Universality of the REM for dynamics of mean-field spin glasses
aa r X i v : . [ m a t h . P R ] J un UNIVERSALITY OF THE REM FOR DYNAMICS OFMEAN-FIELD SPIN GLASSES
G´ERARD BEN AROUS, ANTON BOVIER, AND JI ˇR´I ˇCERN ´Y
Abstract.
We consider a version of a Glauber dynamics for a p -spin Sherrington–Kirkpatrick model of a spin glass that can be seen as a time change of simplerandom walk on the N -dimensional hypercube. We show that, for any p ≥ β >
0, there exist constants γ >
0, such that for allexponential time scales, exp( γN ), with γ ≤ γ , the properly rescaled clock process (time-change process), converges to an α -stable subordinator where α = γ/β < α -stable subordinator. In otherwords, up to rescaling, on these time scales (that are shorter than the equilibrationtime of the system), the dynamics of p -spin models ages in the same way as theREM, and by extension Bouchaud’s REM-like trap model, confirming the latteras a universal aging mechanism for a wide range of systems. The SK model (thecase p = 2) seems to belong to a different universality class. Introduction and results
Aging has become one of the main paradigms to describe the long-time behav-ior of complex and/or disordered systems. Systems that have strongly motivatedthis research are spin glasses , where aging was first observed experimentally in theanomalous relaxation patterns of the magnetization [LSNB83, Cha84]. The theo-retical modeling of aging phenomena took a major leap with the introduction ofso-called trap models by Bouchaud and Dean in the early 1990’ies [Bou92, BD95](see [BCKM98] for a review). These models reproduce the characteristic power lawbehavior seen experimentally while being sufficiently simple to allow for detailedanalytical treatment. While trap models are heuristically motivated to capture thebehavior of the dynamics of spin glass models, there is no clear theoretical, let alonemathematical derivation of these from an underlying spin-glass dynamics. The firstattempt to establish such a connection was made in [BBG02, BBG03a, BBG03b]where it was shown that starting from a particular Glauber dynamics of the Ran-dom Energy Model (REM), at low temperatures and at the time scale slightly shorterthan the equilibration time of the dynamics, the aging of the time-time correlationfunction of the dynamics converged to that given by Bouchaud’s REM-like trapmodel.On the other hand, in a series of papers [B ˇC05, B ˇCM06, B ˇC07a, B ˇC07b] a sys-tematic investigation of a variety of trap models was initiated. In this process, itemerged that there appears to be an almost universal aging mechanism based on α -stable subordinators that governs aging in most of the trap models. It was alsoshown that the same feature holds for the dynamics of the REM at shorter time Date : October 31, 2018.2000
Mathematics Subject Classification.
Key words and phrases. aging, universality, spin glasses, SK model, random walk. scales than those considered in [BBG03a, BBG03b], and that this also happens athigh temperature provided appropriate time scales are considered [B ˇC07a]. For ageneral review on trap models see [B ˇC06].In all models considered so far, however, the random variables describing thequenched disorder were considered to be independent, be it in the REM or in thetrap models. Aging in correlated spin glass models was investigated rigorously onlyin some cases of spherical SK models and at very short time scales [BDG01]. In thepresent paper we show for the first time that the same type of aging mechanism isrelevant also in correlated spin glasses, at least on time scales that are short comparedto equilibration time (but exponentially large in the volume of the system).Let us first describe the class of models we are considering. Our state spaces willbe the N -dimensional hypercube, S N ≡ {− , } N . R N : S N × S N → [ − ,
1] denotesas usual the normalized overlap, R N ( σ, τ ) ≡ N − P Ni =1 σ i τ i . The Hamiltonian of the p -spin SK-model is defined as √ N H N , where H N : S N → R is the centered normalprocess indexed by S N with covariance E [ H N ( σ ) H N ( τ )] = R N ( σ, τ ) p , (1.1)and p ∈ N , p >
2. We will denote by H the σ -algebra generated by the randomvariables H N ( σ ) , σ ∈ S N , N ∈ N . The corresponding Gibbs measure is then given by µ β,N ( σ ) ≡ Z − β,N e β √ NH N ( σ ) , (1.2)where Z β,N denotes the normalizing partition function.We define the classical trap-model dynamics as a nearest neighbor continuous timeMarkov chain σ N ( · ) on S N with transition rates w N ( σ, τ ) = ( N − e − β √ NH N ( σ ) , if dist( σ, τ ) = 1 , , otherwise; (1.3)here dist( · , · ) is the graph distance on the hypercube,dist( σ, τ ) = 12 N X i =1 | σ i − τ i | . (1.4)A simple way to construct this dynamics is as a time change of a simple randomwalk on S N : We denote by Y N ( k ) ∈ S N , k ∈ N , the simple unbiased random walk(SRW) on S N started at some fixed point of S N , say at { , . . . , } . For β > clock-process by S N ( k ) = k − X i =0 e i exp (cid:8) β √ N H N (cid:0) Y N ( i ) (cid:1)(cid:9) , (1.5)where { e i , i ∈ N } is a sequence of mean-one i.i.d. exponential random variables. Wedenote by Y the σ -algebra generated by the SRW random variables Y N ( k ), k ∈ N , N ∈ N . The σ -algebra generated by the random variables e i , i ∈ N will be denotedby E . Then the process σ N ( · ) can be written as σ N ( t ) ≡ Y N ( S − N ( t )) . (1.6)Obviously, σ N is reversible with respect to the measure µ β,N . We will consider allrandom processes to be defined on an abstract probability space (Ω , F , P ). Note thatthe three σ -algebras H , Y , and E are all independent under P . NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 3
We will systematically use the definition of the dynamics given by (1.3) or (1.6).This is the same as was used in the analysis of the REM and in most work on trapmodels. It differs substantially from more popular dynamics such as the Metropolisor the heat-bath algorithm. The main difference is that in these dynamics the tra-jectories are not independent of the environment and are biased against going up inenergy. This may have a substantial effect on the dynamics, and we do not knowwhether our results will apply also (with some modifications) in these cases. Thefact is that we currently do not have the tools to analyze these dynamics even in thecase of the REM!Let V α ( t ) be the α -stable subordinator with the Laplace transform given by E [ e − λV α ( t ) ] = exp( − tλ α ) . (1.7)The main technical result on the dynamics will be the following theorem that providesthe asymptotic behavior of the clock process. Theorem 1.1.
There exists a function ζ ( p ) such that for all p ≥ and γ satisfying < γ < min (cid:0) β , ζ ( p ) β (cid:1) , (1.8) under the conditional distribution P [ ·|Y ] the law of the stochastic process ¯ S N ( t ) = e − γN S N (cid:0)(cid:4) tN / e Nγ / β (cid:5)(cid:1) , t ≥ , (1.9) defined on the the space of c`adl`ag functions equipped with the Skorokhod M -topology,converges, Y -a.s., to the law of γ/β -stable subordinator V γ/β ( Kt ) , t ≥ , where K is a positive constant depending on γ , β and p .Moreover, the function ζ ( p ) is increasing and it satisfies ζ (3) ≃ . and lim p →∞ ζ ( p ) = p . (1.10)We will explain in Section 5 what the M -topology is. Roughly, it is a weaktopology that does not convey much information at the jumps of the limiting process:it can be the case that the approximating processes jumps several times at rathershort distances to produce one bigger jump of the limit process. This will actuallybe the case in our models for p < ∞ , while it is not the case in the REM. Thereforewe cannot replace the M topology with the stronger J -topology in Theorem 1.1.To control the behavior of spin-spin correlation functions that are commonly usedto characterize aging, we need to know more on how these jumps occur at finite N .What we will show, is that if we the slightly coarse-grain the process ¯ S N over blocksof size o ( N ), the rescaled process does converge in the J -topology. What this says,is that the jumps of the limiting process are compounded by smaller jumps thatare made over ≤ o ( N ) steps of the SRW. In other words, the jumps of the limitingprocess come from waiting times accumulated in one slightly extended trap, andduring this entire time only a negligible fraction of the spins are flipped. That willimply the following aging result. Theorem 1.2.
Let A εN ( t, s ) be the event defined by A εN ( t, s ) = { R N (cid:0) σ N (cid:0) te γN (cid:1) , σ N (cid:0) ( t + s ) e γN (cid:1)(cid:1) ≥ − ε (cid:9) . (1.11) Then, under the hypothesis of Theorem 1.1, for all ε ∈ (0 , , t > and s > , lim N →∞ P [ A εN ( t, s )] = sin αππ Z t/ ( t + s )0 u α − (1 − u ) − α d u. (1.12) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 4
Remark.
