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In this paper the celebrated arcsine aging scheme of Ben Arous and Cerný is taken up. Using a brand new approach based on point processes and weak convergence techniques, this scheme is implemented in a broad class of Markov jump processes in random environments that includes Glauber dynamics of discrete disordered systems. More specifically, conditions are given for the underlying clock process (a partial sum process that measures the total time elapsed along paths of a given length) to converge to a subordinator, and consequences for certain time correlation functions are drawn. This approach is applied to Bouchaud's asymmetric trap model on the complete graph for which aging is for the first time proved, and the full, optimal picture,  obtained. Application to  spin glasses are carried out in follow up papers.


Introduction
The term aging qualifies dynamics whose transients towards equilibrium become increasingly slower as time elapses.This phenomenon is measured in the anomalous behavior of certain time correlation functions.Discovered in the physics of spin glasses, aging was successfully accounted for, on a theoretical level, using simple phenomenological models -the so called trap models of Bouchaud [15,18,33,17,16].These are Markov jump processes that describe the behavior of spin glass dynamics on long time scales in terms of thermally activated barrier crossing in landscapes made of random 'traps'.
The first rigorous connection between the microscopic dynamics of a spin system and a trap model was established in [3,4] where it is proved that a particular Glauber dynamics of the REM, known as the Random Hopping Dynamics (hereafter RHD), has the same aging behavior as Bouchaud's symmetric trap model on the complete graph.
λ n (x) ≡ y∈Vn,y =x λ n (x, y) , ∀x ∈ V n . (1.1) We will assume that sup x∈Vn λ n (x) < C P-a.s. for some 0 < C < ∞.A specially important class of such matrices is obtained by choosing the λ n (y, x)'s such that τ n (x)λ n (x, y) = τ n (y)λ n (y, x) , ∀ (x, y) ∈ E n , x = y , (1.2) and λ n (y, x) = 0 for all (x, y) / ∈ E n , x = y.This implies that X n (t) is reversible w.r.t. the random measure on V n that assigns to x the mass τ n (x).Glauber dynamics in particular belong to this class.
Given a family (e n,i , n ∈ N, i ∈ N) of independent mean one exponential r.v.'s the clock process is the partial sum process defined by Here (J n (k) , k ∈ N) is the jump chain of X n , namely, the discrete time Markov chain with one-step transition probabilities p n (x, y) = λ n (x, y)/λ n (x) if (x, y) ∈ E n , x = y, 0, otherwise. (1.4) Note that S n (k) gives the total time spent by X n along the first k steps of J n .Thus, if X n has initial distribution µ n , J n has initial distribution µ n and X n (t) = J n ( S ← n (t)) , t > 0.
(1.5) (Here S ← n denotes the right continuous inverse of S n .) The last expression places the clock process in the limelight.The idea behind the arcsine aging scheme is that if, after appropriate rescaling, the clock process converges to a stable subordinator, then anomalous slowdown of the long term dynamics can be explained in terms of the arcsine law for stable subordinators.To put this scheme into practice one faces two difficulties: the clock process is a random process on the probability space of the environment, and, for fixed realization of the environment, it is a partial sum process whose summands are made dependent through the chain J n .
In [9] this problem is solved for dynamics of RHD type, that is, denoting by d x the degree of x in the graph G n (V n , E n ), for the rates λ n (y, x) = (d x τ n (x)) −1 if (x, y) ∈ E n . (1.6) Using a detailed knowledge of the potential theory of the chain J n (reduced here to the symmetric random walk on G n (V n , E n )) and properties of the environments, a set of abstract conditions is derived that ensure that the clock process converges to a stable subordinator for P-almost all environments.Although independence of the τ n (x) s is not assumed a priori, this approach was only applied to such environments.In particular, it did not allow to deal with the p-spin SK spin glass model.This was done in [2] where, using approximations techniques for Gaussian processes, it is proved that on a certain range of times scales and temperature, and for the rates (1.6), the clock process converges to a stable subordinator, but in P-law only.
In the present paper we adopt yet another approach that allows to both implement the arcsine aging scheme in the general setting of Markov jump processes in random environments described above, and obtain results in the strongest possible convergence mode with respect to the law P of the environment.Our approach is based on a powerful and illuminating method developed by Durrett and Resnick [21] to prove functional limit theorems for dependent variables.By extending the framework of [21] to our random setting, and specializing it to processes of the form (1.3), we give simple sufficient conditions for the properly rescaled sequence S n to converge to a subordinator.This is the content of Subsection 1.2 below.Consequences for aging are drawn in Subsection 1. 3 For later reference we denote by P µn the law of X n and by P µn the law of J n with initial distribution µ n .In view of taking n ↑ ∞ limits we assume that the sequences of chains X n , resp.J n , can be constructed on a common probability space (Ω X , F X , P), resp.(Ω J , F J , P ).Expectation with respect to P, P , and P will be denoted respectively by E, E, and E

