Mean-field caging in a random Lorentz gas
Giulio Biroli, Patrick Charbonneau, Yi Hu, Harukuni Ikeda, Grzegorz Szamel, Francesco Zamponi
MMean-field caging in a random Lorentz gas
Giulio Biroli, Patrick Charbonneau,
2, 3
Yi Hu, ∗ Harukuni Ikeda, Grzegorz Szamel, and Francesco Zamponi Laboratoire de Physique de l’Ecole Normale Sup´erieure, ENS, Universit´e PSL,CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France Department of Chemistry, Duke University, Durham, North Carolina 27708, USA Department of Physics, Duke University, Durham, North Carolina 27708, USA Graduate School of Arts and Sciences, The University of Tokyo 153-8902, Japan Department of Chemistry, Colorado State University, Fort Collins, CO 80523, USA
The random Lorentz gas (RLG) is a minimal model of both percolation and glassiness, which leadsto a paradox in the infinite-dimensional, d → ∞ limit: the localization transition is then expectedto be continuous for the former and discontinuous for the latter. As a putative resolution, we haverecently suggested that as d increases the behavior of the RLG converges to the glassy descrip-tion, and that percolation physics is recovered thanks to finite- d perturbative and non-perturbative(instantonic) corrections [Biroli et al . arXiv:2003.11179]. Here, we expand on the d → ∞ physicsby considering a simpler static solution as well as the dynamical solution of the RLG. Comparingthe 1 /d correction of this solution with numerical results reveals that even perturbative correctionsfall out of reach of existing theoretical descriptions. Comparing the dynamical solution with themode-coupling theory (MCT) results further reveals that although key quantitative features of MCTare far off the mark, it does properly capture the discontinuous nature of the d → ∞ RLG. Theseinsights help chart a path toward a complete description of finite-dimensional glasses.
I. INTRODUCTION
Formulating a first-principle description of glasses re-mains a major challenge of condensed matter and statis-tical physics. A proposal for a constructive approach isto solve for the mean-field theory of glasses in the limit ofinfinite spatial dimensions, d → ∞ , and to then introducesystematic finite- d corrections [1–3]. For standard phasetransitions this approach captures the salient behaviorof physical systems in d = 2 and 3, and the relativelysmooth dimensional evolution of the glass phenomenol-ogy suggests that the same should apply for these richersystems as well [3]. Technically rooted in the celebratedspin glass solution [4] and in the foundational works ofTed Kirkpatrick, Dave Thirumalai and Peter Wolynesin the mid-1980s [1, 2, 5, 6], a mean-field d → ∞ the-ory for structural glasses has been formulated in recentyears [7, 8]. These works, which follow the idea of usingstatic replica methods to find solutions of random Hamil-tonians [5, 9] (see also Refs 10 and 11), have demon-strated – in the d → ∞ limit – that the random first-order transition description of glasses is valid [6], andthat hard spheres are natural archetypes of all simpleliquids [3, 7, 8].The finite- d physics of hard sphere liquids is, however,far from simple. Different non-mean-field effects such ashopping [12], facilitation [13, 14] and nucleation [15] arethen observed. Their competing effects make the analy-sis of finite- d corrections rather difficult. Modified modelshave thus been introduced in an attempt to isolate someof these effects. For example, systems of hard sphereswith randomly shifted pair distances, as proposed by ∗ [email protected] Mari, Kurchan and Krzakala (MKK) [16, 17], can sys-tematically eliminate multi-body interactions, while stillbehaving similarly as hard sphere glasses in the limit d → ∞ [18, 19]. In that vein, we have recently foundthat an even simpler, single-particle model, the randomLorentz gas (RLG), can be solved in the d → ∞ limitand that its solution shares all the features of the equi-librium hard sphere equivalent [20]. Because, by con-struction, most finite- d corrections are then absent – in-cluding nucleation, structural correlations and facilita-tion – those that do persist are especially noticeable. Inparticular, although the RLG has long been known as amodel of continuum percolation [21–28], that descriptionand the d → ∞ mean-field one fundamentally disagreeabout the nature of the localization transition. It shouldbe continuous according to the former, and discontinu-ous according to the latter. To resolve this paradox, wehave proposed that instantonic (non-perturbative) hop-ping events intervene in finite d [20]. Because numericalsimulations of the RLG can be pushed as high as d = 20,we also gathered substantial quantitative support for thisproposal. This analysis, however, leaves many questionsopen. For instance, it remains unclear what the originof the sizable perturbative, 1 /d , corrections identified inRef. 20 might be.We also note that the mode-coupling theory (MCT)of the RLG predicts a continuous localization transi-tion in finite d whereas the intimately related MCT ofglasses [29–31] predicts a discontinuous dynamical glasstransition. Since both mode-coupling theories are ofmean-field flavor and do not include any contributionfrom hopping processes, one wonders whether the pre-diction of a continuous localization transition is a happyartefact or whether some mean-field-like effects influencethe character of the transition.Following Ref. 28, in which some of us investigated the a r X i v : . [ c ond - m a t . d i s - nn ] F e b percolation caging behavior of the RLG, we here inves-tigate the mean-field caging behavior of the RLG as d increases. The plan for the rest of this work is as follows.We first compare static mean-field solutions obtained byvirial expansion by taking the high-asymmetry limit ofbinary hard spheres mixtures and the infinite-radius limitof the non-convex perceptron. We then derive the dy-namical solution of the d → ∞ RLG. Mean-field predic-tions are finally compared with numerical cavity recon-struction results, before briefly concluding.
