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AGING IN THE GREM-LIKE TRAP MODEL VE´RONIQUEGAYRARD,ONURGU¨N ABSTRACT. The GREM-like trap model is a continuous time Markov jump process ontheleavesofafinitevolumeL-leveltreewhosetransitionratesdependonatrapping landscapebuiltontheverticesofthewholetree. Weprovethatthenaturaltwo-time correlationfunctionofthedynamicsagesintheinfinitevolumelimitandidentifythe limiting function. Moreover, we take the limit L → ∞ of the two-time correlation function of the infinite volume L-level tree. The aging behavior of the dynamics is characterizedbyacollectionofclockprocesses,oneforeachlevelofthetree.Weshow 5 thatforanyL,thejointlawoftheclockprocessesconverges. Furthermore,anysuch 1 0 limit can be expressed through Neveu’s continuous state branching process. Hence, 2 thelattercontainsalltheinformationneededtodescribeagingintheGREM-liketrap modelbothforfiniteandinfinitelevels. n a J 3 1 1. INTRODUCTION ] Trap models are main theoretical tools to quantify the out-of-equilibrium dynamics R of spin glasses, and more specifically their aging behavior (see [11] for a review). In- P . troduced in this context by J.P. Bouchaud [10] to model the dynamics of mean field h t spin glasses such as the REM, GREM, and p-spin SK models, trap models are simple a m Markov jump processes that describe dynamics on microscopic spin space in terms of thermally activated barrier crossing between the valleys (or traps) of a random land- [ scape on reduced state space devised so as to retain some of the key features of the 1 freeenergylandscapeoftheunderlyingspinsystem. Activatedagingoccursif,ontime v 6 scales that diverge with the size of the system, the process observed through suitably 8 chosentime-timecorrelationfunctionsbecomesslowerandslowerastimeelapses. 9 2 Thefirstrigorousconnectionbetweenthemicroscopicdynamicsofaspinsystemand 0 a trap model was established in [2, 3] where it is proved that a particular Glauber dy- . 1 namics of the REM has the same aging behavior as Bouchaud’s symmetric trap model 0 on the complete graph or “REM-like trap model” [12]. This result was followed up by 5 1 a series of results yielding a detailed understanding of the aging behavior of the REM : v for a wide range of time scales and temperatures, and various dynamics [4, 21, 23]. i Spin glasses with non-trivial correlations, namely the p-spin SK models, could also be X dealt with albeit in a restricted domain of the time scales and temperature parameters, r a whereitwasprovedthatagingisjustasintheREM[1,5,13,14]. Thisreflectsthefact thatthedynamicsisinsensitivetothecorrelationstructureoftherandomenvironment. Altough we do expect that, on longer time scales, the aging dynamics of the p-spin SKmodelsbelongsto(a)differentuniversalityclass(es),thereisyetnorigorousresult supportingthisidea. At a heuristic level, possible effects of strongly correlated random environments on activated aging were first modeled by Bouchaud and Dean [12] using a trap model Date:January14,2015. 1991MathematicsSubjectClassification. Primary82C44,60K37,60J27;secondary60G55. Keywordsandphrases. Randomwalk,randomenvironment,trapmodels,aging,spinglasses,aging. 