Convergence of the clock process to a subordinator.
The first increment of the clock process plays a special role.For this reason we define (1.7) Given a positive sequences c n and a n we then set, for t ≥ 0, S n (t) = c −1 n S n ( a n t ) , (1.8) and S n (t) = σ n + S n (t) . (1.9) The re-scaled clock processes S n (t) and S n (t) will be called pure and delayed, respectively.
We now state three conditions, (A1)-(A3), that ensure that the sequence of pure processes S n converges to a subordinator.Because this process is a random variable on the probability space (Ω τ , F τ , P) of the landscape (our random environment) we must first decide in which sense to seek convergence on that space.The relevant convergence modes (those that convey the most useful information in applications) are almost sure convergence and convergence in probability.This means that one of the following statements should be in force: Almost sure convergence: There exists a subset Ω τ ⊂ Ω τ such that P( Ω τ ) = 1 and such that, for all ω ∈ Ω τ , for all large enough n, (A1)-(A3) are verified.
We now state our three conditions for fixed ω and make this explicit by adding the superscript ω to landscape dependent quantities.These conditions depend on the initial distribution µ n , and on the sequences a n and c n .We suppose them fixed.

Condition (A3).
There exists a sequence of functions ε n ≥ 0 satisfying lim 0 such that for some 0 < δ 0 ≤ 1, for all 0 < δ ≤ δ 0 and all t > 0, (1.12) Theorem 1.1.For all sequences of initial distributions µ n and all sequences a n and c n for which Conditions (A1), (A2), and (A3) are verified, either P-almost surely or in P-probability, the following holds w.r.t. the same convergence mode: let {(t k , ξ k )} be the points of a Poisson random measure of intensity measure dt × dν; then, where convergence holds weakly in the space D([0, ∞)) of càdlàg functions on [0, ∞) equipped with the Skorohod J 1 -topology 2 .
Remark 1.2.Although we do not make this explicit in the notation, note that the Lévy measure ν of the limiting subordinator S may remain a random variable on the probability space (Ω τ , F τ , P) of the random landscape.We will see an example of this in the context of the asymmetric trap model on the complete graph (cf.Proposition 3.9).
To obtain convergence of the delayed re-scaled clock process S n of (1.9), we still need to control the initial increment σ n .For this we introduce a separate condition. 1The set Ω τ (respectively the sequence of sets Ω τ n ) for which convergence w. r. t. the environment holds almost surely (respectively in probability) is (are) the same for all t > 0 and u > 0.

Condition (A0).
There exists a continuous distribution function F ω on [0, ∞) such that, for all v ≥ 0, . (1.14) Theorem 1.3.For all sequences of initial distributions µ n and all sequences a n and c n for which Conditions (A0), (A1), (A2), and (A3) are verified, either P-almost surely or in P-probability, the following holds w.r.t. the same convergence mode: let σ denote the random variable of (possibly random) distribution function F ; then, for S defined in (1.13), where ⇒ has the same meaning as in (1.13).

Aging.
We now show how the clock process convergence obtained in Theorem (1.3) is useful for deriving aging information, and in particular, for proving the existence of an arcsine aging regime.
We begin with a few definitions.The aging behavior of X n is quantified using a time correlation function, namely, a two-time function C n (t, s), t, s ≥ 0, that measures the dependence of X n (c n (t + s)) and X n (c n t).We then say that: Definition 1.4.A time correlation function C n exhibits normal aging on time scale c n if one of the following three relations is verified: for all ρ ≥ 0, some non trivial limiting function C ∞ : [0, ∞) → [0, 1], and for some convergence mode w.r.t. the probability law P of the random landscape.
In virtually all situations where normal aging was proved so far, the limiting time correlation function is the distribution function of the generalized arcsine law with parameter 0 < α < 1, This motivates the next definition.
Definition 1.5.The process X n has an arcsine aging regime of parameter α whenever one can find a time correlation function C n that exhibits normal aging with (1.20) While the choice of C n (t, s) is model dependent it turns out that the most commonly used time correlation functions (see e.g.[9], [19]) can be approximated, up to error terms that vanish as n → ∞, by the following one This is the probability that the range of the re-scaled clock process S n does not intersect the time interval (t, t + s).For this choice we have, Theorem 1.6.Let the assumptions and the notation be as in Theorem 1.3.Set C ∞ (t, s) = P ({S(u) , u > 0} ∩ (t, t + s) = ∅) , 0 ≤ t < t + s . (1.22) If, for each ω ∈ Ω τ , σ and S in (1.15) are independent random variables on (Ω X , F X , P), then, for all 0 ≤ t < t + s, w.r.t. the same convergence mode as in (1.15), (1.23) In particular, if σ = 0, (  [11] for the first half and Theorem 1 of [12] for the second half).
Theorem 1.8. (1.25) (1.26) Combing Theorem 1.3, Theorem 1.6 and Theorem 1.8 gives sufficient conditions for the process X n to have, or not to have, an arcsine aging regime.This extends the arcsine aging scheme of [9] to situations where the limiting clock process is not necessarily a stable subordinator.This happens for instance in Bouchaud's asymmetric trap model on the complete graph (see Proposition 3.9 and Lemma 3.10) and in the REM [27], when X n is observed on time scales that are of the order of the time scale of stationarity.Let us finally stress that the form of the relation (1.23), where the role of the initial distribution µ n is made explicit, is new.The effect of µ n on the aging phenomenon will be studied elsewhere.
The remainder of the paper is organized as follows.Section 2 contains the proofs of the results of Section 1 and their specialization to asymmetric trap model on the complete graph.Section 3 begins the investigation of Bouchaud's asymmetric trap model on the complete graph proper: there we define the model and state the results.Their proofs occupy the rest of the paper (Section 4-6) up to a short appendix on regular variations and renewal theory.