II. STATIC DERIVATIONS
In Ref. 20, we confirmed that the mean-field RLG be-longs to the same universality class as hard sphere glassesin the d → ∞ limit by an ansatz-free cavity reconstruc-tion calculation. Here, we extend this analogy by con-sidering various mean-field approaches based on differentassumptions and limits. We first consider the virial so-lution of the RLG using a Gaussian cage ansatz, andthen investigate the RLG as a limit case of a binary hardspheres mixture [32–34]. We additionally consider thepossibility of describing the RLG as a special limit ofthe non-convex perceptron, a model that has been par-ticularly informative about the related physics of jam-ming [35–38].For the sake of clarifying the notation, recall that theRLG consists of an infinitesimally small tracer navigat-ing within the space left by N Poisson-distributed hardspherical obstacles in a d -dimensional box of volume V .The unitless obstacle density can thus be given byΦ = ρV d σ d (1)where ρ = N/V is the number density of the obstaclesof radius σ , and V d is the d -dimensional volume of a ballof unit radius, V d = π d/ / Γ(1 + d/ σ . TheRLG can thus also be viewed as a limit case of a binarymixture of hard spheres, with obstacles being infinitelysmaller and heavier than the infinitely-dilute tracer. Inthe d → ∞ limit, the RLG presents a glass-like dynami-cal caging transition at ˆ ϕ d , whose order parameter is thecage size ∆, defined by the long-time limit of the tracermean squared displacement (MSD). For convenience, wealso define the rescaled density ˆ ϕ = Φ /d and cage sizeˆ∆ = d ∆, which reach a finite limit at the caging transi-tion when d → ∞ .The RLG partition function is simply the total freeavailable volume. For a specific realization of quencheddisorder, i.e., obstacle positions, it can be written as Z = (cid:90) d x N (cid:89) i =1 θ ( | x − R i | − σ ) , (2)where the Heaviside step function θ enforces volume ex-clusion by the obstacles, and the d -dimensional vectors x and R i denote the positions of the tracer and of the i -thobstacle, respectively. Assuming that the free-energy isself-averaging and using the replica method, we then get − βF = lim n → log Z n n , (3) Z n = (cid:90) d x N (cid:89) i =1 n (cid:89) a =1 θ ( | x a − R i | − σ ) , where x = { x , · · · , x n } denotes a set of n identical repli-cas of the original tracer, and • denotes averaging overobstacle positions R i . A. Virial expansion
We first consider a solution of the RLG using a liquid-state approach. As is canonical in that context, we in-troduce an external field ψ ( x ) as [39] Z n → Z [ ψ ( x )] ≡ (cid:90) d x e ψ ( x ) N (cid:89) i =1 n (cid:89) a =1 θ ( | x a − R i | − σ )= (cid:90) d x e ψ ( x ) G ( x ) , (4)with auxiliary function G ( x ) ≡ N (cid:89) i =1 n (cid:89) a =1 θ ( | x a − R i | − σ ) . (5)The field ψ ( x ) is conjugate to the density distribution ofthe replicas, ρ ( x ) = (cid:42) n (cid:89) a =1 δ ( x − x a ) (cid:43) = δ log Z [ ψ ( x )] δψ ( x )= 1 Z [ ψ ] e ψ ( x ) G ( x ) , (6)and hencelog ρ ( x ) = ψ ( x ) + log G ( x ) − log Z [ ψ ] . (7)From this relation, we obtain a density functional [5, 40]for ρ ( x )Ω[ ρ ] ≡ log Z [ ψ ] − (cid:90) d x ρ ( x ) ψ ( x )= − (cid:90) d x ρ ( x ) log ρ ( x ) + (cid:90) d x ρ ( x ) log G ( x ) ≈ − (cid:90) d x ρ ( x ) log ρ ( x ) + ρ (cid:90) d x ρ ( x ) (cid:90) d R f ( x , R ) , (8)with Mayer function f ( x , R ) = n (cid:89) a =1 θ ( | x a − R | − σ ) − . (9)Note that this expression is equivalent to that for binarymixtures [41], thus strengthening the liquid-state frame-work analogy.The final step entails optimizing the free-energy withrespect to the density distribution, ρ ( x ). In general thisoperation can be quite challenging, but in the d → ∞ limit the calculation greatly simplifies [8]. The centrallimit theorem indeed guarantees that only the secondmoment of ρ ( x ) matters for the free energy computa-tion. The density ρ ( x ) can thus be taken as a Gaussiandistribution, which for a one-replica symmetry breaking(1RSB) ansatz can be written as ρ ( x ) = n/m (cid:89) k =1 V (cid:90) d X k (cid:89) b ∈ B k γ A ( x b − X k ) , (10)where B k = { mk + 1 , mk + 2 , · · · , m ( k + 1) } is the k threplica block, and γ A ( r ) denotes a d -dimensional Gaus-sian with zero mean and variance A . Within this ansatz,the free-energy expression becomes − βF = lim n → Ω n = 1 m (cid:104) log V + d m −
1) log(2 πA ) + d m + d m − (cid:105) + 1 m ρ (cid:90) d r [ q A/ ( r ) m − , (11)where q A ( r ) = (cid:90) d r (cid:48) γ A ( r − r (cid:48) ) θ ( | r − r (cid:48) | ) . (12)For the RLG, the radial distribution function for obsta-cles is simply a step function, and in that sense is equiv-alent to the Mari-Kurchan model. Hence, q A ( r ) can bedirectly taken from Eq. [S19] of Ref. 19 as q A ( r ) = (cid:90) ∞ σ d u (cid:16) ur (cid:17) d − e ( r − u )24 A √ ru A I d − (cid:16) ru A (cid:17) , (13)where I n ( x ) is the modified Bessel function, and the vec-tor integration q A ( r ) can be transformed into a scalarintegration by using rotational invariance. The saddlepoint condition for Eq. (11) finally gives1 A = 2 ρd ( m − ∂∂A (cid:90) d r q A ( r ) m . (14)The dynamical glass transition corresponds to takingthe limit m →
1, in which case Eq. (14) reduces to1 A = − ρd ∂∂A (cid:90) d r q A ( r ) log q A ( r ) . (15)The dynamical glass transition point is then1 ρ d = − max A Ad ∂∂A (cid:90) d r q A ( r ) log q A ( r ) . (16) Identifying ˆ ϕ and substituting ˆ∆ = 2 d A in Eq. (15)results in the exact same expression as for the ansatz-free cavity derivation of Ref. 20, thus demonstrating theequivalence of the two approaches in the limit d → ∞ .Unlike the cavity derivation, however, this approachalso naturally leads to a model for finite- d corrections [7,42]. Assuming that Gaussian caging holds for all dimen-sions indeed allows for Eq. (13) to be solved in any given d . Although the ensuing correction to ϕ d is marginal [42],that for ˆ∆ is more substantial. Results from this modelare compared with finite- d numerics in later section. B. Binary mixture derivation