1 AGINGINTHEGREM-LIKETRAPMODEL 2 whoserandomtrapsareorganizedaccordingtoahierarchicaltreestructureinspiredby Parisi’sultrametricconstruction[24]. Morerecently,SasakiandNemoto[28,29]intro- ducedatrapmodelonatreewithaviewtomodelingtheagingdynamicsoftheGREM. From a mathematical perspective this GREM-like trap model seems more promising. Indeed a detailed (rigorous) analysis of the statics of the GREM (as well as that of a more general class of continuous random energy models, or CREMs) is available (see [15] and the references therein), making it plausible to expect that the GREM-like trap model correctly predicts the behavior of the aging dynamics of the GREM itself, at leastinsomedomainofthetemperatureandtimescaleparameters. With this in mind, our aim in this paper is two-fold. Firstly, we want to identify the agingbehavioroftheGREM-liketrapmodelasthebranchsizeofthetree,n,diverges, andalsoasthenumberoflevels,L,divergesafterthelimitn → ∞istaken. Secondly, we want to emphasize that Neveu’s continuous state branching process (hereafter abbreviated CSBP) naturally describes aging in the GREM-like trap model, namely, the aging behavior of the dynamics is encoded in a collection of clock processes and all possiblelimitsoftheseclockprocessesareextractedfromNeveu’sCSBP. 1.1. Sasaki and Nemoto GREM-like trap model [28]. We start by specifying the underlying tree structure. For n ∈ N, we write [n] = {1,...,n}. Let L ∈ N be fixed andsetV| = [n]k fork = 1,...,L. WedefinetherootedL-levelperfectn-arytreeby k L (cid:91) T = V| , (1.1) L k k=0 where V| = {∅} is the root. We use the notation µ| = µ µ ···µ for a generic 0 k 1 2 k element of V| . We sometimes simply use µ for µ| ∈ V| . By convention the root k L L belongs to the 0-th generation of the tree and µ| to the k-th generation. For 0 ≤ k < k 1 k ≤ L, we say that µ| ∈ V| is an ancestor of µ| ∈ V| if µ| = µ| . The set 2 k1 k1 (cid:48) k2 k1 k1 (cid:48) k1 V| consistsoftheleavesofthetree,thatis,theverticesthatdonothaveanyoffspring. L Whenever convenient we add the root to the notation by writing µ| = µ µ ···µ , k 0 1 k where µ ≡ ∅. Note that the trees T are parametrized by n ∈ N. However, for 0 L notationalconveniencewedonotkeepninthenotation. ∅ µ1 V1 µ1µ2 µ1µ02 V2 V3 µ1µ2µ3 µ1µ02µ03 FIGURE 1. ArepresentationofTL withL = n = 3. 1 AGINGINTHEGREM-LIKETRAPMODEL 3 Giventwoverticesµ| ,µ| ∈ V| ,wedenoteby k (cid:48) k k (cid:10) (cid:11) (cid:8) (cid:9) µ| ,µ| = max l : µ| = µ| , µ,µ ∈ V| , (1.2) k (cid:48) k l (cid:48) l (cid:48) k the generation of their last common ancestor (hereafter abbreviated g.l.c.a.). The trapping landscape, or random environment, is a collection of independent random vari- T ablesontheverticesofthetree , L (cid:8) (cid:9) λ(µ| ) : µ ∈ V| , k = 1,...,L , (1.3) k L whereλ(∅) ≡ 0and,fork = 1,...,L, P(cid:0)λ 1(µ| ) ≥ u(cid:1) = u αk,L, u ≥ 1. (1.4) − k − Here,theα ’sarerealnumberssatisfying k,L 0 < α < α < ··· < α < 1, (1.5) 1,L 2,L L,L which implies that the λ 1’s are heavy tailed. We assume that the two-parameter se- − quences (in n and L) of random environments are independent and defined on a com- monprobabilityspace(Ω,F,P). WedenotebyEtheexpectationunderP. For k = 1,...,L, let Y| be the discrete time Markov chain on V| with transition k k probabilities Wk(µ|k,µ(cid:48)|k) = (cid:104)µ|k,µ(cid:48)(cid:88)|k(cid:105)∧(k−1) (cid:0)1−λ(µ|l)(cid:1)n(cid:81)k kll(cid:48)−=1l+1λ(µ|l(cid:48)), (1.6) − l=0 with the convention that (cid:81)kj=−k1λ(µ|j) = 1. The GREM-like trap model, denoted by (X (t) : t ≥ 0), is a continuous time Markov jump process on the set of leaves of the L tree,V| ,whosetransitionratesaregivenby L w (µ,µ) = λ(µ)W (µ,µ). (1.7) L (cid:48) L (cid:48) Thus,weseethatafterwaitingataleafµanexponentialtimewithmeanvalueλ 1(µ), − the process jumps to µ with probability W (µ,µ). We also see that X is a reversible (cid:48) L (cid:48) L processwhoseuniqueinvariantmeasureassignstoµ ∈ V| themass(cid:81)L λ 1(µ| ). L k=1 − k A comment on the form of the transition probabilities W (µ,µ) is now in order. L (cid:48) Note that λ(µ|l) ∈ [0,1], 1 ≤ l ≤ L, so that the product (1 − λ(µ|l))(cid:81)Ll(cid:48)=−l1+1λ(µ|l(cid:48)) appearing in the summation in (1.6) is a probability. In physicists’ term, this is the probability that, along the transition from µ to µ, the system is activated from µ to (cid:48) µ| butnotfromµ| toµ| ,meaningthattheprocessjumpsoutofthetrapsattached l+1 l+1 l to the vertices µ| , l + 1 ≤ l ≤ L − 1, but stays stranded in the trap attached to l (cid:48) (cid:48) µ| . The process then choses its next state uniformly at random among the nL l leaves l − descendingfromµ| . l Throughout this paper the initial distribution of X is taken to be the uniform dis- L tribution on V| . We embed the distributions of the chains X (t) for each n ∈ N and L L L ∈ N, on a common probability space whose distribution and expectation we denote by P and E, respectively. We suppress any references to the trapping landscape in the notation. The two-time correlation function that we use to quantify aging in the GREM-like trapmodelisdefinedasfollows. Fork = 1,...,L,wefirstset (cid:0)(cid:10) (cid:11) (cid:1) Π (t,s) = P X (t),X (t+u) ≥ k, ∀u ∈ [0,s] , t,s > 0. (1.8) k L L AGINGINTHEGREM-LIKETRAPMODEL 4 Wenextchooseanon-decreasingsmoothfunctionq : [0,1] → [0,1]withq(0) = 0and q(1) = 1,andthenuseittodefinethetwo-timecorrelationfunction L (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (cid:88) k k −1 C (t,s) = q −q Π (t,s). (1.9) L L L k k=1 Let us point out the key connections between the GREM-like trap model and the GREM(ortheCREM,itsgeneralizationtocontinuoushierarchies). Forthiswerelyon the paper [15] by Bovier and Kurkova, that reviews the state of the art on the statics of (cid:10) (cid:11) thesemodels. Using(1.2)onenaturallydefinesadistance,1−L 1 µ,µ ,µ,µ ∈ V| , − (cid:48) (cid:48) L onthesetofleavesofthetree,whichisnothingbuttheultrametricdistance(see(1.1)in [15])thatendowsthespaceofspinconfigurationsoftheGREM.Observemoreoverthat the function q in (1.9) is the analogue of the function A that enters the definition of the covariance structure of the GREM (see (1.2) in [15]). Thus C (t,s) in (1.8) naturally L playstheroleofthespin-spincorrelationfunctionbetweentwoconfigurationsofsome microscopic GREM dynamics observed at times t and t + s. Let us now turn to the trapping landscape (1.3). Its key features are modeled on those of the point process of extremes of the GREM’s Botzman weights at temperature T. Under certain conditions ontheparametersofthemodel,thisprocessisknowntoconverge,asthesystem’ssize diverges, to Ruelle’s (non-normalized) Poisson cascades, a hierarchical point process constructed from a collection of Poisson point processes of intensity c x (1+T/Tk)dx, k − one for each level, k, of the hierarchy; here T is an associated critical temperature, k and all processes that may result from this limiting procedure must be such that T > 1 ··· > T (see Theorem 2.3, Theorem 3.2, and definition 3.1 in in [15]). Hence, at low L enough temperature, T/T < ··· < T/T < 1. Setting α = T/T in (1.5), the 1 L k,L k non-normalizedinvariantmeasure(cid:81)L λ 1(µ| ),µ ∈ V| ,oftheprocessX