We will in fact prove the stronger statement that aging in the above senseoccurs along almost every random walk trajectory, that islim N →∞ P [ A εN ( t, s ) |Y ] = sin αππ Z t/ ( t + s )0 u α − (1 − u ) − α d u, Y -a.s. (1.13)Let us discuss the meaning of these results. e γN is the time-scale at which we wantto observe the process. According to Theorem 1.1, at this time the random walkwill make of the order of N / e Nγ / β ≪ e γN steps. Since this number is also muchsmaller than 2 N (as follows from (1.10)), the random walk will essentially visit thatnumber of sites.If the random process H N was i.i.d., then the maximum of H N along the trajectorywould be (cid:0) N / e Nγ / β ) (cid:1) / ∼ N / γ/β , and the time spent in that site wouldbe of order e γN . Since Theorem 1.1 holds also in the i.i.d. case, that is in the REM(see [B ˇC07a]), the time spent in the maximum is comparable to the total time andthe convergence to the α -stable subordinator implies that the total accumulated timeis composed of pieces of order e γN that are collected along the trajectory. In fact,each jump of the subordinator corresponds to one visit to a site that has waitingtimes of that order. In a common metaphor, the sites are referred to as traps andthe mean waiting times as their depths.The theorem in the general case states that in the p -spin model, the same isessentially true. The difference will be that the traps here will not consist of asingle site, but consist of a deep valley (along the trajectory) whose bottom that hasapproximately the same energy as in the i.i.d. case and whose shape and width wewill be able to describe quite precisely. Remarkably, the number of sites contributingsignificantly to the residence time in the valley is essentially finite, and differentvalleys are statistically independent.The fact that traps are finite may appear quite surprising to those familiar withthe statics of p -spin models. From the results there (see [Tal03, Bov06]), it is knownthat the Gibbs measure concentrates on “lumps” whose diameter is of order N ǫ p ,with ǫ p >
0. The mystery is however solved easily: the process H N ( σ ) does indeeddecreases essentially linearly with speed N − / from a local maximum. Thus, theresidence times in such sites decrease geometrically, so that the contributions of aneighborhood of size K of a local maximum amounts to a fraction of (1 − c − K ) ofthe total time spend in that valley ; for the support of the Gibbs measure, one needshowever to take into account the entropy, that is that the volumes of the balls ofradius r increases like N r . For the dynamics, at least at our time-scales, this is,however, irrelevant, since the SRW leaves a local minimum essentially ballistically.The proof of Theorem 1.1 relies on the combination of detailed information on theproperties of simple random walk on the hypercube, which is provided in Section 4(but see also [Mat89, BG06, ˇCG06]), and comparison of the process H N on thetrajectory of the SRW to a simpler Gaussian process using interpolation techniques`a la Slepian, familiar from extreme value theory of Gaussian processes.Let us explain this in more detail. On the time scales we are considering, the SRWmakes tN / exp( N γ / β ) ≪ tN / exp( N ζ ( p ) / ≪ N steps. In this regime theSRW is extremely “transient”, in the sense that (i) starting from a given point x , fora times t ≤ ν ∼ N ω , ω <
1, the distance from x grows essentially linearly with speedone, that is there are no backtrackings with high probability; (ii) the SRW will never return to a neighborhood of size ν of the starting point x , with high probability. The NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 5 upshot is that we can think of the trajectory of the SRW essentially as of a straightline.Next we consider the Gaussian process restricted to the SRW trajectory. We expectthat the main contributions to the sums S N ( k ) come from places where Y N is maximal(on the trajectory). We expect that the distribution of these extremes do not feel thecorrelation between points farther than ν apart. On the other hand, for points closerthan ν , the correlation function R N ( Y N ( i ) , Y N ( j )) p can be well approximated by alinear function 1 − p | i − j | /N (using that R N ( Y N ( i ) , Y N ( j )) ∼ − | i − j | /N ). This isconvenient since this process has an explicit representation in terms of i.i.d. randomvariables that allow for explicit computations (in fact, this is one of the famousSlepian processes for which the extremal distribution can be computed explicitly[Sle61, She71]). Thus the idea is to cut the SRW trajectory into blocks of length ν and to replace the original process H N ( Y N ( i )) by a new one U i , where U i and U j are independent, if i, j are not in the same block, and E [ U i U j ] = 1 − p | i − j | /N if they are. For the new process, Theorem 1.1 is relatively straightforward. Themain step is the computation of Laplace transforms in Section 2. Comparing thereal process with the auxiliary one is the bulk of the work and is done in Section 3.The properties of SRW needed are established in Section 4. In Section 5 we presentthe proofs of the main theorems.Our results here show some universality of the REM for dynamics of p -spin modelswith p ≥
3. This dynamic universality is close to the static universality of theREM, which shows that various features of the landscape of energies (that is ofthe Hamiltonian H N ) are insensitive to correlations. This static universality in amicrocanonical context has been introduced by [BM04] (see [BK06a, BK06b] forrigorous results on spin-glasses). The static results closest to our dynamics questionare given in [BGK06, BK07] where it is shown that the statistics of extreme valuesfor the restriction of H N to a random sets X N ⊂ S N are universal, for p ≥ | X N | = e cN , for c small enough.This work was initiated during a concentration period on metastability and agingat the Max-Planck Institute for Mathematics in the Sciences in Leipzig. GBA andAB thank the MIP-MIS and Stefan M¨uller for kind hospitality during this event.AB’s research is supported in part by DFG in the Dutch-German Bilateral ResearchGroup “Mathematics of Random Spatial Models from Physics and Biology”.2. Behavior the one-block sums
In this section we analyze the distribution of the block-sums P νi =1 e i e β √ NU i , where e i are mean-one i.i.d. exponential random variables, and { U i , i = 1 , . . . , ν } is a cen-tered Gaussian process with the covariance E U i U j = 1 − p | i − j | /N ; ν = ν N is afunction of N of the form ν = ⌊ N ω ⌋ , with ω ∈ (1 / , . (2.1)As explained in the introduction, this process will serve as a local approximation ofthe corresponding block sums along a SRW trajectory. We characterize the distri-bution of the block-sums in terms of its Laplace transform F N ( u ) = E h exp n − ue − γN ν X i =1 e i e β √ NU i oi . (2.2) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 6
Proposition 2.1.
For all γ such that γ/β ∈ (0 , there exists a constant, K = K ( γ, β, ω, p ) , such that, uniformly for u in compact subsets of [0 , ∞ ) , lim N →∞ N / ν − e Nγ / β [1 − F N ( u )] = Ku γ/β . (2.3) Proof.
We first compute the conditional expectation in (2.2) given the σ -algebra, U ,generated by the Gaussian process U , E h exp n − ue − γN ν X i =1 e i e β √ NU i o(cid:12)(cid:12)(cid:12) U i = ν Y i =1
11 + ue − γN e β √ NU i = exp (cid:26) − ν X i =1 g (cid:16) ue − γN e β √ NU i (cid:17)(cid:27) , (2.4)where g ( x ) ≡ ln(1 + x ) . (2.5)Note that importantly, g ( x ) is monotone increasing and non-negative for x ∈ R + .We use the well-known fact (see e.g. [Sle61]) that the random variables U i can beexpressed using a sequence of i.i.d. standard normal variables, Z i , as follows. Set Z = ( U + U ν ) / (4 − p ( ν − /N ) / and Z k = ( U k − U k − ) / (4 p/N ) / , k = 2 , . . . , ν .Then Z i are i.i.d. standard normal and U i = Γ Z + · · · + Γ i Z i − Γ i +1 Z i +1 − Γ ν Z ν , (2.6)where Γ = r − pN ( ν −
1) and Γ = · · · = Γ ν = r pN . (2.7)Observe that P νi =1 Γ i = 1. Let us define G i ( z ) = G i ( z , . . . , z ν ) as G i ( z ) = Γ z + · · · + Γ i z i − Γ i +1 z i +1 − · · · − Γ ν z ν . (2.8)Using this notation we get1 − F N ( u ) = Z R ν d z (2 π ) ν/ e − P νi =1 z i n − exp h − ν X i =1 g (cid:16) ue − γN e β √ NG i ( z ) (cid:17) io . (2.9)We divide the domain of integration into several parts according to which of the G i ( z ) is maximal. Define D k = { z : G k ( z ) ≥ G i ( z ) ∀ i = k } . On D k we use thesubstitution z i = b i + Γ i ( γN − log u ) / ( β √ N ) , if i ≤ k , z i = b i − Γ i ( γN − log u ) / ( β √ N ) , if i > k . (2.10)It will be useful to define P kj = i +1 a j as P kj =1 a j − P ij =1 a j , which is meaningful alsofor k < i + 1. Using this definition G k ( b ) − G i ( b ) = 2 k X j = i +1 Γ ν b j . (2.11)Set θ = − log( u ) / ( γN ) and define D ′ k = n b : k X j = i +1 b j + γ √ pβ | k − i | (1 + θ ) ≥ ∀ i = k o . (2.12) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 7
After a straightforward computation we find that (2.9) equals e − Nγ / β u γ/β ν X k =1 Z D ′ k d b (2 π ) ν/ e − P νi =1 b i e − γβ √ NG k ( b )(1+ θ ) × n − exp (cid:16) − ν X i =1 g (cid:16) e β √ NG k ( b ) − β √ p P kj = i +1 b j − pγ | k − i | (1+ θ ) (cid:17) (cid:17)o . (2.13)To finish the proof we have to show that u γ/β is asymptotically the only dependenceof (2.13) on u (or on θ ) and that the sum is of order νN − / . We change variablesonce more to a j = b j / (1 + θ ) in order to remove the dependence of the integrationdomains on u . Then the sum (without the prefactor) in (2.13) can be expressed as ν X k =1 Z D ′′ k (1 + θ ) ν d a (2 π ) ν/ e − (1+ θ ) P νi =1 a i (cid:20) e − γβ √ NG k ( a )(1+ θ ) × n − exp (cid:16) − ν X i =1 g (cid:16) e ( β √ NG k ( a ) − β √ p P kj = i +1 a j − pγ | k − i | )(1+ θ ) (cid:17) (cid:17)o(cid:21) , (2.14)where D ′′ k = (cid:8) a : P kj = i +1 a j + γ √ pβ | k − i | ≥ ∀ i = k (cid:9) . Let δ > δ ) γ/β <