In this subsection we draw a parallel between the RLGand a limit case of a binary hard sphere mixture. Recallfrom Ref. 41 that the free energy of a n -replicated binarymixture of large and small particles, µ = { Large , Small } ,is given in the d → ∞ limit as − βF = (cid:88) µ (cid:90) d x ρ µ ( x )(1 − log ρ µ ( x ))+ 12 (cid:88) µν (cid:90) d x d R ρ µ ( x ) ρ ν ( R ) f µν ( x , R ) , (17)where the density distribution of the replicas is ρ Large ( x ) = (cid:88) i ∈ Large (cid:42) n (cid:89) a =1 δ ( x a − x ai ) (cid:43) , (18a) ρ Small ( x ) = (cid:88) i ∈ Small (cid:42) n (cid:89) a =1 δ ( x a − x ai ) (cid:43) . (18b)In order to model the RLG, we consider a limit thatsatisfies the following three conditions:1. neglect large-large and small-small interactions;2. freeze large particle positions;3. include N − ∼ N large particles and one smallparticle.Under these constraints, the Mayer functions are f Large , Large = f Small , Small = 0 , (19a) f Large , Small = f, (19b) ρ Large ( x ) → ρ (cid:90) d X n (cid:89) a =1 δ ( X − x a ) , (19c) ρ Small ( x ) → ρ ( x ) , (19d)which upon substitution into Eq. (17) immediately gives − βF = ρ (cid:90) d x ρ ( x ) (cid:90) d R f ( x , R ) − (cid:90) d x ρ ( x ) log ρ ( x ) + cnst . (20)The resulting expression is the same as Eq. (8), exceptfor irrelevant additive constants. The binary mixture andthe virial descriptions of the RLG are therefore physicallyequivalent. C. RLG and the spherical perceptron limit
From a completely different viewpoint, the RLG canbe constructed as a limit case of a continuous satisfac-tion problem called the perceptron [43]. In order to un-derstand this relation, we first recall the definition of thespherical perceptron [35, 44] in the following.Consider N points represented by ( d + 1)-dimensionalvectors R i ∈ R d +1 , i = 1 , . . . , N , satisfying the sphericalconstraints R i · R i = R , i = 1 , . . . , N. The the problem is then to find the state vector x ∈ R d +1 that satisfies the exclusion constraints of size κh i = x · R i − κ ≥ , i = 1 , . . . , N, under the spherical constraint x · x = R . The partitionfunction of this constraint satisfaction problem (CSP) isgiven by Z CSP = (cid:90) d x δ ( x · x − R ) N (cid:89) i =1 θ ( x · R i − κ ) , (21)which, in the limits d → ∞ , R → ∞ , and N → ∞ withfixed R /d and N/d , Gardner and Derrida solved for κ ≥ κ < d +1)-dimensional hypersphere of radius R , instead of itstraditional description in d -dimensional Euclidean space.The partition function is then Z RLG = (cid:90) d x δ ( x · x − R ) N (cid:89) i =1 θ ( | x − R i | − σ )= (cid:90) d x δ ( x · x − R ) N (cid:89) i =1 θ ( x · R i + R − σ / . (22)Interestingly, Eqs. (21) and (22) suggest that the RLG onthe hypersphere can be mapped into the spherical versionof the perceptron by setting κ = σ / − R .The RLG in Euclidean space can be identified with theits hypersphere perceptron-like formulation in the ther-modynamic limit R → ∞ , with fixed d and with the num-ber of obstacles N scaling such that a finite density of ob-stacles is maintained. (The curvature of the hypersphereis then negligible.) A (naive) expectation might thus bethat the solution of the RLG in Euclidean space can berecovered by taking the κ → −∞ limit of the solution ofthe perceptron derived by Franz and Parisi [35]. This ex-pectation would be valid, provided the limit d → ∞ with fixed R /d and N/d is equivalent to the limit R → ∞ with fixed d and finite density of obstacles, followed by d → ∞ with a proper scaling of density. This treatment,however, involves a non-trivial exchange of limits, whichsheds some doubt on its validity.To test out the idea, we write the partition function ofthe RLG on the hypersphere as Z RLG = (cid:90) d x δ ( x · x − R ) G ( x ) , (23)where G ( x ) = N (cid:89) i =1 θ ( | x − R i | − σ ) . We write the delta-function constraint using an integralrepresentation, and evaluate it via a saddle point method: Z RLG ∼ (cid:90) d x (cid:90) d λe − λ ( x · x − R ) G ( x ) ∼ e S ( λ ∗ ) , (24)where S ( λ ) = λR (cid:90) d x e − λ x · x G ( x ) , with λ ∗ determined by the (saddle-point) condition ∂S∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ = λ ∗ = 0 → R = (cid:82) d x e − λ x · x G ( x ) x · x (cid:82) d x e − λ x · x G ( x ) . (25)Similarly the distribution function of x is calculated as ρ ( x ) = δ ( x · x − R ) G ( x ) (cid:82) d x δ ( x · x − R ) G ( x ) ∼ e − λ ∗ x · x G ( x ) (cid:82) d x e − λ ∗ x · x G ( x ) . (26)If λ ∗ = 0, then the distribution ρ ( x ) is the same as thatof the RLG in Euclidean space. Is this condition satisfiedin the thermodynamic limit? To answer this question, weintroduce an auxiliary function f ( λ ∗ ) as (cid:104) x · x (cid:105) = d (cid:10) x (cid:11) ≡ df ( λ ∗ ) , (27)where we denote (cid:104)•(cid:105) = (cid:90) d x ρ ( x ) • = (cid:82) d x e − λ ∗ x · x G ( x ) • (cid:82) d x e − λ ∗ x · x G ( x ) , and we note that f ( λ ∗ ) is a decreasing function of λ ∗ because d d f ( λ ∗ )d λ ∗ = − (cid:10) ( x · x ) (cid:11) + (cid:104) x · x (cid:105) ≤ . (28)The equality holds if and only if ρ ( x ) is a delta function.The saddle point condition is now f ( λ ∗ ) = R d . (29)For finite d , the right-hand side of Eq. (29) diverges in thethermodynamic limit R → ∞ . In this case, one concludesthat λ ∗ →
0, because f ( λ ∗ ) is a decreasing function.The distribution of the RLG on the hypersphere can thusbe identified with that in Euclidean space, as expected.However, this does not mean that the solution of thespherical perceptron (See Ref. 37) can be directly used forthe RLG in Euclidean space, because it is derived underthe condition that the ratio R /d is kept finite, whichleads to λ ∗ >
0. It can be explicitly checked within theexact solution of the perceptron model that λ ∗ remainsfinite even in the limit κ → −∞ , see Appendix D inRef. 37. We conclude that the limit κ → −∞ of theperceptron solution does not coincide with the solutionof the RLG in Euclidean space in d → ∞ . III. DYNAMICAL DERIVATION