in(1.7) k=1 − k L L models Ruelle’s (non-normalized) cascade, while each λ 1(µ| ) models the depth of a − k trap at µ| . Based on heuristic ideas derived from metastability, the GREM dynamics k is then replaced in an ad hoc manner by the process X . We refer to Section 1.2 of [3] L for a more precise explanation of the correspondence between microscopic dynamics and trap models in the 1-level (REM-like) trap model. Note finally that in the GREM, the size of the hypercubes of different levels can vary and this correspond to varying branch sizes of the tree in the GREM-like trap model. Our results are valid for a range of trees with varying branch sizes, however, in order to keep the notations simple we optedtoworkwithregulartrees. Remark 1.1. Observe that the two-time correlation function C can be expressed in L termsofthedistributionoftheoverlapobservedbythedynamics. Moreprecisely, (cid:0) (cid:0) (cid:1)(cid:1) C (t,s) = E q T (t,s) , (1.10) L L where (cid:26) (cid:27) 1 T (t,s) = sup l : (cid:104)X (t),X (t+u)(cid:105) ≥ l, ∀u ∈ [0,s] . (1.11) L L L L Indeed,inourproofsweidentifythelimitingdistributionof(1.11),andthus,ourresults arevalidforanychoiceofq. Following the by now well-established strategy to analyze the aging properties of disorderedsystems,foreachlevelk,weintroducetheso-calledk-thlevelclockprocess, S ,thatisthepartialsumprocessdefinedthrough k,L (cid:12) XL(t)(cid:12)k = Y(Sk←,L(t))|k, t ≥ 0. (1.12) AGINGINTHEGREM-LIKETRAPMODEL 5 In contrast with earlier works, in order to fully describe the behavior of the two-time correlation function we need to control a whole collection of clock processes, one for eachlevelofthetree. The following description of S is going to be useful. We say that a jump of X is k,L L fromalevelk ∈ {1,...,L},ifthelastactivatedvertexisonthe(k −1)-thlevelofthe tree, and that a jump is beyond and including the kth level if it is a jump from a level in {1,...,k}. Then, S (i) is the time it takes for the dynamics X to make i jumps k,L L beyondandincludingthek-thlevelofthetree. 1.2. Convergence of the two-time correlation function. To study aging one has to choose a time scale c on which the dynamics is observed. In this paper we work with n c oftheform n c = nρ, ρ > 0. (1.13) n We then say that the system is aging on the k-th level, for θ > 0, if Π (c ,θc ) has k n n a non-trivial limit that depends only on θ, and that it is non-aging on the k-th level if Π (c ,θc )convergesto0. Fork = 1,...,L,weset k n n 1 1 1 d = + +··· −(L−k). (1.14) k,L α α α L,L L 1,L k,L − Notethatby(1.5) d > d > ··· > d > 1. (1.15) 1,L 2,L L,L Foragivenexponentρ,wedefinel (ρ) ∈ {1,...,L}by ∗ l (ρ) = sup{k : ρ < d }. (1.16) ∗ k,L Forα ∈ (0,1)letAsl betheclassicalarcsinelawdistributionfunction α sinαπ (cid:90) u Asl (u) = x α(1−x)α 1dx, u ∈ [0,1]. (1.17) α π − − 0 Theorem 1.1. For any fixed L ∈ N and ρ ∈ (0,d )\{d ,...,d } set l = l (ρ) 1,L L,L 2,L ∗ ∗ and α¯ = (cid:81)l∗ α . There exists a subset Ω ⊆ Ω with P(Ω ) = 1 such that for any k i=k i,L L L environmentinΩ ,foranyθ > 0, L (cid:16) (cid:17) (cid:88)l∗ (cid:20) (cid:18)k(cid:19) (cid:18)k −1(cid:19)(cid:21) (cid:16) 1 (cid:17) lim C c ,θc = q −q Asl . (1.18) n L n n L L α¯k 1+θ →∞ k=1 Remark 1.2. Indeed, we are going to prove that, under the assumptions of Theorem (cid:16) (cid:17) 1.1, for k = 1,...,l , Π (c ,θc ) → Asl 1 , and for k = l (ρ) + 1,...,L, ∗ k n n α¯k 1+θ ∗ Π (c ,θc ) → 0. k n n