1, and let
N > log( u ) / ( γδ ), so that | θ | ≤ δ .We first examine the bracket in the above expression for a fixed k . On D ′′ k exp n − ν X i =1 g (cid:0) e ( β √ NG k ( a ) − β √ p P kj = i +1 a j − pγ | k − i | )(1+ θ ) (cid:1)o ≥ exp (cid:8) − νg (cid:0) e β √ NG k ( a )(1+ θ ) (cid:1)(cid:9) . (2.15)Write G k ( a ) as (recall (2.1)) G k ( a ) = ξ − ω log N (1 + θ ) β √ N . (2.16)The bracket of (2.14) is then smaller than e − γβ ( ξ − ω log N )(1+ θ ) (cid:8) − exp (cid:0) − νg (cid:0) e ξ − ω log N (cid:1)(cid:1)(cid:9) = N γω (1+ θ ) β e − γξβ (1+ θ ) (cid:8) − exp (cid:0) − νg (cid:0) e ξ /ν (cid:1)(cid:1)(cid:9) . (2.17)The function e − γξβ (1+ θ ) (cid:8) − exp (cid:0) − νg (cid:0) e ξ /ν (cid:1)(cid:1)(cid:9) is bounded for ξ ∈ R , uniformly in ν , if (1 + θ ) γ/β <
1. Namely, if ξ ≥ e − γξβ (1+ θ ) (cid:8) − exp (cid:0) − νg (cid:0) e ξ /ν (cid:1) (cid:1)(cid:9) ≤ e − γξβ (1+ θ ) ≤ . (2.18)If ξ <
0, then, since g ( x ) ≤ x , (cid:8) − exp (cid:0) − νg (cid:0) e ξ /ν (cid:1)(cid:1)(cid:9) ≤ (cid:8) − exp (cid:0) − e ξ (cid:1)(cid:9) , (2.19)which behaves like e ξ , as ξ → −∞ . This compensates the exponentially growingprefactor, if (1 + θ ) γ/β <
1. Thus, under this condition, the bracket of (2.14)increases at most polynomially with N .In view of this at most polynomial increase, there exist δ > a i ’s satisfying ν − ν X i =1 a i ∈ (1 − δ, δ ) , | a | ≤ N / , ν X i =1 | a i | ≤ ν δ . (2.20) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 8
The integral over the remaining a i ’s decays at least as e − N δ ′ for some δ ′ > a satisfying (2.20), | G k ( a ) | ≤ N / + N − / ν δ ′ ≪ N / and thus, for any fixed u , uniformly in a , e − γβ √ NG k ( a )(1+ θ ) e − γβ √ NG k ( a ) N →∞ −−−→ , and e − (1+ θ ) P νi =1 a i e − P νi =1 a i N →∞ −−−→ . (2.21)Also, (1 + θ ) ν N →∞ −−−→
1. Hence, up to a small error, we can remove all but the lastoccurrence of θ in (2.14).Finally, taking x i = a i for i ≥ x = N / G k ( a ), and thus a = x − p ( x + · · · + x k − x k +1 − · · · − x ν )Γ √ N , (2.22)(2.14) equals, up to a small error, ν X k =1 Z D ′′ k d x e − P νi =2 x i Γ N / (2 π ) ν/ exp (cid:16) − γβ x − x N (cid:17) exp (cid:16) − a x N (cid:17) × n − exp (cid:16) − ν X i =1 g (cid:16) e (1+ θ ) βx e − ( β √ p P kj = i +1 x j − pγ | k − i | ) (1+ θ ) (cid:17)(cid:17)o . (2.23)The last exponential term on the first line can be omitted. Indeed, − a x N = 4Γ N (cid:2) px ( x + · · · − x ν ) − p ( x + · · · − x ν ) (cid:3) N →∞ −−−→ | x | ≤ N (1+ δ ) / and | x + · · · − x ν | ≤ ν (1+ δ ) / , if δ > x is again at most e − N δ ′ .Now we estimate the integral over x , . . . , x ν , Z ¯ D ′′ k d xe − P νi =2 x i (2 π ) ( ν − / exp (cid:16) − ν X i =1 g (cid:16) e (1+ θ ) βx e − ( β √ p P kj = i +1 x j +2 pγ | k − i | ) (1+ θ ) (cid:17)(cid:17) , (2.25)where ¯ D ′′ k is the restriction of D ′′ k to the last ν − V = ( V , . . . , V ν ) be a sequence of i.i.d. standardnormal random variables. Then, (2.25) equals P [ V ∈ ¯ D ′′ k ] E h exp (cid:16) − ν X i =1 g (cid:16) e (1+ θ ) βx e − ( β √ p P kj = i +1 V j +2 pγ | k − i | ) (1+ θ ) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) V ∈ ¯ D ′′ k i . (2.26)The probability P [ V ∈ ¯ D ′′ k ] is bounded from below by the probability that the two-sided random walk, R i = P ij =0 V j , i ∈ Z , with standard normal increments is largerthan − γ √ p | i | /β for all i . This probability is positive and does not depend on N ,which implies that, for all k , 1 > P [ V ∈ ¯ D ′′ k ] ≥ c > . (2.27)The expectation in (2.26) is bounded by one, since the functions g is positive onthe domain of integration. Moreover, as x → −∞ , the argument of g in (2.26) tends NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 9 to zero (since the first exponential does, and the second is bounded by one on D ′′ k ).Hence g (cid:16) e (1+ θ ) βx e − (2 β √ p P kj = i +1 V j +2 pγ | k − i | )(1+ θ ) (cid:17) ∼ e (1+ θ ) βx e − (2 β √ P kj = i +1 V j +2 pγ | k − i | )(1+ θ ) . (2.28)Therefore, as x i → −∞ , E h exp (cid:16) − ν X i =1 g (cid:16) e (1+ θ ) βx e − ( β √ p P kj = i +1 V j +2 pγ | k − i | ) (1+ θ ) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) V ∈ ¯ D ′′ k i ∼ − e (1+ θ ) βx E h ν X i =1 e − ( β √ p P kj = i +1 V j +2 pγ | k − i | ) (1+ θ ) (cid:12)(cid:12)(cid:12) V ∈ D ′′ k i = 1 − e (1+ θ ) βx ν X i =1 E h e − ( β √ pR k − i +2 pγ | k − i | ) (1+ θ ) (cid:12)(cid:12)(cid:12) R k − i ≥ − γ √ pβ | k − i | i . (2.29)Since R i is a centered normal random variable with variance | i | , a straightforwardGaussian calculation implies that E h e − ( β √ pR k − i +2 pγ | k − i | ) (1+ θ ) (cid:12)(cid:12)(cid:12) R k − i ≥ − γ √ pβ | k − i | i ∼ C β,γ,p p | k − i | e − γ p | k − i | / (2 β ) . (2.30)Hence, (2.29) is essentially a summation of a geometrical sequence and thereforethere exists constants c , c independent of k , such that1 − c e (1+ θ ) βx ≤ (2.29) ≤ − c e (1+ θ ) βx , ∀ x < . (2.31)Bounds (2.27) and (2.31) imply that (2.25) is bounded from above and from below(with different constants) by CN − / exp (cid:16) − γβ x − x N (cid:17) (1 ∧ ce (1+ θ ) βx ) . (2.32)and hence (2.23) is bounded from above and below by CνN − / Z R d x exp (cid:16) − γβ x − x N (cid:17) (1 ∧ ce (1+ θ ) βx ) = CνN − / . (2.33)Moreover, (2.25) is decreasing as function of min( k, ν − k ). As this minimum tendsto infinity, (2.25) behaves as f ( x ) N − / which is of course satisfy the bound (2.32).Due to this convergence, the constants in the lower and the upper bound of (2.33)can be made arbitrarily close. This completes the proof of Proposition 2.1. (cid:3) We close this section with a short description of the shape of the valleys mentionedin the introduction. First, it follows from (2.10) and the following computations thatthe most important contribution to the Laplace transform comes from realizationsfor which max { U i : 1 ≤ i ≤ ν } ∼ γ √ N /β with an error of order N − / . It is the“geometrical” sequence in (2.29) which shows that only finitely many neighbors ofthe maximum actually contribute to the Laplace transform. The same can be seen,at least heuristically, from a simple calculation E h U k + i (cid:12)(cid:12)(cid:12) U k = γβ √ N i = γ √ Nβ − C β,γ,p | i |√ N . (2.34)
NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 10
Which means that, disregarding the fluctuations, the energy decreases linearly withthe distance from the local maximum and thus the mean waiting times decreaseexponentially.3.
Comparison of the real and the block process
We now come to the main task, the comparison of the clock-process sums withthose in which the real Gaussian process is replaced by a simplified process. For agiven realization, Y N , of the SRW, we set X N ( i ) = H N (cid:0) Y N ( i ) (cid:1) (the dependence on Y N will be suppressed in the notation). Then X N ( i ) is a centered Gaussian processindexed by N with covariance matrixΛ ij = E [ X N ( i ) X N ( j )] = R N (cid:0) Y N ( i ) , Y N ( j ) (cid:1) p . (3.1)Now we define the comparison process, X N ( i ), as the centered Gaussian process withthe covariance matrixΛ ij = E [ X N ( i ) X N ( j )] = ( − p | i − j | /N, if ⌊ i/ν ⌋ = ⌊ j/ν ⌋ ,0 , otherwise. (3.2)For h ∈ [0 ,
1] we define the interpolating process X hN ( i ) ≡ √ − hX N ( i ) + √ hX N ( i ).Let ℓ ∈ N , 0 = t < · · · < t ℓ = T and u , . . . , u ℓ ∈ R + be fixed. For any Gaussianprocess X we define a function F N ( X ) = F N (cid:0) X ; { t i } , { u i } (cid:1) as F N (cid:0) X ; { t i } , { u i } (cid:1) ≡ E h exp (cid:16) − ℓ X k =1 u k e γN t k r ( N ) X i = t k − r ( N )+1 e i e β √ NX ( i ) (cid:17)(cid:12)(cid:12)(cid:12) X i ( X )= exp (cid:16) − ℓ X k =1 t k r ( N ) − X i = t k − r ( N ) g (cid:16) u k e γN e β √ NX ( i ) (cid:17)(cid:17) , (3.3)where r ( N ) = N / e Nγ / β . Observe that E [ F ( X ; t, u ) |Y ] is a joint Laplace trans-form of the distribution of the properly rescaled clock process at times t i . Thefollowing approximation is the crucial step of the proof. Proposition 3.1.
If the assumptions of Theorem 1.1 are satisfied, then for all se-quences { t i } and { u i } , lim N →∞ E (cid:2) F N (cid:0) X N ; { t i } , { u i } (cid:1)(cid:12)(cid:12) Y (cid:3) − E (cid:2) F N (cid:0) X N ; { t i } , { u i } (cid:1)(cid:3) = 0 , Y -a.s. (3.4) Proof.
We use the well-known interpolation formula for functionals of two Gaussianprocesses due (probably) to Slepian and Kahane (see e.g. [LT91] E [ F N ( X N ) − F N ( X N ) |Y ] = 12 Z d h tr ( N ) X i,j =1 i = j (Λ ij − Λ ij ) E h ∂ F N ( X hN ) ∂X ( i ) ∂X ( j ) (cid:12)(cid:12)(cid:12) Y i . (3.5)We will show that the integral in (3.5) converges to 0. NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 11
Let k ( i ) be defined by t k ( i ) − r ( N ) < i ≤ t k ( i ) r ( N ). The second derivative in (3.5)is equal to u k ( i ) u k ( j ) β Ne γN e β √ N ( X hN ( i )+ X hN ( j )) g ′ (cid:16) u k ( i ) e γN e β √ NX hN ( i ) (cid:17) g ′ (cid:16) u k ( j ) e γN e β √ NX hN ( j ) (cid:17) F N ( X hN ) ≤ u k ( i ) u k ( j ) β Ne γN e β √ N ( X hN ( i )+ X hN ( j )) × exp h − g (cid:16) u k ( i ) e γN e β √ NX hN ( i ) (cid:17) − g (cid:16) u k ( j ) e γN e β √ NX hN ( j ) (cid:17)i , (3.6)where we used that g ′ ( x ) = (1 + x ) − = exp( − g ( x )) (recall (2.5)), and we omitted inthe summation of F N ( X hN ) all terms different from i and j . To estimate the expectedvalue of this expression we need the following technical lemma. Lemma 3.2.