We now turn to the d → ∞ dynamics of the RLG. Us-ing a dynamical cavity treatment, we here show that thestatic analogy between the RLG and a hard sphere liquidin the limit d → ∞ also holds for the dynamics. Notethat the notation used here follows that of Agoritsas etal. [45], which differs slightly from that of Ref. 46. Morespecifically, m is the tracer’s mass, ζ is the friction co-efficient of an isolated tracer, ξ ( t ) is the Gaussian whitenoise acting on the tracer with auto-correlation function (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = 2 ζ I δ ( t − t (cid:48) ). Note also that, as discussed byManacorda et al . [47], a convenient way to analyze themotion of the tracer among hard obstacles is to considerfirst its motion among purely repulsive yet softened ob-stacles, described by a continuous dimensionless poten-tial V ( r ), and to take the hard sphere limit at the end ofthe computation. This approach sidesteps technical dif-ficulties associated with the singular nature of the hardsphere potential, but without affecting the final result.To simplify the notation, and without loss of generality,we also set the inverse temperature to unity, β = 1.The standard dynamical cavity approach consists of:1. writing equations of motion for the original system;2. adding an additional variable to the problem, insuch a way that its effect on the original system issmall;3. treating this addition perturbatively to derive anequation of motion for the evolution of the newvariable;4. obtaining a self-consistent equation for a typical variable by noting that the new variable is iden-tical to all existing variables in the system.More specifically, in our implementation, the secondstep consists of changing the dimension of the system, d → d + 1. This choice is inspired by Agoritsas et al. ’sanalysis of the perceptron, in which the dimension of theproblem, i.e. , the dimension of the perceptron sphere, is similarly increased. Note that the presentation belowis intended to be physically intuitive. Careful order-of-magnitude estimates of the various terms can be foundin Ref. 48, which presents a related mean-field approachfor a variety of similar problems. Note also that (as inthe perceptron model) there are three sources of random-ness: the initial condition, the thermal noise, and thequenched disorder. Averaging over the initial conditionand noise is here denoted as (cid:104)· · · (cid:105) , and averaging overdisorder as · · · . These averages initially pertain to theoriginal d -dimensional system, as indicated by the sub-script to the averaging notation. Additional averages aredefined later, as needed.Let x µ ( t ) be the tracer coordinates, which satisfy theequation of motion, ζ∂ t x µ ( t ) = − ∂ x µ ( t ) (cid:88) i V ( | x ( t ) − R i | ) + ξ µ ( t ) (30)for µ = 1 , ..., d , where x ≡ ( x , ..., x d ) is the d -dimensional vector, R i ≡ ( R ,i , ..., R d,i ) is the positionof the i th obstacle in d -dimensional space, and ξ µ ( t )is the µ th component of the d -dimensional Gaussianwhite noise ξ ( t ). We here explicitly discuss only theoverdamped case, which corresponds to vanishing mass, m →
0, but the derivation is completely general. It alsoapplies to underdamped (Langevin) dynamics, and theinertial term involving the acceleration could be restoredfollowing Ref. 45.The initial tracer position, x µ , is chosen from the equi-librium Gibbs probability distribution, P d [ x ] = 1 Z d,N exp (cid:34) − (cid:88) i V ( | x − R i | ) (cid:35) , (31)where Z d,N is the partition function for a system withdimension d and N obstacles specified explicitly. In thehard sphere limit distribution Eq. (31) becomes a uniformdistribution in the void space and the partition functioncorresponds to the volume of void space left by obstacles,which are quenched independent random variables.Following the above scheme, we now increase the spa-tial dimension of the problem, which entails adding a newcomponent to the tracer position x ( t ), a new noise com-ponent, ξ ( t ), as well as an additional component to thevectors specifying obstacle locations, R ,i , i = 1 , ..., N .In the presence of this additional variable the equationof motion for the original variables (Eq. (30)) becomes ζ∂ t x µ ( t ) = − ∂ x µ ( t ) (cid:88) i V i ( x ( t )) + ξ µ ( t ) , µ = 1 , ..., d, (32)where we use the shorthand V i ( x ( t )) ≡ V [( | x ( t ) − R i | + | x ( t ) − R ,i | ) / ]in the following.To leading order in | x − R ,i | / | x − R i | , which we im-plicitly treat as an O (1 / √ d ) term, the perturbed equa-tion of motion for the original variables is then ζ∂ t x µ ( t ) = − ∂ x µ ( t ) (cid:88) i (cid:104) V ( | x ( t ) − R i | )+ V (cid:48) ( | x ( t ) − R i | ) | x ( t ) − R i | h i ( t ) (cid:105) + ξ µ ( t ) , (33)where V (cid:48) ( r ) = d V ( r ) / d r and h i ( t ) = | x ( t ) − R ,i | .Note that h i does not have a simple geometric inter-pretation. In particular, it is not a change of the gapbetween the particle and obstacle i . It nevertheless doesnot depend on x and R i , which simplifies some of thefollowing considerations.The initial condition for x µ in Eq. (33) is chosen fromthe equilibrium Gibbs probability distribution, in whichthe additional dimension acts as an external field, P d ( x | x ) ≈ Z d,N exp (cid:40) − (cid:88) i V ( | x − R i | ) − (cid:88) i (cid:20) V (cid:48) ( | x − R i | ) | x − R i | − (cid:28) V (cid:48) ( | x − R i | ) | x − R i | (cid:29) d (cid:21) h i (cid:41) (34)Following Agoritsas et al. [45], we can write the trajec-tories of the original variables as the sum of the unper-turbed trajectories and of the trajectories perturbed bythe change in both the equation of motion and the initialcondition. We then obtain x µ ( t ) = x (0) µ ( t ) + x (dyn) µ ( t ) + x (in) µ ( t ) , (35)where x (0) µ ( t ) satisfies the unperturbed equation of mo-tion with an initial condition drawn from the unper-turbed ensemble given by Eq. (31), x (dyn) µ ( t ) is the trajec-tory change originating from the second term in the per-turbed equation of motion given by Eq. (33), and x (in) µ ( t )is the trajectory change originating from the perturbedinitial condition given by Eq. (34). Agoritsas et al. [45]formally wrote the latter two components as x (dyn) µ ( t ) = (cid:88) i (cid:90) t d t (cid:48) δx (0) µ ( t ) δh i ( t (cid:48) ) h i ( t (cid:48) ) , (36a) x (in) µ ( t ) = (cid:88) i δx (0) µ ( t ) δh i h i , (36b)where in Eq. (36b) h i ≡ h i ( t = 0)Let us now consider the equation of motion for the newcoordinate of the tracer, ζ∂ t x ( t ) = − ∂ x ( t ) (cid:88) i V i ( x ( t )) + ξ ( t ) . Using Eqs. (35)-(36b) we can write ζ∂ t x ( t ) ≈ − ∂ x ( t ) (cid:88) i V i ( x (0) ( t )) − ∂ x ( t ) (cid:88) i,j δV i ( x (0) ( t )) δh j h j − ∂ x ( t ) (cid:88) i,j (cid:90) t d t (cid:48) δV i ( x (0) ( t )) δh j ( t (cid:48) ) h j ( t (cid:48) ) + ξ ( t ) , (37)where we denote again the shorthand V i ( x (0) ( t )) ≡ V [( | x (0) ( t ) − R i | + | x ( t ) − R ,i | ) / ] . The first term on the right-hand side (RHS) of this equa-tion is a fluctuating