Convergence of the clock process and related results
This section is divided in four parts.In Subsection 2.1 we state a result by Durrett and Resnick [21] that is central to the proofs of Theorem 1.1 and Theorem 1.3.The latter are done in Subsection 2.2, and the proof of Theorem 1.6 is done in Subsection 2.3.In Subsection 2.4 we specialize Theorem 1.1 and Theorem 1.3 to the asymmetric trap model on the complete graph.We also give sufficient conditions for convergence of the re-scaled clock process to a partial-sum process in the case, not covered by the theorems of Section 1, where the auxiliary time scale a n is a constant (see Theorem 2.4 and Theorem 2.5).

A result by Durrett and Resnick.
In [21] a method is developed for proving convergence of partial sums processes with dependent increments to Lévy processes.This method consists of two steps.In the first step, one shows that a sequence of point processes associated with the increments converges weakly to a two dimensional Poisson process.Then, applying appropriate functionals (to 'sum up the points') and continuity arguments, one obtains weak convergence of the sum to a limiting Lévy process.
In this section we specialize this result, namely Theorem 4.1 of [21], to the case of processes with non-negative increments.Our framework is the following.Let {Z n,i , n ≥ 1, i ≥ 1}, Z n,i ≥ 0, be an array of random variables defined on a probability space (Ω, F, P) and let {F n,i , n ≥ 1, i ≥ 0} be an array of sub-sigma fields of F such that for each n and i ≥ 1, Z n,i is F n,i measurable and F n,i−1 ⊂ F n,i .Let k n (t) be a nondecreasing right continuous function with range {0, 1, 2, . . .} and assume that for each t > 0 k n (t) is a stopping time.Set for k ≥ 1, S n,0 = 0, and define S n (t) = S n,kn(t) . (2. 2) The next theorem gives conditions for S n to converge to a subordinator.To state it we will need the following extra notation: for δ ≥ 0 set Z δ n,i = Z n,i 1 {Zn,i≤δ} ; further set for k ≥ 1, S δ n,0 = 0, and define S δ n (t) = S δ n,kn(t) .
This does not seem to be correct.

Convergence to subordinators.
In this subsection we prove Theorem 1.1 and the first assertion of Theorem 1.3, and give an alternative to Condition (A3).
So far we kept ω ∈ Ω τ fixed, i.e. we worked with a fixed realization of the environment.Let us now introduce the subsets Ω τ n,1 , Ω τ n,2 ⊂ Ω τ (with the notation of (2.12)) ) . By definition of weak convergence what we have just established is that for each ω ∈ Ω τ n , and large enough n, Proof of Theorem 1.3.As in the proof of Theorem 1.1 we first establish (1.15) for a fixed realization ω ∈ Ω τ of the environment.Note that the additional Condition (A0) is designed to guarantee that σ n converges in distribution to σ.Indeed, since . Thus, supplementing Conditions (A1) and (A2) with Condition (A0), it follows from Theorem 1.1 that, viewing σ n as a constant function in D([0, ∞)), the pairs (σ n , S n ) jointly converge, weakly, to the pair (σ, S), in D 2 ([0, ∞)).It next follows from the continuous mapping theorem, upon adding σ n and S n , that σ n + S n ⇒ S = σ + S in D([0, ∞)) (see [36], p. 84, last paragraph of Section 3.3, for the continuity of the addition of an arbitrary element of D([0, ∞)) and the constant function).Eq. (1.15) being established for a fixed realization ω ∈ Ω τ , we conclude the proof proceeding exactly as in the proof of Theorem 1.1 4 , introducing the extra subsets Condition (A3) may not always be easy to handle.Here is an alternative:

Convergence of the time-time correlation function.
We will now exploit the convergence of S n established above to prove convergence of the time-time correlation function, using the continuous-mapping theorem.
Proof of Theorem 1.6.This pattern of proof is classical (see [36] section 9.7.2) and relies on the continuity property of a certain function of the inverse mapping on D([0, ∞)), the so-called overshoot, which we now define.Let η ∈ D([0, ∞)).For t > 0 let L t be the time of the first passage to a level beyond t; i.e., (2.23) Similarly, (1.22) can be rewritten as ( From this and the definition of weak convergence it follows that lim n→∞ P µn θ t ( S n ) ≥ s = P θ t ( S) ≥ s in P-probability.Since the sequence of subsets Ω τ n does not depend on t and s, convergence holds uniformly in 0 ≤ t < t + s, in P-probability.
It remains to express P θ t ( S) ≥ s in terms of C ∞ (t, s) and F .If σ = 0 then S = S, and by (2.24), P θ t ( S) ≥ s = C ∞ (t, s), which proves (1.24).Otherwise, from the assumption that σ and S in (1.15) are independent r.v.'s on (Ω X , F X , P) for each fixed ω ∈ Ω τ , we get, conditioning on σ, that (2.26) Since (2.26) holds true for each ω ∈ Ω τ uniformly in 0 ≤ t < t + s, (1.23) obtains uniformly in 0 ≤ t < t + s, and inherits the convergence mode of P µn θ t ( S n ) ≥ s , that is to say, the convergence mode of S n .The proof of Theorem 1.6 is now complete.