Let c ∈ [ − , and let U , U be two standard normal variables withthe covariance E [ U U ] = c and λ a small constant, < λ < − γ/β (which willstay fixed). Define Ξ N ( c ) = Ξ N ( c, β, γ, u, v ) and ¯Ξ N ( c ) = ¯Ξ N ( c, β, γ, u, v, λ ) by Ξ N ( c ) = uvβ Ne γN E h exp n β √ N ( U + U ) − g (cid:0) ue β √ NU − γN (cid:1) − g (cid:0) ve β √ NU − γN (cid:1)oi (3.7) and ¯Ξ N ( c ) = ( C ( γ,β,u,v,λ )(1 − c ) / exp n − γ Nβ (1+ c ) o , if c > ( γ/β ) + λ − , C ′ ( γ, β, u, v ) N exp (cid:8) N ( β (1 + c ) − γ ) (cid:9) , if c ≤ ( γ/β ) + λ − , (3.8) where C ( γ, β, u, v, λ ) and C ′ ( γ, β, u, v ) are suitably chosen constants, independent of N and c . Then Ξ N ( c ) ≤ ¯Ξ N ( c ) . (3.9) Proof.
Define κ ± = p ± c ). Let ¯ U , ¯ U be two independent standard normalvariables. Then U and U can be written as U = 12 ( κ + ¯ U + κ − ¯ U ) , U = 12 ( κ + ¯ U − κ − ¯ U ) . (3.10)Hence, U + U = κ + ¯ U . Using g ( x ) + g ( y ) = g ( x + y + xy ) ≥ g ( x + y ) and ue x + ve − x ≥ min( u, v ) e | x | , we get g (cid:0) ue β √ NU − γN (cid:1) + g (cid:0) ve β √ NU − γN (cid:1) ≥ g (cid:16) min( u, v ) exp (cid:16) κ + β √ N ¯ U (cid:12)(cid:12)(cid:12) κ − β √ N ¯ U (cid:12)(cid:12)(cid:12) − γN (cid:17)(cid:17) . (3.11)Denoting min( u, v ) by ¯ u , we find that Ξ N ( c ) is bounded from above by uvβ Ne γN Z R d y π exp n − y + y β √ N κ + y − g (cid:0) ¯ ue κ + β √ Ny / κ − β √ N | y | / − γN (cid:1)o . (3.12)Substituting z = y − β √ N κ + , z = y we get uvβ Ne γN e β κ N/ Z R d z π exp (cid:16) − z + z (cid:17) × exp (cid:16) − g (cid:16) ¯ u exp n √ N h(cid:16) β κ − γ (cid:17) √ N + βκ + z + βκ − | z | io(cid:17)(cid:17) . (3.13) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 12
The function exp( − g (¯ ue √ Nx )) converges to the indicator function x< , as N → ∞ .The rˆole of x will be played by the bracket in the expression (3.13).If this bracket remains negative for z close to zero, that is if γ ≥ − λ ′ + β κ / c ≤ ( γ/β ) + λ − c :Ξ N ( c ) ≤ uvβ Ne γN e β κ N/ = C ′ ( γ, β, u, v ) N exp (cid:8) N ( β (1+ c ) − γ ) (cid:9) = ¯Ξ N ( c ) . (3.14)If this is not the case, that is γ < − λ ′ + β κ /
2, then we need another substitution, z = 1 √ N h v − κ − κ + | v | − N (cid:16) βκ + − γβκ + (cid:17)i ,z = v √ N . (3.15)This substitution transforms the domain where the bracket of (3.13) is negativeinto the half-plain v <
0: The expression inside of the braces in (3.13) equals βκ + v /
2. Substituting (3.15) into ( z + z ) / (cid:0) − ( β κ − γ ) N β κ (cid:1) . Another prefactor N − comes from the Jacobian.The remaining terms can be bounded from above by Z R d v π exp n(cid:16) βκ + − γβκ + (cid:17)(cid:16) v − κ − κ + | v | (cid:17) − g (¯ ue βκ + / ) o , (3.16)which can be separated into a product of two integrals. The integration over v givesa factor (cid:16)(cid:16) βκ + − γβκ + (cid:17) κ − κ + (cid:17) − ≤ C ( λ ) κ − − ≤ C ( λ )(1 − c ) − / . (3.17)Using properties of g , the integrand of (3.16) behaves as exp {− v γ/βκ + } as v →∞ , and as exp { ( βκ + − (2 γ/βκ + )) v } as v → −∞ . Therefore, the integral over v isbounded uniformly by some λ -dependent constant for all values of c ≥ − γ/β ) + λ . Putting everything togetherΞ N ( c ) ≤ C (1 − c ) − / uvβ Ne γN e β κ N/ N exp (cid:16) − ( β κ − γ ) N β κ (cid:17) = C ( γ, β, u, v, λ )(1 − c ) − / exp n − γ Nβ (1 + c ) o = ¯Ξ N ( c ) . (3.18)This finishes the proof of Lemma 3.2. (cid:3) Let k d k = min( d, N − d ) and D ij = dist( Y N ( i ) , Y N ( j )). Define, with a slight abuseof notation, Λ d = (1 − dN − ) p . That is Λ d is the covariance of X N ( i ) and X N ( j )if D ij = d . The next proposition, which will be proved in Section 4, will be used tocontrol the correlations of the process X N . Proposition 3.3.
Let γ and β satisfy the hypothesis of Theorem 1.1, and let ν be asin (2.1) . Then, for any η > , there exists a constant, C = C ( β, γ, ν, η ) , such that, Y -a.s. for N large enough, for all d ∈ { , . . . , N } tr ( N ) X i,j =1 ⌊ i/ν ⌋6 = ⌊ j/ν ⌋ { D ij = d } ≤ C (cid:20) t r ( N ) − N (cid:18) Nd (cid:19) + tr ( N ) ν − e η k d k (cid:21) , (3.19) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 13 tr ( N ) X i,j =1 ,i = j ⌊ i/ν ⌋ = ⌊ j/ν ⌋ { D ij = d } (Λ d − Λ ij ) ≤ Cd tr ( N ) N { d ≤ ν } . (3.20)We now conclude the proof of Proposition 3.1, that is we prove that the right-handside of (3.5) tends to 0. Observe first that D ij is smaller than | i − j | . Hence, for ⌊ i/ν ⌋ = ⌊ j/ν ⌋ Λ ij = (cid:2) − N − D ij (cid:3) p ≥ [1 − N − | i − j | ] p ≥ Λ ij . (3.21)Since Λ ij = 0 for ( i, j ) with ⌊ i/ν ⌋ 6 = ⌊ j/ν ⌋ , Λ ij − Λ ij < ij < ij − Λ ij ) + E h ∂ F N ( X hN ) ∂X ( i ) ∂X ( j ) (cid:12)(cid:12)(cid:12) Y i − (Λ ij ) − E h ∂ F N ( X hN ) ∂X ( i ) ∂X ( j ) (cid:12)(cid:12)(cid:12) Y i . (3.22)We bound this expression using Lemma 3.2. For given { u i } let˜Ξ N ( c ) = max { ¯Ξ N ( c, β, γ, u i , u j ) : 1 ≤ i, j ≤ ℓ } . (3.23)Then ˜Ξ N ( c ) satisfies (3.8) for some constants C and C ′ and it is therefore increasingin c . The absolute value of the right-hand side of (3.5) is then bounded from aboveby tr ( N ) X i,j =1 i = j (Λ ij − Λ ij ) + E h ∂ F N ( X N ) ∂X ( i ) ∂X ( j ) (cid:12)(cid:12)(cid:12) Y N i + tr ( N ) X i,j =1 i = j (Λ ij ) − E h ∂ F N ( X N ) ∂X ( i ) ∂X ( j ) i ≤ N X d =0 ( tr ( N ) X i,j =1 ⌊ i/ν ⌋6 = ⌊ j/ν ⌋ { D ij = d } (Λ d ) + Z ˜Ξ( h Λ d )d h + tr ( N ) X i,j =1 ,i = j ⌊ i/ν ⌋ = ⌊ j/ν ⌋ { D ij = d } (Λ d − Λ ij )˜Ξ (cid:0) Λ d (cid:1) + tr ( N ) X i,j : | i − j |≥ N/ { D ij = d } (Λ d ) − ˜Ξ (cid:0) (cid:1)) . (3.24)From the definition of ˜Ξ it follows that, Z ˜Ξ( hc )d h ≤ C exp n − γ Nβ (1 + c ) o Z (1 − hc ) − / d h. (3.25)The last integral can be easily evaluated and is smaller than 2 for all c ∈ [ − , C N X d =0 t r ( N ) − N (cid:18) Nd (cid:19) Λ d exp n − γ Nβ (1 + Λ d ) o (3.26)and C N X d =0 tr ( N ) e η k d k ν Λ d exp n − γ Nβ (1 + Λ d ) o . (3.27) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 14
The second line of (3.24) is bounded by C ν X d =0 tr ( N ) d N ˜Ξ(Λ d ) . (3.28)The third line is non-zero only if p is odd, and in that case it is bounded by N X d = N/ C (cid:20) t r ( N ) − N (cid:18) Nd (cid:19) + tr ( N ) ν − e η k d k (cid:21)(cid:16) dN − (cid:17) p ˜Ξ(0) , (3.29)We estimate (3.26) first. Let I ( u ) be defined by I ( u ) = u log u + (1 − u ) log(1 − u ) + log 2 , (3.30)and let J N ( u ) = 2 − N (cid:18) N ⌊ N u ⌋ (cid:19)r πN e NI ( u ) . (3.31)Stirling’s formula yields J N ( u ) N →∞ −−−→ (4 u (1 − u )) − uniformly in u on compact subsetsof (0 , J N ( u ) ≤ CN / for all u ∈ [0 , r ( N ) and˜Ξ, we find that(3.26) = C N X d =0 t N / (cid:16) − dN (cid:17) p exp n N Υ p,β,γ (cid:16) dN (cid:17)o J N (cid:16) dN (cid:17) , (3.32)whereΥ p,β,γ ( u ) = ( γ β − I ( u ) − γ β (1+(1 − u ) p ) , if (1 − u ) p ≥ γβ + λ − γ β − I ( u ) + β (1 + (1 − u ) p ) − γ, if (1 − u ) p ≤ γβ + λ − Lemma 3.4.