potential field at position x ( t ). Itsfluctuations are due to the unperturbed “gap” variables, r (0) i ( t ) = | x (0) ( t ) − R i | , which evolve on their own anddue to the quenched randomness, R ,i . The second andthird terms describe a feedback process. The additionalcoordinate perturbs the tracer evolution in the original d -dimensional system, which in turn influences its evolu-tion in the additional dimension.In order to complete the derivation, we make twoassumptions whose justification will be presented else-where 48:1. The influence of the presence of a specific obsta-cle on the distance between the tracer and anotherspecific obstacle is negligible. This implies that thecontributions to the force originating from differ-ent obstacles are uncorrelated and therefore onlythe diagonal i = j terms contribute to the doublesummations in Eq. (37).2. The summations in the second and third terms ofEq. (37) concentrate around their averages. Notethat these averages include averaging over the dis-order, including the 0th coordinates of the obsta-cles, i.e. disorder averages are d + 1 dimensional.The second assumption leads to the following (cid:88) i δV i ( x (0) ( t )) δh i ( t (cid:48) ) h i ( t (cid:48) ) → (cid:88) i δ (cid:104) V i ( x (0) ( t )) (cid:105) d δh i ( t (cid:48) ) h i ( t (cid:48) ) d +1 , (38a) (cid:88) i δV i ( x (0) ( t )) δh i h i → (cid:88) i δ (cid:104) V i ( x (0) ( t )) (cid:105) hd δh i h id +1 , (38b)where (cid:104) . . . (cid:105) hd denotes averaging over distribution ofEq. 34 and the functional derivatives are evaluated at h i = 0.The two functional derivatives above are relatedthrough the fluctuation-dissipation relation. To intro-duce this relation we first note due to Eq. (34) the func-tional derivative in Eq. (38b) can be expressed in terms ofan equilibrium time-dependent correlation function. Fora given obstacle i we then get (recalling that β = 1) δ (cid:104) V i ( x (0) ( t )) (cid:105) hd δh i = − (cid:68) V i ( x (0) ( t )) (cid:104) V (cid:48) ( | x (0) − R i | ) | x (0) − R i | − (cid:68) V (cid:48) ( | x (0) − R i ) | x (0) − R i | (cid:69) d (cid:105)(cid:69) d , (39)where x (0) ≡ x (0) ( t = 0).More generally, let us define the following correlation function A i ( t, t (cid:48) ) = − (cid:68) V i ( x (0) ( t )) (cid:104) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | − (cid:68) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | (cid:69) d (cid:105)(cid:69) d . (40)In terms of this new correlation function, the fluctuation-dissipation relation reads, δ (cid:10) V i ( x (0) ( t )) (cid:11) d δh i ( t (cid:48) ) = − ∂ t (cid:48) (cid:68) V i ( x (0) ( t )) (cid:104) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | − (cid:68) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | (cid:69) d (cid:105)(cid:69) d . (41)Using the fluctuation-dissipation relation and integrating by parts we obtain ζ∂ t x ( t ) = − ∂ x ( t ) (cid:88) i V i ( x (0) ( t )) − ∂ x ( t ) (cid:88) i A i ( t, t ) h i ( t ) d +1 − ∂ x ( t ) (cid:90) t d t (cid:48) (cid:88) i (cid:68) V i ( x (0) ( t )) (cid:16) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | − (cid:68) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | (cid:69) d (cid:17)(cid:69) d ∂ t (cid:48) h ν ( t (cid:48) ) d +1 + ξ ( t ) (42)It follows from the original equation of motion for the additional coordinate that the differentiation ∂ x ( t ) acts onlyon x ( t ) in V i ( x (0) ( t )).Because every quantity except x ( t ) is integrated out, by translational symmetry the expression (cid:80) i A i ( t, t ) h i ( t ) d +1 in the second term of Eq. (42) is a constant and hence its contribution vanishes. The subtracted contribution in thepenultimate term vanishes as well, because it involves averaging over the tracer position, in the presence of one specificobstacle. In the overwhelming majority of phase space, the tracer is far from this obstacle. Even when the tracer islocalized, it spends most of its time localized away from that obstacle. The penultimate term can thus be rewrittenas − (cid:88) i ∂ x ( t ) (cid:68) V i ( x (0) ( t )) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | (cid:69) d ∂ t (cid:48) h ν ( t (cid:48) ) d +1 = − (cid:88) i ∂ x ( t ) (cid:68) V i ( x (0) ( t )) V (cid:48) ( | x (0) ( t (cid:48) ) − R i | ) | x (0) ( t (cid:48) ) − R i | (cid:69) d ( x ( t (cid:48) ) − R ,i ) d +1 ∂ t (cid:48) x ( t (cid:48) ) , (43)where the average can be identified with the auto-correlation function of the force acting along the 0th co-ordinate of the tracer, F ,i ( t ) = − V (cid:48) ( | x (0) ( t ) − R i | ) x ( t ) − R ,i | x (0) ( t ) − R i |≈ − ∂ x ( t ) V i ( x (0) ( t )) . (44)The average is therefore given by (cid:88) i (cid:104) F ,i ( t ) F ,i ( t (cid:48) ) (cid:105) dd +1 , (45)which depends on the displacement of the tracer alongthe additional coordinate, x ( t ) − x ( t (cid:48) ). We note thatduring the decay of the auto-correlation function inEq. (45) the tracer displacement along the 0th coordi-nate is vanishingly small and therefore, to leading order in 1 /d , we can average Eq. (45) over all possible valuesof x ( t ) − x ( t (cid:48) ), which results in (cid:88) i (cid:104) F ,i ( t ) F ,i ( t (cid:48) ) (cid:105) d +1 d +1 . (46)(A more careful discussion of this point will be given inRef. 48.)The equation of motion for the additional coordinateis then ζ∂ t x ( t ) = (cid:88) i F ,i ( t ) (47) − (cid:88) i (cid:90) t d t (cid:48) (cid:104) F ,i ( t ) F ,i ( t (cid:48) ) (cid:105) d +1 d +1 ∂ t (cid:48) x ( t (cid:48) ) + ξ ( t ) . In the d → ∞ limit, this coordinate is indistinguishablefrom any other coordinate. We can then write the mem-ory function as (cid:88) i (cid:104) F ,i ( t ) F ,i ( t (cid:48) ) (cid:105) d +1 d +1 = 1 d (cid:88) i (cid:104) F i ( t ) · F i ( t (cid:48) ) (cid:105) d +1 d +1 . (48)Next, in this limit the random force, i.e. the first termon the RHS of Eq. (47) becomes a Gaussian randomvariable η . Finally, recognizing once more that in the d → ∞ limit all coordinates are equivalent we can writethe equation of motion for the tracer in vector notation, ζ∂ t x ( t ) = η ( t ) − (cid:90) t d t (cid:48) M ( t − t (cid:48) ) ∂ t (cid:48) x ( t (cid:48) ) + ξ ( t ) , (49) (cid:104) η ( t ) η ( t (cid:48) ) (cid:105) = I M ( t − t (cid:48) ) . (50)The memory function in Eq. (48) depends on the statis-tics of r µ,i ( t ) ≡ x µ ( t ) − R µ,i only. To complete thederivation we need to repeat the above treatment for r , ( t ) ≡ x ( t ) − R , and once again recognize the equiv-alence of all coordinates. We start with the equation ofmotion for r , , ζ∂ t r , ( t ) = − ∂ r , ( t ) V (cid:18)(cid:113) | r ( t ) | + ( r , ( t )) (cid:19) − (cid:88) i (cid:54) =1 ∂ r ,i ( t ) V (cid:18)(cid:113) | r i ( t ) | + ( r ,i ( t )) (cid:19) + ξ ( t ) . (51)The second term on the RHS can be analyzed as before.Because the tracer interacts with an average of d obsta-cles, in the limit d → ∞ excluding one specific obstacledoes not matter. The resulting equation of motion is ζ∂ t r , ( t ) = F , ( t ) + (cid:88) i (cid:54) =1 F ,i ( t ) − (cid:88) i (cid:54) =1 (cid:90) t d t (cid:48) (cid:104) F ,i ( t ) F ,i ( t (cid:48) ) (cid:105) dd +1 ∂ t (cid:48) r , ( t (cid:48) )+ ξ ( t ) , (52)where F ,i ( t ) = − ∂ r ,i ( t ) V (cid:18)(cid:113) | r (0) i ( t ) | + ( r ,i ( t )) (cid:19) . (53)Again, replacing Eq. (45) by Eq. (46) gives the standardexpression, ζ∂ t r , ( t ) = F , ( t ) + (cid:88) i (cid:54) =1 F ,i ( t ) − (cid:88) i (cid:54) =1 (cid:90) t d t (cid:48) (cid:104) F ,i ( t ) F ,i ( t (cid:48) ) (cid:105) d +1 d +1 ∂ t (cid:48) r , ( t (cid:48) ) + ξ ( t ) . (54) The central argument of the cavity method can thenbe invoked. In the limit d → ∞ , r , ( t ) is equivalentto any r µ, ( t ), (cid:80) i (cid:54) =1 F i ( t ) becomes a random Gaussianforce with autocorrelation, (cid:10) F i ( t ) F i ( t (cid:48) ) (cid:11) d +1 d +1 , self-consistently determined by the statistics of r µ,i ( t ) = x µ ( t ) − R µ,i , which are the same as for r µ, ( t ), etc . Wethus have the self-consistent stochastic process, ζ∂ t r ( t ) = F ( r ( t )) + η ( t ) − (cid:90) t d t (cid:48) M ( t − t (cid:48) ) ∂ t (cid:48) r ( t (cid:48) ) + ξ ( t ) , (55)where, consistently with Eq. (50), (cid:104) η ( t ) η ( t (cid:48) ) (cid:105) = I M ( t − t (cid:48) ) (56)but with an explicit expression for the memory function, M ( t ) = nd (cid:90) d r e − V ( r ) (cid:104) F ( r ( t )) · F ( r ) (cid:105) r , (57)where in turn the average (cid:104) . . . (cid:105) r is over the stochasticprocess defined in Eq. (55).Equations (55)-(57) allow one to evaluate the memoryfunction, which can then be used to analyze the motionof the tracer from Eqs. (49)-(50). One can then follow thederivation in one of the recent references on the topic [8,45, 49], and derive the self-consistent equation for thelocalization length from Eqs. (55)-(57). The conditionof Eq. (16), which gives that a discontinuous dynamicaltransition arrests the tracer dynamics in the limit d → ∞ ,is straightforwardly recovered. In the d → ∞ limit, thedynamics of the RLG is therefore completely equivalentto that of an equilibrium hard sphere liquid, after a merefactor of two rescaling. IV. NUMERICAL RESULTS AND DISCUSSION
The above theoretical approaches provide a consistentmean-field, d → ∞ description of RLG caging. How-ever, they all generically leave out perturbative correc-tions of leading order 1 /d , let along non-perturbative cor-rections. As discussed in the context of the static deriva-tions above, the direct cavity reconstruction scheme doesnot readily provide an estimate of the prefactor for thiscorrection (and neither does the dynamical scheme of dy-namic derivation), but a finite- d Gaussian cage ansatzcan straightforwardly be implemented in the virial treat-ment. Although this ansatz was found not to hold infinite- d glasses in the vicinity of the dynamical arrest [50],and clearly fails to account for non-perturbative hoppingcorrections near the mean-field dynamical transition atˆ ϕ d [20], it is unclear whether similar problems affect RLGcaging at high obstacle densities. In this regime, hoppingis indeed strongly suppressed and the Gaussian ansatzmight well predict the scale of perturbative corrections.In this section, we use finite- d simulations to provide anoverall evaluation of the Gaussian ansatz. -6 -3 -6 -3 -10 -5 -3 -1 -4 -2 -4 -2 -5 -9 -3 -4 -1 -4 -1 (a) (b)(c) (d) FIG. 1. (a) Self van Hove function and (b) cage size dis-tribution at ˆ ϕ = 5 for various d . The small displacement G s ( ˆ∆) ∼ ˆ∆ d/ − (colored dashed lines) and the d → ∞ pre-diction from Eq. 15 (vertical black dashed line) are included.Insets in (a,b): Same results given on a lin-log scale. (c) Selfvan Hove function and (d) cage size distribution in d = 3 forvarious ˆ ϕ . At small obstacle density, close to the percolationthreshold, a shoulder emerges in G s ( ˆ∆) (arrow in (c)). Thisfeature is associated with a fat tail that makes the mean cagesize ˆ∆ diverge for ˆ ϕ < ˆ ϕ p . The cage size distribution alsobroadens upon decreasing ˆ ϕ . The challenge of assessing the Gaussian ansatz is thatit implicitly assumes that all cages are the same size.Although this typical cage does dominate in the d →∞ limit, strong deviations from caging uniformity areobserved in finite d . More quantitatively, the mean cagesize is obtained by averaging over cages (i.e., disorder)and over initial tracer positions within a cageˆ∆ = d C (cid:88) C |{ i }||{ j }| (cid:88) i,j ∈ C ( r i − r j ) , (58)where C is a cage realization. The Gaussian ansatz, how-ever, only accounts for the latter averaging and replacesthe disorder average by a concentration assumption. Twodistribution naturally assess these two effects, and thusallow us to discern the dominant contribution.1. The long-time limit of the self part of the van Hovefunction G s ( r ) ∼ (cid:88) C ,i,j δ ( | r i − r j | − r ) , (59)normalized as (cid:82) ∞ G s ( r )d r = 1, can be expressed interms of the squared displacement ˆ∆ = r d througha change of variables, G s ( ˆ∆) = G s ( r ) / (2 r ). The re-sulting distribution accounts for both types of av-eraging. 2. The cage size distribution P ( ˆ∆) ∼ (cid:88) C δ |{ i }||{ j }| (cid:88) i,j ( r i − r j ) − ˆ∆ /d , (60)normalized as (cid:82) ∞ P ( ˆ∆)d ˆ∆ = 1, accounts for thedisorder distribution only.Numerically, these observables can be efficiently obtainedby cavity reconstruction and Monte Carlo integration asin Refs. 20 and 28. Sample results averaged over 5 × (in d = 2 ,