The special case of the asymmetric trap model on the complete graph (Convergence to renewal processes).
Our aim in this section is twofold: specialize the results of Theorems 1.1, 1.3 and 1.6 to the asymmetric trap model on the complete graph defined through (3.2)-(3.3)(we do not however specify the distribution of the landscape variables, i.e. we do not assume (3.1)), and give sufficient conditions for convergence of the re-scaled clock processes to a renewal process in the case where the auxiliary time scale a n of the re-scaled clock process (1.8) is a constant.For such time scales the sample paths of S n are increasing functions on [0, ∞) that have discontinuities at all integer time points.The natural topological space in which to interpret weak convergence of S n is, here, the space R ∞ of infinite sequences equipped with the usual Euclidean topology (see e.g.[13] section 3).We will use the arrow to denote weak convergence in that space.As in Theorem 1.1, weak convergence in Skorohod topology on D([0, ∞)) will be denoted by ⇒.Set and define , u ≥ 0 .(i) If there exists a sequence a n satisfying a n ↑ ∞ as n ↑ ∞, a σ-finite measure ν on (0, ∞) satisfying (0,∞) (1 ∧ u)ν(du) < ∞, and a function ε ≥ 0 satisfying lim δ→0 ε(δ) = 0, such that, either P-almost surely or in P-probability, and, for all 0 < δ ≤ δ 0 , for some 0 < δ 0 ≤ 1, lim sup then, w.r.t. the same convergence mode, where {(t k , ξ k )} are the marks of a Poisson process on [0, ∞)×(0, ∞) with mean measure dt × dν.
(ii) If, taking a n = 1, there exists a probability distribution ν on (0, ∞) such that, either P-almost surely or in P-probability, (2.28) is verified for all u ≥ 0, then, w.r.t. the same convergence mode, where {ξ k , k ≥ 1} are independent r.v.'s with identical distribution ν.
In the sequel we will adopt the terminology used in [24] and call the sequence {R(k) , k ∈ N} a renewal process of inter-arrival distribution ν (equivalently, of interarrival times ξ k ).As in Theorem 1.3 the extra Condition (A0) on the convergence of the initial increment σ n enables us to deduce convergence of the full clock process S n from that of S n .
(i) If, in addition to the assumptions of assertion (i) of Theorem 2.4, Condition (A0) is satisfied w.r.t. the same convergence mode as in (2.28), then, in this convergence mode, denoting by σ the random variable of (possibly random) distribution function F , the following holds: For S defined in (2.30), where σ and S are independent.Moreover for C ∞ (t, s) defined in (1.22), for all 0 ≤ t < t + s, (2.33) (2.34) (ii) Substituting the assumptions of assertion (ii) of Theorem 2.4 to those of assertion (i) in the statement of assertion (i') above, and leaving the definition of σ unchanged, the following holds: For R defined in (2.31), where σ and R are independent.Moreover, (2.33)-(2.34)hold true with C ∞ (t, s) defined through (2.36) Thus, when a n diverges, S n converges to a delayed subordinator, and it converges to a delayed renewal process otherwise.
Remark 2.6.As in Theorem 1.6, the statement that σ and S are independent in (2.32) has the precise meaning that for each fixed ω ∈ Ω τ , σ and S are independent random variables on the probability space (Ω X , F X , P).The same remark applies to the statement that σ and R in (2.35) are independent.
Specializing the previous theorem to the case where the initial distribution µ n is the invariant measure π n of the jump chain (see ( where the r.h.s. is chain independent.Thus, if a n is a diverging sequence, (1.10) and (1.11) of Conditions (A1) and (A2) of Theorem 1.1 reduce, respectively, to ) . (2.41) In other words the sum appearing in Condition (A1) of Theorem 1.1 is 'ergodic'.A similar observation holds for Condition (A2).
The new part of Theorem 2.4 is assertion (ii), whose elementary proof we now give.
Assume first that there exists a probability distribution ν such that, for all u ≥ 0, (2.28) sequence on the probability space (Ω X , F X , P) since, by (3.8), the chain variables (J n (i), i ∈ N) form an i.i.d.sequence, and since P µn (ξ n,i > u) = ν n (u, ∞) does not depend on i.This means that S n has stationary positive increments.To prove (2.31) it thus suffices to prove that, in P-probability, for each integer k (finite and independent of n), S n (k) d → R(k) (see e.g.[13] p. 30).To this end consider the Laplace transforms Λ n (k, θ) = E µn e −θSn(k) and Λ(k, θ) = Ee −θR(k) , θ > 0. From the assumption that, for all u ≥ 0, (2.28) holds in Pprobability, it follows that there exists a sequence Ω τ n ⊂ Ω τ satisfying lim n→∞ P( Ω τ n ) = 1, and such that, for all large enough n, for all ω ∈ Ω τ n .Let now ω ∈ Ω τ n be fixed, where n will be taken as large as needed.By independence, Λ n (k, θ) = Ee −θξn,i k .From the integration by parts formula E µn e −θξn,i = 1 − θ ∞ 0 e −θu P µn (ξ n,i > u)du, it follows that E µn e −θξn,i − Ee −θξi ≤ sup Proof of Theorem 2.5.We first deal with assertion (i').Eq. (2.32) is proved just as (1.15) of Theorem 1.3.Assuming that for each ω ∈ Ω τ , σ and S in (2.32) are independent random variables on the probability space (Ω X , F X , P), (2.33) is proved in the same way as (1.23) of Theorem 1.6, and the special case σ = 0 of (2.34) is nothing but (1.24).
Let us show that the above independence assumption is verified.For this let ω ∈ Ω τ be fixed.Note that by (3.8) the jump chain (J n (i), i ∈ N) becomes stationary in exactly one step.Namely, for any initial distribution µ n , for all i ≥ 1, P µn (J n (i) = x) = π n (x), x ∈ V n .Thus, for each n, σ n and {S n (k) , k ≥ 1} in (1.7) are independent r.v.'s on (Ω X , F X , P).This in turn implies that, for each n, σ n and {S n (t) , t > 0} in the r.h.s. of (1.9) are independent r.v.'s on (Ω X , F X , P).Thus σ and S(•) are independent, and since this is true for each ω ∈ Ω τ , the claim follows.
We skip the proof of assertion (i"), which is a re-run of the proof of assertion (i') (and, upstream from it, of Theorems 1.3 and 1.6) in the simpler setting of discrete time process.
Proof of Corollary 2.7.Since 1 − P µn (σ n < v) = x∈Vn µ n (x)e −vcnλn(x) (see e.g. the proof of assertion (i) of Theorem 1.3) it follows from (2.27) and the choice µ n = π n that (2.44) Suppose first that the assumptions of assertion (i) of Theorem 2.4 are verified.In view of (2.28) and (2.44), 1 − P µn (σ n < v) → 0 for all v ≥ 0, so that Condition (A0) is satisfied with F (v) = 1, v ≥ 0, w.r.t. the same convergence mode as in (2.28).Eq. (2.37) then follows from (2.34).Suppose next that the assumptions of assertion (ii) of Theorem 2.4 are verified.Reasoning as above we readily see that Condition (A0) is satisfied with . the same convergence mode as in (2.28).Thus, by (2.35), the first increment σ of the limiting renewal process R has the same distribution as the inter-arrival times ξ k of R. Hence, for all 0 ≤ t < t + s, where the last equality is (2.36).Since σ and R in (2.35) are independent, we also have, conditioning on σ and using (2.33), that, for all 0 ≤ t < t + s, (2.46) Equating the r.h.s. of (2.44) with the r.h.s. of (2.45) gives (2.37).The proof of Corollary 2.7 is done.The proof of Corollary 2.7 is done.