There exists a function ζ ( p ) such that for all p ≥ , and γ , β satisfying γ ≤ ζ ( p ) β and γ < β , there exist positive constants δ , δ ′ and c such that Υ p,β,γ ( u ) ≤ − δ for all u ∈ [0 , \ (1 / − δ ′ , / δ ′ ) , (3.34) and Υ p,β,γ ( u ) ≤ − c ( u − / for all u ∈ (1 / − δ ′ , / δ ′ ) . (3.35) Moreover ζ ( p ) is increasing and satisfies (1.10) , that is ζ (2) = 2 − / , ζ (3) = 1 . , and lim p →∞ ζ ( p ) = p . (3.36) Proof.
Since γ/β <
1, the second line of the definition of Υ p,β,γ is used only for p odd and u ≥ u c ( p, β, γ, λ ) = (1 + (1 − λ − γ/β ) /p ) / > /
2. Furthermore,Υ p,β,γ (1 /
2) = Υ ′ p,β,γ (1 /
2) = 0 andΥ ′′ p,β,γ (1 /
2) = ( (cid:0) γ β − (cid:1) , if p = 2, − . (3.37)The second derivative is always negative for β , γ , p satisfying the assumptions ofTheorem 1.1. Therefore (3.35) holds.The second line of the definition of Υ p,β,γ ( u ) is decreasing in u . Hence for u ≥ u c Υ p,β,γ ( u ) ≤ Υ p,β,γ ( u c ) = − γ (1 − γ/β ) − I ( u c ) (3.38)which is obviously strictly negative and (3.34) is proved for u ≥ u c . NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 15 - - - - U - - - - U - - - - U Figure 1.
Function Υ p,γ,β for p = 2, 3, 4 and various values of γ/β .For any δ ′ > u < / − δ ′ the function I ( u ) is strictly positive, and thefunction Φ( u ) ≡ − / (1 + (1 − u ) p ) is bounded. Therefore, if γ/β is sufficientlysmall, then Υ p,β,γ ( u ) < − δ . If p is even, the function Υ p,β,γ is symmetric around u = 1 /
2. If 1 / < u < u c ( p, β, γ ) and p is odd, thenΥ p,β,γ ( u ) < Υ p, , ( u ) = − I ( u ) < γ ≤ − / β . However, Υ ,β,γ ( u ) is increasing in γ /β and I ( u ) ≥ (1 − u ) /
2. Thus, for γ ≤ − / β , Υ ,β,γ ( u ) ≤ (cid:16) −
11 + (1 − u ) (cid:17) −
12 (1 − u ) . (3.40)The right-hand side of the last inequality is equal 0 for u = 1 / − u ) (cid:16) − − u ) ) (cid:17) > u < / . (3.41)The symmetry of Υ ,β,γ around 1 / /
2, Φ ′ (0) = − p , I (0) = log 2 and I ′ (0) = −∞ . Hence,for γ/β = √ log 2 there exists u small such that Υ p,β,γ ( u ) is positive. This implies ζ ( p ) < √ u ∈ (0 , /
2) then lim p →∞ Φ( u ) = 0. This yield the second half of(3.36).For illustration you find the graphs of function Υ p,β,γ for p = 2 , , β = 1,and γ = 0 (solid lines), γ = p / γ = 1 (dash-dotted lines) and γ = √ ζ (3) was calculated numericallyusing the figure for p = 3. (cid:3) We can now finish the bound on (3.26). Lemma 3.4 and bounds on the function J N yield that for d/N / ∈ (1 / − δ ′ , / δ ′ ) the summands decrease exponentially in N . Therefore they can be neglected. The remaining part can be bounded by C (1 / δ ′ ) N X d =(1 / − δ ′ ) N t N / (cid:16) − dN (cid:17) p exp( − cN ( d/N − / ) ≤ Ct N / Z δ ′ − δ ′ x p e − c ′ Nx d x ≤ Ct N / N − ( p +1) / Z ∞−∞ u p e − c ′ u d u N →∞ −−−→ , (3.42)if p ≥ NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 16
Similarly, for (3.27) we have(3.27) ≤ C N/ X d =0 tN / ν − (cid:16) − dN (cid:17) p exp( N ˜Υ( d/N )) , (3.43)where, setting k u k = min( u, − u ),˜Υ p,β,γ ( u ) = ( γ β − γ β (1+(1 − u ) p ) + η k u k , if (1 − u ) p ≥ γβ + λ − γ β + β (1 + (1 − u ) p ) − γ + η k u k , if (1 − u ) p ≤ γβ + λ − p,β,γ is always strictly negative.It is also easy to be checked that it is possible to choose δ , δ ′ and η small such thatthe first part of the definition of ˜Υ( u ) < δ for all k u k ≥ δ ′ . Therefore such d can beneglected. Around d = 0 the function ˜Υ( x ) can be approximated by a linear function − cx , c >
0, and the summation by an integration. As an upper bound we get
CtN / ν − Z δ ′ e − cNx d x ≤ CtN / ν − N →∞ −−−→ . (3.45)An analogous bound works for d close to N and p even.For (3.28) we have(3.28) ≤ C ν X d =0 tN − / d [1 − (1 − dN − ) p ] − / exp( N ˜Υ( d/N )) . (3.46)The linear approximation of ˜Υ and of the bracket in the last expression yields anupper bound CtN / Z ε x / e − c ′ Nx d x ≤ CtN − N →∞ −−−→ . (3.47)Finally, since ˜Ξ(0) = Ce − Nγ /β , it is easy to see that the second half of (3.29)tends to 0. The first half equals (up to constant) N X d = N/ (cid:16) dN − (cid:17) p t N − N (cid:18) Nd (cid:19) ≤ Ct n X d ≥ N/ N / N − N (cid:18) Nd (cid:19) + N / X i =1 (cid:16) N + iN − (cid:17) p N / e − i / N o , (3.48)where we used the known approximation of (cid:0) Nd (cid:1) ≤ CN − / N e − i / N for d = ( N + i ) / i ≪ N / . The first term in (3.48) tends to 0 by a standard moderate deviationargument. The second one can be approximated by Ct N − ( p/ Z ∞ x p e − x / d x N →∞ −−−→ p ≥
3. This completes the proof of Proposition 3.1. (cid:3)
NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 17 Random walk properties
In this section we prove Proposition 3.3. For A ⊂ S N let T A = min { k ≥ Y N ( k ) ∈ A } be the hitting time of A . We write P x for the law of the simple randomwalk Y N conditioned on Y N (0) = x . Let Q = Q i , i ∈ N , be a birth-death process on { , . . . , N } with transition probabilities p i,i − = 1 − p i,i +1 = i/N . We use P k and E k todenote the law of (the expectation with respect to) Q conditioned on Q = k . Under P , Q i has the same law as dist( Y N (0) , Y N ( i )). Define T k = min { i ≥ Q i = k } thehitting time of k by Q . It is well-known fact that for k < l < mP l [ T m < T k ] = P l − i = k (cid:0) N − i (cid:1) − P m − i = k (cid:0) N − i (cid:1) − . (4.1)Finally, let p k ( d ) = P ( Q k = d ). We need the following lemma for estimating p k ( d )for large k . Lemma 4.1.
There exists K large enough such that for all k ≥ KN log N =: K ( N ) and x, y ∈ S N (cid:12)(cid:12)(cid:12)(cid:12) P y [ Y N ( k ) = x ∪ Y N ( k + 1) = x ]2 − − N (cid:12)(cid:12)(cid:12)(cid:12) ≤ − N (4.2) and thus (cid:12)(cid:12)(cid:12)(cid:12) p k ( d ) + p k +1 ( d )2 − − N (cid:18) Nd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − N . (4.3) Proof.
The beginning of the argument is the same as in [Mat87]. We constructcoupling between Y N (which by definition starts at site = (1 , . . . , ∈ S N ) andanother process Y ⋆N . This process is a simple random walk on S N with the initialdistribution µ ⋆N being uniform on those x ∈ S N with dist( x, ) even. The couplingis the same as in [Mat87]. This coupling gives certain random time T N which can beused to bound the variational distance between µ ⋆ and the distribution µ kN of Y N ( k ):for k even d ∞ ( µ ⋆N , µ kN ) ≡ max A ⊂S N | µ ⋆N ( A ) − µ kN ( A ) | ≤ P [ T N > k ] . (4.4)The law of T N is as follows. Let U = dist( Y ⋆N (0) , ). That is U is a binomial randomvariable with parameters N and 1 / Y U on S U started from . The distribution of T N is then thesame as the distribution of the hitting time of { x ∈ S U : dist( , x ) = U/ } . It isproved in [Mat87] that P ( T N > N log N ) → c <
1. It is then easy to see that, P [ T N ≥ K ( N )] ≤ c KN/ ≤ − N , (4.5)if K is large enough. Thus, for even k ≥ K ( N ), d ∞ ( µ ⋆N , µ kN ) ≤ − N and thus | µ ⋆N ( x ) − µ kN ( x ) | ≤ − N for all x ∈ S N . A similar claim for k odd is then not difficultto prove. The second part of the lemma is a direct consequence of the first part. (cid:3) Lemma 4.2.
Let γ , β , ν satisfy the hypothesis of Proposition 3.3. Then, there existsa constant, C = C ( β, γ, ν ) , such that for all N large enough, Y -a.s. tr ( N ) X i,j =1 ,i = j ⌊ i/ν ⌋ = ⌊ j/ν ⌋ { D ij = d } ≤ Ctr ( N ) { d ≤ ν } , (4.6) and for all d ∈ { , . . . , N } . NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 18
Proof.