3) to 300 (in d = 7) cavities are given in Fig. 1.As expected, both distributions narrow as d increases,and seemingly converge to a δ distribution as d → ∞ (Fig. 1(a, b)). Conversely, upon decreasing d distri-butions not only broaden but also become increasinglyasymmetric. For instance, the small ˆ∆ limit of thevan Hove function scales as G s ( r ) ∼ r d − and thus G s ( ˆ∆) ∼ ˆ∆ ( d − / , while in the large ˆ∆ limit, G s ( ˆ∆)decays quickly. The cage size distribution P ( ˆ∆) is alsolopsided, although slightly less. From this comparison,we conclude that the non-Gaussian character of cagingis significantly affected both by cage-to-cage fluctuationsand by the non-Gaussian character of individual cages,even at high densities.Upon approaching ˆ ϕ p the static cage size distributionfurther broadens at large ˆ∆ (Fig. 1(c, d)), as in the MKmodel [19]. This deviation, which is clearly distinct fromthe non-Gaussian caging correction, contributes to thedramatic growth of the mean cage size in that regime. Italso eventually leads to void percolation. The Gaussianansatz, which assumes that cages are local and closed,thus completely fails in this regime.Perturbative 1 /d corrections to the mean cage size, ˆ∆,arise both from the distribution mode shifting with d andfrom its increasing anisotropy. It is therefore strongly af-fected by the percolation physics as well as other caginganisotropy, which are bound to lead to deviations fromthe Gaussian ansatz. In order to minimize this contribu-tion, we also consider the modal cage size [20],ˆ∆ mode = arg max( ˆ∆ · P ( ˆ∆)) , (61)where P (∆) is the probability of having a cage ofsize ∆ (as shown in Fig. 1(b) for example), with (cid:82) ∞ ˆ∆ P ( ˆ∆)d ˆ∆ = ˆ∆ mean . As argued in Ref. 20, this quan-tity is closer in spirit to the saddle-point evaluation in amean-field calculation and might thus be a better finite- d estimator of local caging. For each of these differentcaging estimators (est), we define the deviation from the d → ∞ result in Eq. (62) as δ ˆ∆ est = | ˆ∆ est − ˆ∆ d →∞ | ∼ ( d ˆ∆ est ) /d, (62)where d ˆ∆ est is the prefactor of the perturbative correc-tion.Figure 2 shows the evolution of different cage size es-timators with packing fraction ˆ ϕ . As expected, both the0 -4 -3 -2 -1 -4 -2 -3 -2 -1 -4 -2 -3 -2 -1 -4 -2 (a) (b) (c) FIG. 2. Comparison between various cage sizes: (a) modal cage size extracted from the mode of P ( ˆ∆); (b) mean cage sizereported in Ref. 20; (c) Gaussian-ansatz cage size from Eq. (15). All three quantities converge to the d → ∞ result fromEq. (16) (black dashed lines). The insets show that in all three cases deviations from the d → ∞ results scale as 1 /d , albeitwith maarkedly different prefactors. The mode and the mean have different signs and are both at least an order of magnitudelarger than the Gaussian-ansatz prediction. modal and the mean cage sizes converge nicely to ˆ∆ d →∞ at densities far above the percolation threshold, and onlythe former exhibit a perturbative scaling in the surround-ings of ˆ ϕ p . The contribution of percolation physics canthus be reasonably well isolated. The deviation pref-actors of these two estimators, however, have differentsigns. They are also at least an order of magnitude larger(in absolute value) than the Gaussian-ansatz correctionfrom Eq. (13). This mismatch reflects the pronouncedanisotropy of the van Hove and cage size distributions, aneffect that is completely absent in the Gaussian ansatz.In addition, as density approaches ˆ ϕ d , the modal cor-rection appears to diverge, while the Gaussian ansatzremains finite. This discrepancy is not a mere quanti-tative concern, but qualitatively wrong. This effect islikely related to the expectation that the dynamical sus-ceptibility should diverge around ˆ ϕ d in the d → ∞ limit.Hence, the Gaussian ansatz not only misses out on per-colation physics, but seems to omit key features of mean-field physics as well. V. CONCLUSION
This article contains two main sets of results. The firstis methodological: we develop several complementary ap-proaches to analyse the RLG in the d → ∞ limit. Weshow that the static approach based on replica theory, forwhich we present two different derivations, agrees witha full dynamical treatment obtained through the cavitymethod. However, we also unveil that despite the appar-ent similarity between perceptron dynamics and RLG,the mean-field solution of the perceptron does not allowto recover the RLG in the d → ∞ limit. The secondmain outcome of our work has been obtained by con-trasting results of numerical simulations in very high di-mension with the d → ∞ analytical solution of the RLG.This comparison allowed us to confirm the distinct roleof percolation and glassy physics in the d → ∞ limit and to clarify to what extent the mean-field solution isable to capture the latter but not the former. Our workhighlights that corrections to the mean-field solution arecrucial to describe dynamics even in large but finite di-mension and opens the way for their analysis. In fact,we have found that cage formation and caging dynamicsare well described by the mean-field solution for d → ∞ ,and that finite-dimensional corrections can be capturedperturbatively but require analytical treatments beyondthe Gaussian-ansatz employed until now. Cage escape,which contributes to percolation, is a more subtle phe-nomenon that takes place on timescales that seeminglydiverge exponentially with d . It thus corresponds to anon-perturbative instantonic correction [20], which mayonly be captured by large-deviation calculations (in con-trast to the saddle point calculations presented in theprevious sections). In conclusion, the RLG – despite itssimplicity and its differences with models of supercooledliquids – offers a new way to look at the problem of glasstransition and provides the long-sought framework andguideline to tackle some of the important perturbativeand non-perturbative effects in glassy dynamics.The dynamical solution of the RLG is also particu-larly interesting in the context of the mode-coupling the-ory (MCT) of the same system. MCT has long beenunderstood as a mean-field theory of the glass and lo-calization transitions. This analogy would imply that itshould fare particularly well in the d → ∞ limit, wherea proper mean-field theory becomes exact. In the con-text of the glass transition, however, this expectation hasbeen shown to be only partially true. MCT correctlypredicts the critical features of the infinite dimensionaldynamic glass transition ( i.e. its discontinuous nature)but fails rather spectacularly at predicting the locationof this