Thecollectionin(1.14)canbeseenascriticaltimescaleexponents. Moreprecisely, as a consequence of Theorem 1.1, if ρ < d , the dynamics is aging on the k-th level, k,L andifρ > d ,itisnon-agingonthek-thlevel. Then,thelevels1,...,l (ρ)areaging k,L ∗ whereas the levels below l (ρ) are non-aging, and the inequalities in (1.15) reflect that ∗ asthetimescaleofobservationgetslongertheagingbehaviordisappearsfrombottom tothetopofthetree. Next, we take the L → ∞ limit of the limiting two-time correlation function in (1.18). In order to do this, we need to define a family of collections of α parameters AGINGINTHEGREM-LIKETRAPMODEL 6 satisfying (1.5). Let R : [0,1] → [0,∞) be a strictly increasing, strictly concave, smoothfunctionwithR(0) = 0. ForL ∈ N,set (cid:110) (cid:16) (cid:17)(cid:111) (cid:0) (cid:1) (cid:0) (cid:1) α = exp − R k/L −R (k −1)/L , k = 1,...,L. (1.19) k,L As a result, the collection {α ,...,α } satisfies (1.5) for all L ∈ N. Note that 1,L L,L although for a given collection {α ,...,α } as in (1.5), one can find a function 1,L L,L R satisfying (1.19), it is not necessarily possible to do that for a given family of such collections {α ,...,α : L ∈ N}. Also observe that, in the light of Theorem 1.1, 1,L L,L the choice of (1.19) is natural for an infinite levels limit because the infinite product of α parametersisnon-trivial. Foragivenexponentρ > 0,weset r (ρ) = sup{s ≥ 0 : R(1)−R(s)+1 > ρ}. (1.20) ∗ We will see in Section 3 (see (3.23) and the paragraph before it) that r (ρ) is the limit ∗ of the ratio l (ρ)/L as L diverges. As a result, there is aging for ρ such that r (ρ) > 0. ∗ ∗ SinceR(0) = 0,thelatterisequivalenttoρ < R(1)+1. Corollary 1.2. Let ρ ∈ (0,R(1) + 1) \ {d ,...,d } for all L large enough. Set L,L 2,L (cid:8) (cid:9) r = r (ρ) and let α(x) = exp − (R(r ) − R(x)) for x ∈ [0,r ]. There exists a ∗ ∗ ∗ ∗ subsetΩ ⊆ ΩwithP(Ω) = 1suchthatforanyenvironmentinΩ,foranyθ > 0, (cid:48) (cid:48) (cid:48) (cid:16) (cid:17) (cid:90) r∗ (cid:16) 1 (cid:17) lim lim C c ,θc = q (x)Asl dx. (1.21) L→∞n→∞ L n n 0 (cid:48) α(x) 1+θ Remark 1.3. The result in Corollary 1.20 can be extended to functions q that are right continuous and left limits. However, since the formula in 1.21 is very transparent and neat,weuseonlysmoothfunctions. Letusdiscusstheconnectionofourresultswithearlierliteratureandcommentabout some directions for future research. Observe that our results do not cover the time scalesc ∼ ndk,L. Moreover,varyingbranchsizescanyieldtimescalesthatarecritical n for several levels simultaneously. For these levels the inequalities in (1.15) become equalities. The cases where the branch sizes are fine tuned in such a way that all the critical time scale exponents are the same were studied in [19]. There, the authors constructed a K process in an infinite L-level tree and proved that for any critical time scale, the scaling limit of the GREM-like trap model exists and is given by such an L- levelK process. In[18],theauthorsconstructed,foraspecialchoiceofparameters,an infinite levels, infinite volume K process and showed that it is the limit L → ∞ of the L-levelK processobtainedin[19]. Aninvestigationofintermediatecases,wherethere are times scales that are critical for some levels and aging for others, would require a non-trivialcombinationofanalysisofthispaperandthatof[19]. 