ( ≪ τ N ) time transitioning or tunneling between such clusters. - Next cluster to visit is chosen (effectively) uniformly.

Setup Motivation Results Proof Idea Pre-equilibrium Timescales What about pre-equilibrium timescales, that is when t ≪ τ N ( β ) ? Theorem (Cortines, Svejda, L. ’18+) Let β > β c (and a = 0 ). There exists 0 < γ 0 < γ 1 < ∞ and θ 0 > 0 such that �� �� � � � � � � � � 1 lim � X N ( 1 + θ ) t − X N t � < r = Asl β c /β . P 1 + θ N →∞ r →∞ � τ N / (log N ) γ 1 , τ N / (log N ) γ 0 � uniformly in t ∈ and θ ∈ ( 0 , θ 0 ] . Asl κ is the CDF of the generalized arcsine distribution (on [ 0 , 1 ] ): � u sin κπ v κ − 1 ( 1 − v ) − κ d v , Asl κ ( u ) = Asl κ ( 0 ) = 0 , Asl κ ( 1 ) = 1 . π 0 θ 0 , γ 0 can be chosen arbitrarily large, resp. small. True also conditionally on h N (in probability).