Bouchaud's asymmetric trap model on the complete graph.
We now begin the investigation of Bouchaud's asymmetric trap model on the complete graph, which will occupy the rest of the paper.The results we present are the first aging results for a trap model of mean field type which is not a time change of a simple random walk.
This section is organized as follows.In Subsection 3.1 we describe the model and some of its static properties.We then state our main results, first, on the convergence of the time correlation function (Subsection 3.2), and next, on the clock process (Subsection 3.3).

The model.
This model appeared in [16] where it was proposed and studied on various graphs that represent the depths of traps, and whose distribution belongs to the domain of attraction of a positive stable law with parameter α ∈ (0, 1).This means that there exists a function L, slowly varying at infinity, such that Given a parameter 0 ≤ a < 1, the Markov jump process X n has holding time parameters and its jump chain, J n , has one-step transition probabilities p n (x, y) = τ a n (y) y:(x,y)∈En τ a n (y) p n (x, y) = 0 otherwise.When a = 0, J n simply is the homogeneous random walk on G n , whereas when a > 0, J n favors jumps to the neighboring traps of largest depths.Models with a > 0 are called asymmetric as opposed to the symmetric ones where a = 0.
The first rigorous results for the asymmetric trap model were obtained for the graph

Z in [?]
. There, it is shown that the time-time correlation function (1.21) does not exhibit an arcsine aging regime but is sub-aging, and has the same (a-dependent) aging regime for all a ∈ [0, 1].The recent work [1] suggests that on the contrary, on the graphs Z d , d ≥ 3, the asymmetry parameter, a, has no relevance on the aging phenomenon.These results contrast with the case of the complete graph where the asymmetry parameter triggers a dynamical phase transition.More precisely, we show that there exists a positive threshold value in a below which the model exhibits an (a-dependent) arcsine aging regime, whereas above it arcsine aging is interrupted.This phenomenon occurs "on all time scales", i.e. from time scale one up to, and including, the time scale of stationarity.We also show how, on the time scale of stationarity, the model can be driven from an arcsine aging regime to its stationary regime.
From now on we focus on the model where G n (V n , E n ) is the complete graph on V n ≡ {1, . . ., n} that has a loop at each vertex.Clearly X n has a unique reversible invariant measure, denoted by G α,n , which is the Gibbs measure of the model, i.e.
Clearly also, the jump chain J n has a unique reversible invariant measure, π n , given by , x ∈ V n .10).This readily implies that most of the mass of the Gibbs measure is supported by the points x(k) with largest weights (i.e. with deepest traps).In contrast, when α > 1, no single point carries a positive mass asymptotically.In particular, it is not hard to show that lim n→∞ sup x∈Vn G α,n (x) = 0 in P-probability.Here the Gibbs measure "resembles a uniform measure".
It is now easy to see why the chain X n should undergo a dynamical phase transition at the value a = α.By (3.3) and (3.5), where, because π n = d G α/a,n , π n undergoes a transition at the value a = α.Thus, when a > α the jump chain should resemble a symmetric random walk, and may explore the entire landscape.In contrast, when a < α the jump chain will quickly go and visit a trap of extreme depth from which it will not be able to escape, unless time is measured on the scale of stationarity.

Aging of C n (t, s).
We now state our main results on the asymptotic behavior of the time-time correlation function C n (t, s) of (1.21).We cover all choices of a and α with 0 < α < 1, 0 ≤ a < 1, and a = α, and any choice of the time scale c n up to and including the time scale of stationarity.All these results are obtained for a special choice of the initial distribution −a) .This relation prompts us to call r n ≡ c 1/(1−a) n a space scale.We will distinguish three types of space scales: the constant scales (which simply are constant sequences), the intermediate, and the extreme scales.Our first theorem establishes that if a < α then C n (t, ρt) ages on all time scales.Theorem 3.3 (Arcsine aging regime).Assume that a < α and take µ n = π n .