1.3. Convergence of the clock processes. In this section we state our results on the convergence of the clock processes. We express the limiting clock processes using Neveu’s CSBP which is a time-homogeneous Markov process (W(r) : r ≥ 0) whose semigroupischaracterizedby E(cid:2)e−κW(r)|W(0) = t(cid:3) = exp(cid:0)−tκe−r(cid:1), κ > 0,t ≥ 0. (1.22) We write W(r,t)forW startingfromt ≥ 0, thatis,W(0,t) = t. UsingKolmogorov’s extension theorem one can construct a process(W(r,t) : r,t ≥ 0) such thatW(·,0) ≡ 0 and, W(·,t + s) − W(·,t) is independent of (W(·,c) : 0 ≤ c ≤ t) and has the AGINGINTHEGREM-LIKETRAPMODEL 7 same law as W(·,s). Hence, for any fix r ≥ 0, the right continuous version of W(r,·) has independent, stationary increments.From its Laplace transform we see that it is an e−r-stable subordinator with Laplace exponent κe−r. Moreover, by (1.22), W(r +p,·) has the same distribution as the Bochner subordination of W(r,·) with the directing processW (p,·),whereW (p,·)isanindependentcopyofW(p,·),thatis, (cid:48) (cid:48) d W(r+p,·) = W(r,W (p,·)). (1.23) (cid:48) The above description is taken from [7] where it is pointed out that, in general, (1.23) allowstoconnectCSBPsandBochner’ssubordination,aconnectiondevelopedfurther inthefollowing: Proposition 1.3 (Proposition 1 in [7] applied to Neveu’s CSBP). On some probability (cid:0) (cid:1) spacethereexistsaprocess Z (t) : 0 ≤ p ≤ r, andt ≥ 0 suchthat: p,r (cid:0) (cid:1) (i)Forevery0 ≤ p ≤ r,Z = Z (t) : t ≥ 0 isane (r p)-stablesubordinatorwith p,r p,r − − Laplaceexponentκe−(r−p). (ii)Foranyintegerm ≥ 2and0 ≤ r ≤ ··· ≤ r thesubordinatorsZ ,Z ,..., 1 m r1,r2 r2,r3 Z areindependentand rm 1,rm − Z (t) = Z ◦···◦Z (t), ∀t ≥ 0 a.s. (1.24) r1,rm rm 1,rm r1,r2 − (cid:8) (cid:9) (cid:8) (cid:9) Finally, Z (t) : r ≥ 0,t ≥ 0 and W(r,t) : r,t ≥ 0 have the same finite 0,r dimensionaldistributions. Toeachlevelofthetreeweassignapairofsequences,a (k)andc (k),definedby n n (cid:26) n1+ρ dk,L k = L,...,l +1, a (k) = − ∗ (1.25) n nα¯k(1+ρ−dl∗,L) k = l∗,...,1, and c (k) = a (k +1), k = 1,...,L−1, c (L) = c . (1.26) n n n n Note that as n → ∞, a (k) (cid:28) n and c (k) = a (k)1/αk,L for all k ≤ l (i.e. for all the n n n ∗ aging levels), whereas a (k) (cid:29) n and c (k) = a (k)n1/αk,L 1 for all k ≥ l +1 (i.e. n n n − ∗ forallthenon-aginglevels). Moreover,undertheassumptionsofTheorem1.1,inboth casesthedecayorgrowthofa (k)/nisatleastpolynomial. n Fork = 1,...,L,wedefinetherescaledclockprocesses S ((cid:98)ta (k)(cid:99)) S(n)(t) = k,L n , t ≥ 0, (1.27) k,L c n where we set S (0) = 0. Hence, S(n) ∈ D([0,∞)) where D([0,∞)) denotes the k,L k,L space of ca`dla`g functions on [0,∞). The following is our main result on the conver- genceofclockprocesses. Theorem 1.4. For any L ∈ N there exists a subset Ω(cid:101) ⊆ Ω with P(Ω(cid:101) ) = 1 such that L L forsomepositiveconstantsb ,...,b setting 1,L l ,L ∗ Z(cid:101) (·) = Z (b ·), (1.28) k,l R((k 1)/L),R(l /L) k,L ∗ − ∗ foranyenvironmentinΩ(cid:101) ,asn → ∞ L (cid:16) (cid:17) (cid:16) (cid:17) Sk(n,L) : k = 1,...,l∗ =⇒ Z(cid:101)k,l∗ : k = 1,...,l∗ (1.29) weaklyonthespaceDl∗([0,∞))equippedwiththeproductSkorohodJ1 topology. AGINGINTHEGREM-LIKETRAPMODEL 8 Remark 1.4. Note that by (1.19) and the definition of Neveu’s CSBP, for any i = 1,...,L and b > 0, Z (b·) is a stable subordinator with index α . R((i 1)/L),R(i/L) i,L − Therefore,by(1.24)thedistributionoftherighthandsideof(1.29)isgivenbycompo- sitionsofstablesubordinators. Remark 1.5. Together with the previous remark, Theorem 1.4 implies that S(n) con- k,L vergesweaklytoanα¯ -stablesubordinator,whereα¯ isgivenasinTheorem1.1. Ifone k k is only interested in marginal distributions of the clock processes (which is enough to obtainourresultsontwo-timecorrelationfunctions)ashorterproofisavailablethrough a technique based on Durrett and Resnick [17], which has been recently proved to be very useful in the context of dynamics in disordered systems, see [13, 14, 20, 21, 22]. However, in this paper we choose to prove the stronger result of the joint convergence ofclockprocessesinordertomaketheconnectiontoNeveu’sCSBPmoretransparent. In [7], Bertoin and Le Gall gave a representation of the genealogical structure of CSBPs using Bochner’s subordination. We say that an individual t at generation r has anancestorcatgenerationp ∈ [0,r]ifcisajumpingtimeofZ and p,r Z (c−) < t < Z (c). (1.30) p,r p,r SincetheLe´vymeasureofZ hasnoatoms,thesetofindividualsatgenerationdwho r,p do not have an ancestor at generation r < d has a.s. Lebesgue measure 0. In view of this,forindividualst andt atgenerationr,welet 1 2 T (t ,t ) = sup{p ≥ 0 : t andt haveacommonancestoratgenerationp}, r 1 2 1 2 and set T (t ,t ) = −∞ if t and t do not have a common ancestor. Using T we can r 1 2 1 2 r express the limiting two-time functions in Theorem 1.1 and Corollary 1.2 as follows. Notethat,bydefinition,wehave P(T (t ,t ) ≥ p) = P({Z (t) : t ≥ 0}∩[t ,t ] = ∅), (1.31) r 1 2 p,r 1 2 and since Z is a stable subordinator with index e (r p), the right hand side of (1.31) p,r − − is nothing but Asl (b/d), where α = e (r p). Thus, we can rewrite the Asl terms in α − − Theorem1.1andCorollary1.2as,respectively, (cid:16) 1 (cid:17) (cid:16) (cid:17) Asl = P T (1,1+θ) ≥ R((k −1)/L) , (1.32) α¯k 1+θ R(l∗/L) and (cid:16) 1 (cid:17) (cid:16) (cid:17) Asl = P T (1,1+θ) ≥ R(x) . (1.33) α(x) 1+θ R(r∗) Remark 1.6. Neveu’s CSBP was first used in the study of the statics of the GREM and CREM [25]. Namely, the limiting geometric structure of the Gibbs measure of these models can be expressed in terms of the genealogy of Neveu’s CSBP (see Section 5 of [15]). However,inthecontextofthedynamics,itappearsinadifferentway,describing thelimitsoftheclockprocesses. The rest of this paper is organized as follows. In Section 2 we describe clock processes through a certain cascade of point processes associated to the dynamics, prove that they converge weakly to a cascade of Poisson point processes, and finally, using littlemorethanthecontinuousmappingtheorem,weestablishTheorem1.4. InSection 3weproveTheorem1.1andCorollary1.2. AGINGINTHEGREM-LIKETRAPMODEL 9 2. CONVERGENCE OF THE CLOCK PROCESSES. 2.1. Description of the clock processes through a cascade of point processes. We firstgivedefinitionsandintroducenotationforgeneralcascadeprocesses. For a complete, separable metric space A, we designate by M(A) the space of point measures on A, and by ε the Dirac measure at x ∈ A, i.e. ε (F) = 1 if x ∈ F and x x ε (F) = 0ifx ∈/ F. Weset x H = (0,∞)×(0,∞), and M = M(H1)⊗···⊗M(Hl), l ∈ N. (2.1) 1,l All the point measures we use are indexed by Nk and we use the notation j| = k j j ···j foramemberofNk. 