transition [46, 51–54]. For the RLG the situa-tion is a bit more involved. The high-dimensional limitof the full wavevector-dependent MCT predicts a discon-tinuous localization transition [31], which qualitativelyagrees with the exact dynamical transition. As for the1dynamical glass transition, MCT predicts an incorrect d → ∞ scaling of the localization transition. However,in finite dimensions MCT predicts a continuous localiza-tion transition, which agrees with numerical simulations.In contrast, as we have shown here, the exact infinitedimensional dynamic theory, when generalized, perhapstoo naively, to finite dimensions, predicts a discontinu-ous localization transition. It would be very interestingto look for a less naive generalization of the infinite di-mensional theory that correctly predicts the character ofthe finite dimensional localization transition. Conversely,it would also be instructive to reformulate MCT to prop-erly capture the scaling of the localization threshold, itsmean-field criticality as well as the rich interplay betweencontinuous and discontinuous caging at high yet not di-verging d . ACKNOWLEDGMENTS
We thank Dave Thirumalai for inviting us to con-tribute to this special issue in his honor. We also wantto acknowledge the key role of his (and his collabora-tors’) ideas on our own research path. We also thank allour own collaborators on these topics for their numer-ous and invaluable inputs. In particular, we acknowl-edge many stimulating discussions with E. I. Corwinand with B. Charbonneau. This work was supported bythe Simons Foundation grants [1] T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. A ,3072 (1987).[2] T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. B ,8552 (1987).[3] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, andF. Zamponi, Annu. Rev. Condens. Matter Phys , 265(2017).[4] M. M´ezard, G. Parisi, and M. Virasoro, Spin glass the-ory and beyond: An Introduction to the Replica Methodand Its Applications , Vol. 9 (World Scientific PublishingCompany, 1987).[5] T. R. Kirkpatrick and D. Thirumalai, J. Phys. A ,L149 (1989).[6] T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes,Phys. Rev. A , 1045 (1989).[7] G. Parisi and F. Zamponi, Rev. Mod. Phys. , 789(2010).[8] G. Parisi, P. Urbani, and F. Zamponi, Theory of simpleglasses: Exact Solutions in Infinite Dimensions (Cam-bridge University Press, Cambridge CB2 8BS, UnitedKingdom, 2020) Chap. 4.[9] T. Kirkpatrick and D. Thirumalai, in
Structural Glassesand Supercooled Liquids (Wiley Online Library, 2012) pp.223–236.[10] R. Monasson, Phys. Rev. Lett. , 2847 (1995).[11] M. M´ezard and G. Parisi, J. Phys. A , 6515 (1996).[12] M. P. Ciamarra, R. Pastore, and A. Coniglio, Soft Matter , 358 (2016).[13] L. Berthier and G. Biroli, Rev. Mod. Phys. , 587(2011).[14] G. Biroli and J. P. Garrahan, J. Chem. Phys. ,12A301 (2013).[15] M. Dzero, J. Schmalian, and P. G. Wolynes, Phys. Rev.B , 100201(R) (2005).[16] R. Mari, F. Krzakala, and J. Kurchan, Phys. Rev. Lett. , 025701 (2009).[17] R. Mari and J. Kurchan, J. Chem. Phys. , 124504(2011).[18] M. M´ezard, G. Parisi, M. Tarzia, and F. Zamponi, J. Stat. Mech. Theory Exp. , P03002 (2011).[19] P. Charbonneau, Y. Jin, G. Parisi, and F. Zamponi, Proc.Natl. Acad. Sci. U.S.A , 15025 (2014).[20] G. Biroli, P. Charbonneau, E. I. Corwin, Y. Hu, H. Ikeda,G. Szamel, and F. Zamponi, arXiv preprint (2020),arXiv:2003.11179.[21] A. R. Kerstein, J. Phys. A , 3071 (1983).[22] W. T. Elam, A. R. Kerstein, and J. J. Rehr, Phys. Rev.Lett. , 1516 (1984).[23] F. H¨ofling, T. Franosch, and E. Frey, Phys. Rev. Lett. , 165901 (2006).[24] F. H¨ofling, T. Munk, E. Frey, and T. Franosch, J. Chem.Phys. , 164517 (2008).[25] T. Bauer, F. H¨ofling, T. Munk, E. Frey, and T. Franosch,Eur. Phys. J Spec. Top. , 103 (2010).[26] M. Spanner, F. H¨ofling, S. C. Kapfer, K. R. Mecke, G. E.Schr¨oder-Turk, and T. Franosch, Phys. Rev. Lett. ,060601 (2016).[27] C. F. Petersen and T. Franosch, Soft Matter , 3906(2019).[28] B. Charbonneau, P. Charbonneau, Y. Hu, and Z. Yang,In preparation (2021).[29] W. G¨otze, E. Leutheusser, and S. Yip, Phys. Rev. A ,2634 (1981).[30] E. Leutheusser, Phys. Rev. A , 2765 (1984).[31] Y. Jin and P. Charbonneau, Phys. Rev. E , 042313(2015).[32] B. Coluzzi, M. M´ezard, G. Parisi, and P. Verrocchio, J.Chem. Phys. , 9039 (1999).[33] I. Biazzo, F. Caltagirone, G. Parisi, and F. Zamponi,Phys. Rev. Lett. , 195701 (2009).[34] H. Ikeda, K. Miyazaki, and A. Ikeda, J. Chem. Phys. , 216101 (2016).[35] S. Franz and G. Parisi, J. Phys. A , 145001 (2016).[36] S. Franz, G. Parisi, P. Urbani, and F. Zamponi, Proc.Natl. Acad. Sci. U.S.A 10.1073/pnas.1511134112 (2015).[37] S. Franz, G. Parisi, M. Sevelev, P. Urbani, and F. Zam-poni, SciPost Phys. , 019 (2017).[38] S. Franz, A. Sclocchi, and P. Urbani, Phys. Rev. Lett. , 115702 (2019).[39] J.-P. Hansen and I. R. McDonald, Theory of simple liq-uids (Elsevier, 1990).[40] Y. Singh, J. Stoessel, and P. Wolynes, Phys. Rev. Lett. , 1059 (1985).[41] H. Ikeda, K. Miyazaki, H. Yoshino, and A. Ikeda, arXivpreprint (2017), arXiv:1710.08373.[42] M. Mangeat and F. Zamponi, Phys. Rev. E , 012609(2016).[43] F. Rosenblatt, Psychol. Rev. , 386 (1958).[44] E. Gardner and B. Derrida, J. Phys. A , 271 (1988).[45] E. Agoritsas, G. Biroli, P. Urbani, and F. Zamponi, J.Phys. A , 085002 (2018).[46] T. Maimbourg, J. Kurchan, and F. Zamponi, Phys. Rev.Lett. , 015902 (2016).[47] A. Manacorda, G. Schehr, and F. Zamponi, J. Chem.Phys. , 164506 (2020).[48] C. Liu, G. Biroli, D. R. Reichman, and G. Szamel, Inpreparation (2020).[49] G. Szamel, Phys. Rev. Lett. , 155502 (2017).[50] P. Charbonneau, A. Ikeda, G. Parisi, and F. Zamponi, Proc. Natl. Acad. Sci. U.S.A , 13939 (2012).[51] A. Ikeda and K. Miyazaki, Phys. Rev. Lett. , 255704(2010).[52] B. Schmid and R. Schilling, Phys. Rev. E , 041502(2010).[53] J.-P. Bouchaud, Journal Club for Condensed MatterPhysics , june (2010).[54] P. Charbonneau, A. Ikeda, G. Parisi, and F. Zamponi,Phys. Rev. Lett. , 185702 (2011).[55] R. Pordes, D. Petravick, B. Kramer, D. Olson,M. Livny, A. Roy, P. Avery, K. Blackburn, T. Wenaus,F. W¨urthwein, I. Foster, R. Gardner, M. Wilde, A. Blate-cky, J. McGee, and R. Quick, in J. Phys. Conf. Ser. , 78,Vol. 78 (2007) p. 012057.[56] I. Sfiligoi, D. C. Bradley, B. Holzman, P. Mhashilkar,S. Padhi, and F. Wurthwein, in2009 WRI WorldCongress on Computer Science and Information Engi-neering