(i) If r n is a constant space scale then, P-almost surely, for all ρ > 0, (3.11) (ii) If r n is an intermediate space scale then for all t ≥ 0 and all ρ > 0, (3.12) This holds P-a.s. if b n is regularly varying at infinity with index ζ < 1, and in P-probability otherwise.
(iii) If r n is an extreme space scale then, for all ρ > 0, in P-probability,   At a heuristic level Theorem 3.4 is easy to understand.For a > α the initial distribution µ n behaves like a "low temperature" Gibbs measure, namely µ n = d G β,n , β = α/a < 1.This means that almost all its mass is carried by traps whose size is of the order of extreme space scales.Now the mean waiting time in such deep traps diverges with n whenever time is measured on a scale which is small compared to extreme scales: the chain gets stranded.
The last theorem below is valid for all 0 ≤ a < 1.It states that, as expected, on extreme time scales, taking the infinite volume limit first, the process reaches stationarity as t → ∞.As before let {γ k } denote the marks of PRM(µ) on (0, ∞), and define where r n is an extreme space scale.The following holds for all 0 ≤ a < 1, a = α: where d = denotes equality in distribution.
(ii) If µ n = π n , for all s > 0, Comparing (3.13) and (3.18) we see that when time goes from 0 to ∞, for a < α, the chain moves out of an arcsine aging regime and crosses over to its stationary regime.
Aging is then interrupted.

Convergence of the clock process.
This section contains the ingredients needed in the proofs of Theorem 3.3, 3.4 and 3.5.Consider the pure clock process S n of (1.8).For a < α let a n be any sequence such that a n r a n /(b n Eτ a ) ∼ 1, and for a > α, take a n ∼ 1.As before µ n = π n . .We show below that S n has different limits depending on the choice of space scales (constant, intermediate, and extremal) and the value of a.The proofs of these results rely on Theorem 2.4 whose notations we now use.Proposition 3.6.Let r n be a constant space scale.For a < α and τ ≡ τ (1), let ν cst,− be the measure defined through , u > 0 . ( (i) If a < α then S n R cst,− P-a.s., where R cst,− is the renewal process of inter-arrival distribution ν cst,− .
(ii) If a > α then, S n R cst,+ in P-probability, where R cst,+ is the degenerate renewal process of inter-arrival distribution ν cst,+ = δ ∞ .
The lemma below shows that ν cst,− (u, ∞) is regularly varying at infinity with index 1−a for some function (u) slowly varying at infinity.Proposition 3.8.Let r n be an intermediate space scale.For a < α, let ν int,− be the measure on (0, ∞) defined through (i) If a < α then, S n ⇒ S int,− where S int,− is the stable subordinator of Lévy measure ν int,− .Convergence holds P-a.s. if b n = n ζ for some 0 < ζ < 1, and in P-probability otherwise.
(ii) If a > α then, S n R int,+ in P-probability, where R int,+ is the degenerate renewal process of inter-arrival distribution ν int,+ = δ ∞ .
To formulate the results on extreme scales recall that for µ defined in (3.6), {γ k } denote the marks of PRM(µ) on (0, ∞), and introduce the re-scaled landscape variables: Proposition 3.9.If r n is an extreme space scale then both the sequence of re-scaled landscapes (γ n (x), x ∈ V n ), n ≥ 1, and the marks of PRM(µ) can be represented on a common probability space (Ω, F, P) such that, in this representation, denoting by S n the process (1.8), the following holds.For a < α, resp.a > α, let ν ext,− , resp.ν ext,+ be the random measures on (0, ∞) defined on (Ω, F, P) through Then, P-almost surely, where S ext,− is the subordinator of Lévy measure ν ext,− , and R ext,+ is the renewal process of inter-arrival distribution ν ext,+ .
Here the subordinator S ext,− is not stable.However ν ext,− (u, ∞) is regularly varying at 0 + with index α−a 1−a : Lemma 3.10.P-a.s., By Corollary 2.7, Theorem 3.3, Theorem 3.4 and Theorem 3.5 are direct consequences of the above results and Theorem 1.8 (see also Theorem 7.2 of Appendix 7.1 on renewal theory for the discrete time version of Theorem 1.8; for the proof of Theorem 3.5, (i), see Theorem 7.3).Similarly, Proposition 3.6, (ii), and Proposition 3.8, (ii), are simple consequences of Theorem 2.4, (ii).We omit their proofs.Details can be found in [26].
The rest of the paper is devoted to the proofs of Proposition 3.6, (i), Proposition 3.8, (i), and Proposition 3.9.As they rely on very different tools, we give them in three separate sections (Section 4, 5, and 6 respectively).