1 2 k Thesetofl-levelcascadepointmeasures,M ,isthesubsetofM whereforeach 1,l 1,l m = (m ,...,m ) ∈ M therecorrespondsacollectionofpointsinH oftheform 1 l 1,l (cid:8)(t ,x ) : j| ∈ Nl,k = 1,...,l(cid:9) (2.2) jk jk l | | suchthatforeachk = 1,...,l (cid:88) m = ε . (2.3) k (tj1,xj1,...,tj|k,xj|k) jk Nk | ∈ Werefertothecollectionin(2.2)asthemarksofm. Throughoutthispaperweassume thatallthepointmeasuresaresimple. LetM(cid:102) bethesubsetofM suchthatforeach 1,l 1,l m ∈ M(cid:102) ,k = 1,...,l,j| ∈ Nk 1 andt > 0, 1,l k 1 − − (cid:16) (cid:17) (cid:0) (cid:1) m {(t ,x ,...,t ,x )}× (0,t]×(0,∞) < ∞. (2.4) k j1 j1 j|k−1 j|k−1 Then,form ∈ M(cid:102) ,thereexistsauniquelabelingofthemarkssothatforanyj| ∈ 1,l k 1 Nk 1 − − t < t < ··· . (2.5) j|k−11 j|k−12 Fromnowonweonlyusethislabelingforthemarksofm ∈ M(cid:102) . 1,l WedefineT : M(cid:102) → D([0,∞))by l 1,l (cid:88) (cid:88) (cid:88) T (m)(t) = ··· x . (2.6) l jl | tj1≤t tj1j2≤xj1 tj|l≤xj|l−1 Next,weintroduceamapT : M(cid:102) → M(cid:102) . Form ∈ M(cid:102) let l 1,l 1,l 1 1,l − (cid:16) (cid:17) Z(i) = m {(t ,x )}×(cid:0)(0,x ]×(0,∞)(cid:1) , i ∈ N, (2.7) 2 i i i wherewesetZ(0) = 0, g(i) = max{r : Z(0)+···+Z(r) < i}, h(i) = Z(0)+···+Z(g(i)), i ∈ N, (2.8) and s(i) = x +···+x , i ∈ N, s(0) = 0. (2.9) 1 i Then,wedefineT (m)asapointinM whosemarksaregivenby l 1,l (cid:8)(t ,x ) : j| ∈ Nl 1,k = 1,...,l−1(cid:9), (2.10) j|k j|k l−1 − where,forj ∈ N, 1 t = s(g(j ))+t , x = x , (2.11) j1 1 (g(j1)+1)(j1 h(j1)) j1 (g(j1)+1)(j1 h(j1)) − − AGINGINTHEGREM-LIKETRAPMODEL 10 andfork > 1andj| ∈ Nk, k t = t , x = x . (2.12) jk (g(j1)+1)(j1 h(j1))j2...jk jk (g(j1)+1)(j1 h(j1))j2...jk | − | − It is clear that T (m) ∈ M(cid:102) and the marks in (2.10) is already ordered in jump times, l 1,l thatis,(2.5)issatisfied. Fork ≤ l,wedefineT : M(cid:102) → D([0,∞))by k,l 1,l T = T (T ◦···◦T ◦T ), (2.13) k,l l k+1 l k+2 l 1 l − − − whereitisunderstoodthatT = T . 1,l l Forl ≤ l andm = (m ,...,m ) ∈ M(cid:102) , letm| = (m ,...,m ) ∈ M(cid:102) . We 1 2 1 l2 1,l2 l1 1 l1 1,l1 usethefollowingpropertylater: fork < l ≤ l andm ∈ M(cid:102) , 1 2 1,l2 T (m) = T (m)◦T (m| ), (2.14) k,l2 l1,l2 k,l1 1 l1 1 − − wherethecompositionintheabovedisplayisonthespaceD([0,∞)). WenowdescribeacascadeofpointprocessesassociatedtothedynamicsX whose L image under the functionals T yields the clock processes. A cascade of simple ran- k,L domwalkswithLlevelsonT isacollectionofrandomvariables L (cid:110) (cid:111) J = J (j ;j| ) : j| ∈ NL, k = 1,...,L (2.15) k k k 1 L − thatischaracterizedasfollows: (i) For each k = 1,...,L and j| fixed, {J (j ;j| ) : j ∈ N} is a collection of k 1 k k k 1 k i.i.d. randomvariablesdistributed−uniformlyon[n]. − (ii)Thefamilies(cid:8)J (j ;j| ) : j ∈ N(cid:9)fork = 1,...,Landj| areindependent. k k k 1 k k 1 − − Fork = 1,...,L,thejumpchainJ(j| )onV| isdefinedby k k J(j| ) = J (j )J (j ;j| )···J (j ;j| ). (2.16) k 1 1 2 2 1 k k k 1 − In (i) above we make use of the fact that a simple random walk on the complete graph [n] starting from a uniform distribution is the same as a sequence of i.i.d. uniform distributions on [n]. For a fixed realization of the random environment, consider the collectionofrandomvariables (cid:8)(t ,ξ ) : j| ∈ NL,k = 1,...,L(cid:9) (2.17) jk jk L | | given as follows: for a given realization of J, it is an independent collection with ξ =d G(λ(J(j| ))) for k = 1,...,L − 1, and ξ =d λ 1(J(j| ))e. Here, G(p) jk k jL − L de|notes a geometric random variable with success p|robability p and e is a mean one exponential random variable. We also set t = j . Using the collection in (2.17) we jk k defineζL = (ζL,...,ζL)by | 1 L (cid:88) ζL = ε . (2.18) k j1,...,jk N (tj1,ξj1,...,tj|k,ξj|k) ∈ Clearly ζL ∈ M(cid:102) and the marks of it are already ordered in jump times. Finally, the 1,L clockprocessS isgivenby k,L S = T (ζL), k = 1,...,L. (2.19) k,L k,L