The lemma is trivially true for d > ν . For d ≤ ν , let ρ ( d ) = E ν X i =1 { Q i = d } . (4.7)We have ρ (0) ≥ N − and ρ ( d ) ≥ P [ T d ≤ ν ]. This probability is decreasing in d and P [ T ν ≤ ν ] = NN · N − N . . . N − ν + 1 N ≥ e − ν /N . (4.8)Thus ρ ( d ) ≥ e − ν /N for all d ≤ ν . To get an upper bound on ρ ( d ) we write ρ ( d ) ≤ E h T ν X i =1 { Q i = d } i = 1 + E d h T ν X i =1 { Q i = d } i = 1 + 1 P d [ T ν < T d ] . (4.9)However, using (4.1), P d [ T ν < T d ] = N − dN P d +1 [ T ν < T d ] = N − dN (cid:0) N − d (cid:1) − P ν − i = d (cid:0) N − i (cid:1) − = 1 − O ( νN − ) . (4.10)Since ν ≪ N , ρ ( d ) ≤ ν X i,j =1 { D ij = d } =: ν ˜ Z. (4.11)Of course, ˜ Z ∈ [0 ,
1] and, using the results of the previous paragraph, e − ν /N (2 ν ) − ≤ E [ ˜ Z ] ≤ ν − . (4.12)The left-hand side of (4.6) is stochastically smaller than ν P mk =1 ˜ Z k , where ˜ Z k arei.i.d. copies of ˜ Z and m = ⌈ tr ( N ) /ν ⌉ . By Hoeffding’s inequality [Hoe63], P h m X i =1 ˜ Z k ≥ m E [ ˜ Z k ] i ≤ exp {− m E [ ˜ Z k ] } ≤ exp {− m e − ν /N (2 ν ) − } , (4.13)where we used the lower bound from (4.12). Since ν/N ≪ N , by the Borel-Cantellilemma, the left-hand side of (4.6) is a.s. bounded by ν m E [ ˜ Z ] ≤ Ctr ( N ) (4.14)for all N large enough and d ≤ ν . This completes the proof of Lemma 4.2. (cid:3) Proof of Proposition 3.3.
We prove (3.20) first. Observe that for i, j in the sameblockΛ d − Λ ij = (cid:16) − dN (cid:17) p − (cid:16) − p | i − j | N (cid:17) = 2 p ( | i − j | − d ) N + O (cid:16) d N (cid:17) . (4.15)The contribution of the error term is smaller than the right-hand side of (3.20), asfollows from Lemma 4.2.To compute the contribution of the main term, let˜ ρ ( d ) = E h ν X i =1 ( i − d ) { Q i = d } i . (4.16) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 19
Let T d = T d and T kd = min { i > T k − d : Q i = d } . Then˜ ρ ( d ) = E h ∞ X j =1 ( T jd − d ) { T jd < ν } i = E h ∞ X j =1 ( T jd − T d + T d − d ) { T jd < ν } i ≤ E [( T d − d ) { T d < ν } ] (cid:16) ∞ X i =1 E d [ T id { T id < ν − d } ] (cid:17) . (4.17)Using (4.8), P [ T d = d ] ≤ Ce − d /N and further P [ T d ≥ d + 2 k ] ≤ (cid:18) d + 2 kk (cid:19)(cid:16) dN (cid:17) k ≤ C d k N k . (4.18)Hence, cd N − ≤ E [( T d − d ) { T d < ν } ] ≤ Cd N − .For the second term in (4.17) we write1+ ∞ X i =1 E d [ T id { T id < ν − d } ] ≤ E d [ T d { T d < ν − d } ] (cid:16) ∞ X i =1 E d [ T id { T id < ν − d } ] (cid:17) = ∞ X k =0 (cid:8) E d [ T d { T d < ν − d } ] (cid:9) k . (4.19)Using the well-known estimate (cid:0) kk (cid:1) ≤ Ck − / k and k < k , E d [ T d { T d < ν − d } ] ≤ ν/ X k =1 k (cid:18) kk (cid:19)(cid:16) νN (cid:17) k ≤ C ∞ X k =1 (cid:16) νN (cid:17) k ≤ C νN (4.20)and (4.19) is finite. Thus ˜ ρ ( d ) ≤ Cd N − for all d ∈ { , . . . , ν } .The one-block contribution of the first term of (4.15) to (3.20) is then given by2 pN ν X i,j =1 ( | i − j | − d ) { D ij = d } =: 2 pN ν ˜ Z, (4.21)with ˜ Z ∈ [0 ,
1] and cd N − ν − ≤ E [ ˜ Z ] ≤ Cd N − ν − . (4.22)Therefore, as in the proof of Lemma 4.2, Hoeffding’s inequality and (4.22) imply thatthe contribution of the first term of (4.15) to (3.20) is smaller than Ctr ( N ) d N − ,which was to be shown.Finally, we prove (3.19). Since we are interested in an upper bound only we can,without loss of generality, restrict the summation on i < j . We first consider thecontribution of pairs ( i, j ) such that j − i ≥ K ( N ). Then necessarily, ⌊ i/n ⌋ 6 = ⌊ j/n ⌋ .Let R = tr ( n ). Lemma 4.1 yields E h R X j − i ≥K ( N ) { D ij = d } i = R X j − i ≥K ( N ) p j − i ( d ) ≤ CR − N (cid:18) Nd (cid:19) . (4.23) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 20
Further,Var (cid:20) R X j − i ≥K ( N ) { D ij = d } (cid:21) = R X j − i ≥K ( N ) R X j − i ≥K ( N ) P (cid:2) D i ,j = D i ,j = d (cid:3) − P (cid:2) D i ,j = d (cid:3) P (cid:2) D i ,j = d (cid:3) . (4.24)We can again suppose that i ≤ i . The right-hand side of (4.24) is non-null onlyif i ≤ i ≤ j < j or i ≤ i < j ≤ j . We will consider only the first case. Thesecond one can be treated analogously. In is not difficult to see using Lemma 4.1that if i − i j ≥ K ( N ) or j − j ≥ K ( N ) then the difference of probabilities in theabove summation is at most 2 − N . Therefore, the contribution of such ( i , i , j , j )to the variance is at most R − N .If i − i < K ( N ) and j − j < K ( N ) then, using Lemma 4.1 again, P (cid:2) D i ,j = D i ,j = d (cid:3) ≤ C − N (cid:18) Nd (cid:19) . (4.25)We choose ε >
0. For k d k ≤ (1 − ε ) N/ X j − i ≥K ( N ) i − i < K ( N ) X j − i ≥K ( N ) j − j < K ( N ) P (cid:2) D i ,j = D i ,j = d (cid:3) ≤ C K ( N ) R − N (cid:18) Nd (cid:19) ≤ C K ( N ) R e − NI ((1 − ε/ / ≪ N − R ν − , (4.26)say. For k d k ≥ (1 − ε ) N/
2, that is | d − N/ | ≤ εN/
2, we have for ε small enough(how small depend on γ and β ) that 2 − N (cid:0) Nd (cid:1) ≫ N R − . Then, X j − i ≥K ( N ) i − i < K ( N ) X j − i ≥K ( N ) j − j < K ( N ) P (cid:2) D i ,j = D i ,j = d (cid:3) ≤ CN R − N (cid:18) Nd (cid:19) ≪ N − R − N (cid:18) Nd (cid:19) . (4.27)We have thus found that the expectation of the summation over j − i > K ( N ) issmaller than the right-hand side of (3.19) and the variance of the same summationis much smaller than N − times the right-hand side of (3.19) squared. A straight-forward application of the Chebyshev inequality and the Borel-Cantelli Lemma thengives the desired a.s. bound for pairs j − i ≥ K ( N ) and all d ∈ { , . . . , N } .Choose again ε >
0. For j − i < K ( N ), observe first that if k d k ≥ (log N ) ε ≫ log N then the summation over such pairs ( i, j ) in (3.19) is always smaller than K ( N ) R ≪ Rν − e η k d k for all η >
0. For the remaining d ’s, that is k d k < (log N ) ε ′ ,let K N ≥ K be the smallest constant such that K N N log N is a multiple of ν . Since ν ≪ N , K N − K ≪
1. As the difference between K and K N is negligible, we willuse the same notation K ( N ) for K N N log N and we will simply suppose that K ( N )is a multiple of ν . The summation in (3.19) for j − i ≤ K ( N ) can be bounded from NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 21 above by tr ( N ) X
For j ≥ d and N large enough, Z ℓ ( j, d ) ≤ Z ℓ ( d + 6 , d ). We have, cN − ≤ K ( N ) E [ Z ℓ ( d + 6 , d )] ≤ CN − . (4.36)Indeed, the lower bound is trivial and for the upper bound we use the fact that theprobability that Y N reaches d before returning to d + 6 is smaller than CN − andbefore the time K ( N ) there are at most K ( N ) tries. Hence, for j ≥ d the probability P h K ( N ) ⌈ R/ K ( N ) ⌉ X ℓ =0 ˜ Z ℓ ( k, d ) ≥ RN K ( N ) i (4.37)decreases at least exponentially in N and thus the interior inequality is not valida.s. for all N large. Summing over k we get K ( N ) − X k =0 ⌈ R/ K ( N ) ⌉ X ℓ =0 K ( N ) Z ℓ ( j k , d ) ≤ d K ( N ) ν − R K ( N ) + K ( N ) RN K ( N ) ≤ CRν − e ηd , (4.38)since γ/β < (cid:3) Convergence of clock process
We will prove the convergence of the rescaled clock process to the stable sub-ordinator on space D ([0 , T ] , R ) equipped with the Skorokhod M -topology. Thistopology is not commonly used in the literature, therefore we shortly recall some ofits properties and compare it with the more standard Skorokhod J -topology, whichwe will need later, too. For more details the reader is referred to [Whi02] for bothtopologies and to [Bil68] for detailed account on J -topology.5.1. Topologies on the Skorokhod space.