Constant scales.
Proof of Proposition 3.6, (i).It suffices to check that the conditions of Theorem 2.4, (ii), are satisfied P-almost surely.For all a < α Eτ a < ∞ so that Eτ a e −u/τ (1−a) ≤ Eτ a < ∞ for all u ≥ 0. Thus, for all u ≥ 0, the strong law of large numbers applies to both the numerator and denominator of (2.27), yielding lim n→∞ ν n (u, ∞) = ν cst,− (u, ∞) P-almost surely.Together with the monotonicity of ν n (u, ∞) and the continuity of the limiting function ν cst,− (u, ∞), this implies that there exists of a subset Ω τ 1 ⊂ Ω τ of the sample space Ω τ of the τ 's with the property that P(Ω τ 1 ) = 1, and such that, on Ω τ The proof of Proposition 3.6 is done.
Proof of Lemma 3.7.Let a < α.For u ≥ 0 and y ≥ 0 set ϕ u (y) = y a e −u/y (1−a) . Integrating by parts, Eϕ u (τ ) = To deal with I n (u) we use that h n (z) → z −α , n → ∞, where the convergence is uniform in 0 ≤ z ≤ 1, since for each n, h n (z) is a monotone function, and since the limit, z α , is continuous.Thus for all ε > 0 there exists n(ε) such that for all n ≥ n(ε) , ∀u > 0 .
. Combining these observations with (5.5) we conclude that for all u > 0 and all large enough n there exist constants 0 ≤ c 0 , c 4 < ∞ and 0 < c 2 ≤ 1, that depend only on α and a, such that I n (u) ≤ c 0 + c 4 u −1+c2 .
To bound I n (u) we note that h n (z) = x −α (L(r kn z)/L(r kn )) and use that by Lemma 7.5, for each x > 1 and large enough n, Integrating by parts I n (u) ≤ (1+δ n ) Using Lemma 5.5, (i), we easily see that for all u > 0, which is valid for all u > 0. Indeed, if u ≤ v := ( i b kn ) 1−a , then the last ineqality follows from Lemma 5.5, (ii); if on the contrary u > v, then, by Lemma 5.5,(iii) Inserting our bounds on I n (u) and I n (u) in (5.4) and (5.2) successively yields (5.8) Consider the first sum in the right hand side of (5.8).Integration by parts yields where h n is given in (5.3).Write J n (u) = J n (u) + J n (u), where J n (u) = 1 0 ϕ u (x)h n (x)dx and J n (u) = ∞ 1 ϕ u (x)h n (x)dx.On the one hand, proceeding as we did to establish (5.6), we obtain that lim n→∞ J n (u) = 1 0 x α dx for all u > 0. On the other hand, using the bounds on h n from the paragraph above (5.6), x α−εn dx , (5.10) where , which is finite for u > 0, dominated convergence applies and yields, lim n→∞ J n x α dx.Collecting our results we get ( (5.12) To bound J n (ϕ −1 u (i/b n )) we use the upper bound (5.10) (which is valid for all n large enough) and, proceeding as in the paragraph below (5.6) (but replacing ϕ 2 u by ϕ u ), we readily obtain that J . (5.13) Turning to the last term in the right hand side of (5.12), we have where φ(y) is defined in (5.1).Using furthermore that r α n P(τ (5.15) Consider the last quotient in the r.h.s. of (5.14).The bound φ u i bn ≥ 1 of Lemma 5.5 (valid for all u ≥ 0) together with Lemma 7.5 imply that there exist positive sequences ε n and δ n that verify ε n ↓ 0, δ n ↓ 0 as n ↑ ∞ and such that, for all n large enough, this quotient is bounded above by (1 + δ n )(i/b n ) εn a .Inserting this in (5.15) we obtain, Proof of Lemma 5.2.It suffices to show that for some n 0 < ∞, (5.17) Also note that for all fixed y > 1, b n ϕ u (y/r kn ) is an increasing function of n for all n ≥ n 0 and some n 0 < ∞.Therefore, taking n 0 = max(n 0 , n 0 ), (5.18) Proceeding as in (5.14)-(5.16) to bound the last probability we get that, for some ε n ↓ 0, i≥i0 where α−εn a > 1 for all n large enough.From this and the assumption that b n < n ζ , ζ > 0, it follows that the r.h.s. of (5.19) is bounded above by which tends to zero as i 0 → ∞, proving (5.17).The proof of Lemma 5.2 is done.
To conclude the proof of Proposition 5.1 it remains to handle the intermediate values Observe that for such values of m, we have: (5.21) the Continuous Mapping appropriate continuous functionals of Υ n converge to the corresponding functionals of Υ.This convergence however is in distribution only, and this is not enough for our purposes.The usual way out of this difficulty is to think of weak convergence from Skorohod's representation Theorem and replace the sequence (γ n (x), x ∈ V n ) by a new sequence with identical distribution, but almost sure convergence properties.This strategy was first implemented in the context of an aging system by Fontes et al. [25], and often used since.We in turn adopt it using, however, an explicit representation of the re-scaled landscape.The latter is given in Subsection 6.1.In Subsection 6.2 we consider the model obtained by substituting the representation for the original landscape and prove Proposition 3.9.The final Subsection 6.3 contains the proof of Lemma 3.10.