Consider space D = D ([0 , T ] , R ) ofc`adl`ag functions. The J -topology is the topology given by the J -metric: for f, g ∈ Dd J ( f, g ) = inf λ ∈ Λ {k f ◦ λ − g k ∞ ∨ k λ − e k ∞ } , (5.1)where Λ is the set of strictly increasing functions mapping [0 , T ] onto itself such thatboth λ and its inverse are continuous, and e is the identity map on [0 , T ].Also the M -topology is given by a metric. For f ∈ D let Γ f be its completedgraph, Γ f = { ( z, t ) ∈ R × [0 , T ] : z = αf ( t − ) + (1 − α ) f ( t ) , α ∈ [0 , } . (5.2)A parametric representation of the completed graph Γ f (or of f ) is a continuousbijective mapping φ ( s ) = ( φ ( s ) , φ ( s )), [0 , Γ f whose first coordinate φ isincreasing. If Π( f ) is set of all parametric representation of f , then the M -metricis defined by d M ( f, g ) = inf {k φ − ψ k ∞ ∨ k φ − ψ k ∞ : φ ∈ Π( f ) , ψ ∈ Π( g ) } . (5.3)The space D equipped with both M - and J -topologies is Polish. The M -topologyis weaker than the J -topology: As an example, consider the sequence f n = { [1 − /n, } + 2 · { [1 , T ] } , (5.4)which converges to f = 2 · { [1 , T ] } in the M -topology but not in the J -topology.One often says that the M -topology allows “intermediate jumps”. NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 23
We will need a criterion for tightness of probability measures on D . To this endwe define several moduli of continuity, w f ( δ ) = sup (cid:8) min (cid:0) | f ( t ) − f ( t ) | , | f ( t ) − f ( t ) | (cid:1) : t ≤ t ≤ t ≤ T, t − t ≤ δ (cid:9) ,w ′ f ( δ ) = sup (cid:8) inf α ∈ [0 , | f ( t ) − ( αf ( t ) + (1 − α ) f ( t )) | : t ≤ t ≤ t ≤ T, t − t ≤ δ (cid:9) ,v f ( t, δ ) = sup (cid:8) | f ( t ) − f ( t ) | : t , t ∈ [0 , T ] ∪ ( t − δ, t + δ ) (cid:9) . (5.5)The following result is a restatement of Theorem 12.12.3 of [Whi02] and Theorem 15.3of [Bil68]. Theorem 5.1.
The sequence of probability measures { P n } on D ([0 , T ] , R ) is tight inthe J -topology if (i) For each positive ε there exist c such that P n [ f : k f k ∞ > c ] ≤ ε, n ≥ . (5.6)(ii) For each ε > and η > , there exist a δ , < δ < T , and an integer n suchthat P n [ f : w f ( δ ) ≥ η ] ≤ ε, n ≥ n , (5.7) and P n [ f : v f (0 , δ ) ≥ η ] ≤ ε and P n [ f : v f ( T, δ ) ≥ η ] ≤ ε, n ≥ n . (5.8) The same claim hold for the M -topology with w f ( δ ) in (5.7) replaced by w ′ f ( δ ) . Proof of Theorem 1.1.
To prove the convergence of the rescaled clock process¯ S N ( · ) = e − γN S N ( · r ( N )) to the stable subordinator V γ/β , we check first the conver-gence of finite-dimensional marginals. As can be guessed, Proposition 3.1 will serveto this purpose. Let ℓ , { u i } and { t i } be as above. Then, E h exp n − ℓ X i =1 u i (cid:0) ¯ S N ( t k ) − ¯ S N ( t k − ) (cid:1)o(cid:12)(cid:12)(cid:12) Y N i = E (cid:2) F N ( X N ; { t i } , { u i } ) (cid:12)(cid:12) Y N (cid:3) = E (cid:2) F N ( X N ; { t i } , { u i } ) (cid:3) + o (1) , (5.9)as follows from Proposition 3.1.The value of E (cid:2) F N ( X N ; { t i } , { u i } ) (cid:3) is not difficult to calculate. Define j N ( i ) = ⌊ t i r ( N ) /ν ⌋ . Then E (cid:2) F N ( X N ; { t i } , { u i } ) (cid:3) = E h exp (cid:16) − ℓ X k =1 u k e γN t k r ( N ) − X i = t k − r ( N ) e i e β √ NX N ( i ) (cid:17)i ≥ E h ℓ Y k =1 j ( k ) Y j = j ( k − exp (cid:16) − u k e γN ν − X i =0 e jν + i e β √ NX N ( jν + i ) (cid:17)i (5.10)Since the process X N is a piece-wise independent process, the product in (5.10) is aproduct of independent random variables. Then expectations of all of them can be NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 24 then bounded using Proposition 2.1. We get, for δ > N large enough, E (cid:2) F N ( X N ; { t i } , { u i } ) (cid:3) ≥ ℓ Y k =1 j N ( k ) Y j = j N ( k − F N ( u k ) ≥ ℓ Y k =1 (cid:0) − (1 + δ ) νN − / e − Nγ / β Ku γ/β k (cid:1) j N ( k ) − j N ( k − − ≥ ℓ Y k =1 exp (cid:8) − (1 + 2 δ )( t k − t k − ) Ku γ/β (cid:9) , (5.11)which is (up to 1 + 2 δ term) the Laplace transform of V γ/β ( K · ). A correspondingupper bound can be constructed analogously.To check the tightness for ¯ S N in D ([0 , T ] , R ) equipped with the Skorokhod M -topology we use Theorem 5.1. Since the processes ¯ S N are increasing, it is easy to seethat condition (i) is equivalent to the tightness of the distribution of ¯ S N ( T ), whichcan be checked easily from the convergence of the Laplace transform of the marginalat time T (the limiting Laplace transform tends to 1 as u → w ′ ¯ S N ( δ ) is always equal to zero. So checking (ii) boils down to controllingthe boundary oscillations v ¯ S N (0 , δ ) and v ¯ S N ( T, δ ). For the first quantity (using againthe monotonicity of ¯ S N ) this amounts to check that P [ ¯ S N ( δ ) ≥ η ] < ε if δ is smallenough and N large enough. Using the convergence of of marginal at time δ , it issufficient to take δ such that P [ V γ/β ( Kδ ) ≥ η ] ≤ ε/
2, and take n such that for all n ≥ n (cid:12)(cid:12) P [ ¯ S N ( δ ) ≥ η ] − P [ V γ/β ( Kδ ) ≥ η ] (cid:12)(cid:12) ≤ ε/ . (5.12)The reasoning for v ¯ S N ( T, δ ) is analogous. (cid:3)
Coarse-grained clock process.
To prove our aging result, that is Theo-rem 1.2, we need to modify the result of Theorem 1.1 slightly. Let ˜ S N be the“coarse-grained” clock processes,˜ S N ( t ) = 1 e γN S N ( ν ⌊ tr ( N ) ν − ⌋ ) . (5.13)For these processes we can strengthen the topology used in Theorem 1.1, that is wecan replace the M - by the J -topology. Theorem 5.2.
If the hypothesis of Theorem 1.1 is satisfied, then ˜ S N ( t ) N →∞ −−−→ V γ/β ( Kt ) Y − a.s., (5.14) weakly in the J -topology on the space of c`adl`ag functions D ([0 , T ] , R ) . Unfortunately, we cannot prove the theorem with estimates we have already atdisposition. We should return back and improve some of them. First we show thattraps with energies “much smaller” than γ √ N /β almost do not contribute to theclock process. Let B m = γ √ N /β − m/ ( β √ N ) and let¯ S mN ( t ) = e − γN ⌊ tr ( N ) ⌋ X i =0 e i exp (cid:8) β √ N X N ( i ) (cid:9) { X N ( i ) ≤ B m } . (5.15) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 25
Lemma 5.3.
For every T and η , ε > there exists m large enough such that P [ ¯ S mN ( T ) ≥ η |Y ] ≤ ε, Y -a.s. (5.16) Proof.
To prove this lemma we should improve/modify slightly the calculations ofSections 2 and 3. With the notation of Section 2 define F mN = E h exp n − e − γN ν X i =1 e i e β √ NU i { U i ≤ B m } oi . (5.17)(comparing with (2.2) observe that we set u = 1). We will show thatlim N →∞ f ( N ) e Nγ / β [1 − F mN ] = K m , (5.18)with K m → m → ∞ . The proof of this claim is completely analogous to theproof of Proposition 2.1. One should only modify the domains of integrations. Moreprecisely, the definition of D k which appears after (2.9) should be replaced by D mk = D k ∩ { z : G k ( z ) ≤ B m } . Hence, D ′ k becomes D ′ mk = D ′ k ∩ { b : G k ( b ) ≤ − m/ ( β/ √ N ) } ,which then restricts the domain of integration in (2.33) to ( −∞ , − m/β ]. Hence, theconstant K m can be made arbitrarily small by choosing m large.Further, as in Section 3, define F mN ( X ) = exp (cid:16) − T r ( N ) − X i =0 g (cid:16) e − γN e β √ NX ( i ) { X ( i ) ≤ B m } (cid:17)(cid:17) . (5.19)Then, as in Proposition 3.1, we will showlim N →∞ E (cid:2) F mN ( X N ) (cid:12)(cid:12) Y (cid:3) − E (cid:2) F mN ( X N ) (cid:3) = 0 , Y -a.s. (5.20)We use again (3.5) to show this claim. Although the indicator function is not dif-ferentiable, we will proceed as if it was, setting ( { x ≤ B } ) ′ = − δ ( x − M ), where δ denotes the Dirac delta function. As usual, this can be justified e.g. by using smoothapproximations of the indicator function. The second derivative of F mN ( X ) equals u β Ne γN e β √ N ( X ( i )+ X ( j )) g ′ (cid:0) ue β √ NX ( i ) − γN (cid:1) g ′ (cid:0) ue β √ NX ( j ) − γN (cid:1) F mN ( X ) × (cid:16) { X ( i ) ≤ B m } − δ B m ( X ( i )) β √ N (cid:17)(cid:16) { X ( j ) ≤ B m } − δ B m ( X ( j )) β √ N (cid:17) ≤ u β N e β √ N ( X hN ( i )+ X hN ( j )) − γN exp (cid:0) − g (cid:0) ue β √ NX hN ( i ) − γN (cid:1) − g (cid:0) ue β √ NX hN ( j ) − γN (cid:1)(cid:1) × (cid:16) { X ( i ) ≤ B m } − δ B m ( X ( i )) β √ N (cid:17)(cid:16) { X ( j ) ≤ B m } − δ B m ( X ( j )) β √ N (cid:17) . (5.21)We should now bound the contributions of four terms. The one with the product oftwo indicator functions is easy, because we can use directly the result of Lemma 3.2.For remaining three terms, those with the product of one indicator and one deltafunction, and this with two delta function, the calculation should be repeated. How-ever, in the end we find that (5.21) is bounded by ¯Ξ(Cov( X ( i ) , X ( j ))) as before. Thepresence of the delta functions makes actually the calculations slightly less compli-cated. The proof then proceed as in Section 3. NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 26
We can now finish the proof of Lemma 5.3. By (5.17) and (5.20), E (cid:2) exp( − ¯ S mN ( T )) (cid:12)(cid:12) Y (cid:3) = E (cid:2) F mN ( X N ) (cid:12)(cid:12) Y (cid:3) = E (cid:2) F mN ( X N ) (cid:12)(cid:12) Y (cid:3) + o (1)= (1 − K m f ( N ) − e − Nγ / β ) T r ( N ) /ν + o (1) = e − K m T + o (1) . (5.22)Since K m → m → ∞ , P [ ¯ S mN ( T ) ≥ η |Y ] ≤ − E (cid:2) exp( − ¯ S mN ( T )) (cid:12)(cid:12) Y (cid:3) − e − η (5.23)can be made arbitrarily small by taking m large enough. (cid:3) We study now how the blocks where the process visits sites with energies largerthan B m are distributed along the trajectory. To this end we set for any Gaussianprocess X s mN ( i ; X ) = {∃ j : iν < j ≤ ( i + 1) ν, X ( j ) > B m } . (5.24)and we define point process H mN ( X ) on [0 , T ] by H mN ( X ; d x ) = T r ( N ) /ν X i =0 s mN ( i ; X ) δ iν/r ( N ) (d x ) . (5.25) Lemma 5.4.
For every m ∈ R the point processes H mN ( X N ) converge to a homoge-neous Poisson point process on [0 , T ] with intensity ρ m ∈ (0 , ∞ ) , Y -a.s.Proof. To show this lemma we use Proposition 16.17 of Kallenberg [Kal02]. Ac-cording to it, to prove the convergence of H mN ( X N ) to a Poisson point process withintensity ρ m it is sufficient to check that for any interval I ⊂ [0 , T ]lim N →∞ P [ H mN ( X N ; I ) = 0 |Y ] = e − ρ m | I | (5.26)and lim sup N →∞ E [ H mN ( X N ; I ) |Y ] ≤ ρ m | I | , (5.27)where | I | denotes the Lebesgue measure of I .The proof of the first claim is completely similar to the previous ones. We startwith a one-block estimate for (5.26):lim N →∞ N / ν − e Nγ / β E [ s mN (0 , U )] = ρ m , (5.28)Using the notation of Section 2, we get E [ s mN (0 , U )] = Z A m d z (2 π ) ν/ e − P νi =1 z i , (5.29)where A m = { z : ∃ k ∈ { , . . . , ν } G k ( z ) > B m } . Dividing the domain of integrationaccording to the maximal G k ( z ), this is equal ν X k =1 Z D k d z (2 π ) ν/ e − P νi =1 z i , (5.30)where D k = { z : G k ( z ) > B m , G i ( z ) ≤ G k ( z ) ∀ i = k } . Using the substitution z i = b i ± Γ i B m on D k (where + sign is used for i ≤ k and − sign for i > k ) we get e − Nγ / β e mγ/β ν X k =1 Z D ′ k d b (2 π ) ν/ e − P νi =1 b i e − B m G k ( b ) , (5.31) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 27 where D ′ k = { b : G k ( b ) > , P kj = i +1 b j + | k − i | Γ ν B m ≥ ∀ i = k } . The same reasoningas before then allows to show that the last expression behaves like ρ m νN − / e − γ N/ β as N → ∞ .To compare the real process with the block-independent process, let F N ( I ; X ) = { max { X ( i ) : iν/r ( N ) ∈ I } ≤ B m } . (5.32)The difference between E [ F N ( I ; X N ) |Y ] and E [ F N ( I ; X N )] is again given by theGaussian comparison formula (3.5). This time the second derivative equals δ ( X ( i ) − B m ) δ ( X ( j ) − B m ) Y k = i,j { X ( k ) ≤ B m } ≤ δ ( X ( i ) − B m ) δ ( X ( j ) − B m ) . (5.33)If covariance of X ( i ) and X ( j ) equals c , the expectation of the last expression isgiven by the value of the joint density of X ( i ), X ( j ) at point ( B m , B m ) which is(2 π (1 − c )) − e − B m / (1+ c ) ≤ C (1 − c ) − exp n − γ Nβ (1 + c ) o . (5.34)The exponential term is the same as in ¯Ξ( c ). The polynomial prefactor is howeverdifferent, it diverges faster as c →
1. We should thus return to (3.24) with ˜Ξ replacedby the right-hand side of (5.34). First Z (1 − c ) − = c − arg tanh( c ) ≈ −
12 log(1 − c ) (5.35)as c →
1, which is not bounded for all c as before. The estimates (3.26) and (3.27) areinfluenced by this change. For (3.26) we can actually neglect this change, becausethe main contribution to this term came from the neighborhood of d = N/ c = 0) and was exponentially small in the neighborhood of d = 1 (or c ∼ /N ). Inthe treatment of (3.27), the change has more effect, after some computations (3.45)becomes CtN / ν − Z δ ′ log( c/x ) e − cNx d x ≤ CtN / ν − log N N →∞ −−−→ . (5.36)Finally, the change of polynomial prefactor of ¯Ξ implies change in the control of(3.28). The equation (3.46) becomes(3.28) ≤ C ν X d =0 tN − / d [1 − (1 − dN − ) p ] − exp( N ˜Υ( d/N )) . (5.37)and the linearization of ˜Υ gives new form of (3.47) CtN / Z ε xe − c ′ Nx d x ≤ CtN − / N →∞ −−−→ . (5.38)Therefore, using (5.28) P [ H mN ( X N ; I ) = 0 |Y ] = E [ F N ( I ; X N ) |Y ] = E [ F N ( I ; X N )] + o (1)= (1 − E [ s mN (0 , U )]) | I | r ( N ) /ν → e − ρ m | I | . (5.39)This completes the proof of (5.26).It is easy to check (5.27). By definition, E [ H mN ( X N ; I ) |Y ] = X i : iν/R ∈ I E [ s mN ( i, X N ) |Y ] . (5.40) NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 28
Since Λ ij ≥ Λ ij for i , j in the same block, E [ s mN ( i, X N ) |Y ] ≤ E [ s mN ( i, X N )]. Therefore,(5.40) ≤ | I | r ( N ) /ν E [ s mN (0 , U )] = ρ m | I | . (5.41)This completes the proof of Lemma 5.4. (cid:3) Proof of Theorem 5.2.
Checking the convergence of finite-dimensional marginals aswell of condition (i) and the second part of (ii) of Theorem 5.1 is analogous as for theoriginal clock process ¯ S N . We should thus only prove the first part of condition (ii).Namely that, for any η and ε there exist δ such that P [ w ¯ S N ( δ ) ≥ η ] ≤ ε, (5.42)for all N large enough.Let w f ([ τ, τ + δ ]) = sup { min( | f ( t ) − f ( t ) | , | f ( t ) − f ( t ) | ) : τ ≤ t ≤ t ≤ t ≤ τ + δ } . (5.43)Fix m such that P [ ¯ S mN ( T ) ≥ η/ ≤ ε/
2, which is possible according to Lemma 5.3.If H mN ( X n ; [ τ, τ + δ ]) ≤ w ¯ S N ([ τ, τ + δ ]) ≤ ¯ S mN ( τ + δ ) − ¯ S mN ( τ ) ≤ ¯ S mN ( T ) . (5.44)Hence, P [ w ¯ S N ([ τ, τ + δ ]) ≥ η | i ¯ S mN ( T ) ≤ η/ ≤ P [ H mN ( X N ; [ τ, τ + δ ]) ≥ ≤ Cρ m δ . (5.45)We can now show (5.42). Estimate w ˜ S N ( δ ) ≤ max { w ˜ S N ([ τ, τ + 2 δ ]) : 0 ≤ τ ≤ T, τ = kδ, k ∈ N } (5.46)yields P [ w ˜ S N ( δ ) ≥ η |Y ] ≤ T δ − X k =0 P [ w ˜ S N ([ kδ, ( k + 2) δ ]) ≥ ε |Y ] ≤ P [ ¯ S mN ( T ) ≥ η/
2] +
T δ − X k =0 P [ H mN ( X N ; [ kδ, ( k + 2) δ ]) ≥ ≤ ε/ CT δ − ρ m δ ≤ ε (5.47)if δ is chosen small enough. This completes the proof. (cid:3) Proof of Theorem 1.2.
Let R N be the range of the coarse grained process ˜ S N . Ob-viously, for any 1 > ε > A εN ( t, s ) ⊃ {R N ∩ ( t, s ) = ∅} , (5.48)because if the above intersection is empty, then σ N makes less than ν steps in timeinterval [ te γN , se γN ], and thus the overlap of σ N ( te γN ) and σ N ( se γN ) is O ( ν/N ).If R N ∩ ( t, s ) = ∅ , than there exist u such that ˜ S N ( u ) ∈ ( t, s ). Moreover, it followsfrom Theorem 5.2 that for any δ there exist η such than P [ ˜ S N ( u + η ) ∈ ( s, t )] ≥ − δ. (5.49)This however means that the process σ N make at least ηr ( N ) steps between times t and s and thus the overlap between σ N ( te γN ) and σ N ( se γN ) is with high probabilityclose to 0. NIVERSALITY OF THE REM FOR DYNAMICS OF MEAN-FIELD SPIN GLASSES 29
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G. Ben Arous, Courant Institute of the Mathematical Sciences, New York Uni-versity, 251 Mercer Street, New York, NY 10012, USA
E-mail address : [email protected] A. Bovier, Weierstrass Institute for Applied Analysis and Stochastics, Mohren-strasse 39, 10117 Berlin, Germany, and, Mathematics Institute, Berlin Universityof Technology, Strasse des 17. Juni 136, 10269 Berlin, Germany
E-mail address : [email protected] J. ˇCern´y, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzer-land
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