A representation of the re-scaled landscape.
The representation we now introduce is due to Lepage et al. [32] and relies on an elementary property of order statistics.Let τn (1) ≥ • • • ≥ τn (n) and γn (1) ≥ • • • ≥ γn (n) denote, respectively, the landscape and re-scaled landscape variables, (τ (x), x ∈ V n ) and (γ n (x), x ∈ V n ), arranged in decreasing order of magnitude.For u ≥ 0 set G(u) = P(τ (x) > u) and G −1 (u) := inf{y ≥ 0 : G(y) ≤ u} . (6.1) Let (E i , i ≥ 1) be a sequence of i.i.d.mean one exponential random variables defined on a common probability space (Ω E , F E , P).We will now see that both the ordered landscape variables and the limiting point process Υ can be expressed in terms of this sequence.Set, for k ≥ 1, and, for 1 Proof.Note that G is non-increasing and right-continuous so that G −1 is non-increasing and right-continuous.It is well known that if the random variable U is a uniformly distributed on [0, 1] we may write τ (0) In turn it is well known (see [24], Section III.3) that if (U (k), 1 ≤ k ≤ n) are independent random variables uniformly distributed on [0, 1] then, denoting by Ūn (1) ≤ • • • ≤ Ūn (n) their ordered statistics, ( Ūn (1), . . ., Ūn (n))  Note that for this it is enough to take the limit along the integers since, ϕ(m 1/α γ k ) being a strictly increasing function of m, m m ϕ( m 1/α γ k ) .In the worst situation α < la for all l > 1 (indeed if α ≥ la, then σ (l) M ≤ α 1−a Γ α−la 1−a < ∞).Let us thus assume that α < la for all l > 1.In this case, σ (recall that by assumption 2a > α and a < α).Choosing ε sufficiently small so as to guarantee that δ 2 m m/σ (2) M ≤ log(4/3) , (6.30) the bound (6.28) becomes P (A m ) ≤ 2 m 2 .Thus m P (A m ) ≤ ∞ which, invoking the first Borel-Cantelli Lemma, proves (6.20).
The proof of Lemma 3.10 is complete.
We summarize here what we need to know about renewal theory for subordinators and renewal processes.Recall first that:

( 2 . 22 )
With this definition the time-time correlation function (1.21) may be rewritten as

(2. 27 ) 2 . 4 .
Theorem Consider the asymmetric trap model on the complete graph on time scale c n .The following holds for any choice of the initial distribution µ n .

Definition 3 . 1 .( 3 . 9 )
We say that a positive and diverging sequence r n is: (i) an intermediate space scale if there exists an increasing and diverging sequence b n > 0 such that b n n = o(1) and b n P(τ (x) ≥ r n ) ∼ 1 , (ii) an extreme space scale if there exists an increasing and diverging sequence 0 < b n ≤ n such that b n n ∼ 1 and b n P(τ (x) ≥ r n ) ∼ 1 .

(3. 10 ) 3 . 2 .
Remark These scales are well separated.Namely, if r cst n , constant, an intermediate and an extreme space scale, then r cst n

(3. 13 )Theorem 3 . 4 (
When a > α, none of the time scale and limiting procedures of Theorem 3.3 yields aging: Stranded in deep traps).Assume that a > α and take µ n = π n .(i)If r n is a constant or intermediate space scale then, for all 0 ≤ t < t + s, lim n→∞ C n (t, s) = 1 in P − probability.
whereas by Lemma 5.5, (ii), for all y ≥ i b kn , ϕ −1 v (y) ≤ (e i b kn ) 1/a .Collecting our bounds we get that for all large enough n, b kn I n (u) ≤ c 5 b kn + c 6 b
d → R(k) in P-probability.
The proof of Lemma 5.3 is done.