Dynamical Instantons and Activated Processes in Mean-Field Glass Models
DDynamical Instantons and Activated Processesin Mean-Field Glass Models
V. Ros , G. Biroli , C. Cammarota Laboratoire de Physique de l’Ecole normale sup´erieure, ENS, Universit´e PSL, CNRS,Sorbonne Universit´e, Universit´e Paris-Diderot, Sorbonne Paris Cit´e, Paris, France Department of Mathematics, King’s College London, Strand London WC2R 2LS, UK* [email protected] 7, 2020
Abstract
We focus on the energy landscape of simple mean-field models of glasses andanalyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, weconsider Langevin dynamics at low temperature in the energy landscape of thespherical p -spin model. We select as initial condition for the dynamics one ofthe many unstable index-1 saddles in the vicinity of a reference local minimum.We show that the associated dynamical mean-field equations admit two solutions:one corresponds to falling back to the original reference minimum, and the otherto reaching a new minimum past the barrier. By varying the saddle we scanand characterize the properties of such minima reachable by activated barrier-crossing. Finally, using time-reversal transformations, we construct the two-pointfunction dynamical instanton of the corresponding activated process. Rough high-dimensional energy landscapes are central in many different contexts. In physics,they are one of the key ingredients of the theory of glasses, and more generally of disorderedsystems [1]. In computer science, they are studied to characterize algorithmic phase transitionsfor inference and signal processing [2], and they have attracted a lot of attention in the fieldof deep neural networks [3]. In biology, they appear in analysis of evolution and in the studyof protein folding [4, 5].In all these disparate contexts the system under study explores a rough landscape by stochasticdynamics, and the main aim is to characterize the complex dynamical behavior that ensuesfrom it. The mean-field theory of glasses and spin-glasses has been instrumental in thisrespect. It provided the first quantitative analysis of rough high-dimensional landscapes [6],in particular of the number and the properties of the critical points, and of the associatedstochastic dynamics [7]. It was shown theoretically that starting from a random high-energyinitial condition mean-field glass models very slowly approach metastable states, which are a r X i v : . [ c ond - m a t . d i s - nn ] J u l ciPost Physics Submission typically the most numerous ones and are marginally stable, meaning that their Hessianmatrix is characterized by arbitrary small eigenvalues [8]. The intuitive explanation of thisphenomenon is that these metastable states, called the threshold states, are the most numerousand the wider ones, and hence naturally correspond to the largest basins of attraction. Thisparadigmatic behavior has been applied and transposed with success in a variety of contextsin the last twenty years, in particular to explain the glassy dynamics of three-dimensionalinteracting particle systems (super-cooled liquids, colloidal glasses, etc.) [1].Calling N the dimension of the energy landscape, which in physics contexts is proportional tothe number of degrees of freedom, mean-field glass models display two dynamical regimes: a slow descent regime that corresponds to time-scales that do not diverge with N in which thesystem approaches (or more precisely ages toward) the threshold states. An activated regime in which the systems jumps over increasingly larger barriers and is able to explore fully theenergy landscape. To observe such activated processes in mean-field models one has to probethe dynamics on time-scales which are exponentially large in N since the barriers between lowenergy metastable states scale as N [9, 10, 11, 12]. Whereas a theory of the first dynamicalregime has been progressively developed in the last twenty years, constructing a theoreticalframework to understand the second one remains an open problem—a central one in many ofthe contexts in which rough energy landscapes play a role.The main reason for this state of affairs is that activated dynamics is well understood mainlyin low dimensional cases, where the number of minima and of saddles connecting them is finite and possibly small. The standard methods to tackle this problem were developedquite independently in statistical physics to analyze the phenomenon of nucleation at first-order phase transitions [13], in quantum field-theory for tunneling between degenerate vacua(where Planck’s constant plays the role of temperature) [14], and in probability theory [15].On the contrary, rough high-dimensional energy landscapes are characterized by diverging (exponentially in N ) number of paths that connect a diverging number of metastable states.In this case standard frameworks are not adapted, and new ideas and methods are needed.The main technical difficulty in establishing a theory of activated processes for mean-fieldglassy systems and high-dimensional rough landscapes is that the correct order parameterthat describes glassy dynamics is the correlation function between two different times [7].This is different with respect to standard phase transitions where one instead focuses on one-time (or point) functions. In consequence, contrary to known situations in which to describean activated process one has to find the rare trajectory, called instanton, that connects twominima and that corresponds to the optimal change of the one-point function correspondingto the order parameter [16], in this case one has to find the instanton on a two-point function.This is a quite different, less intuitive and more complex mathematical object. Henceforth, inorder to highlight this difference, we will call it dynamical instanton . Although some resultshave been given in the past literature [17, 18, 19, 20], the problem of finding the dynamicalinstanton corresponding to the activated jump out of a given minimum of the energy landscapeof a mean-field glass model remains a largely unsolved challenge. Here we provide the firstcomputation of such dynamical instanton and characterize the properties of the new minimareached after barrier-crossing.In order to achieve this goal we make use of the results we obtained recently on the numberof the stationary points constrained to be at fixed overlap q (or distance d ) from a givenminimum in a prototypical energy landscape [21, 22] (see also [23] for a related analysis).These studies showed that, given an arbitrary local minimum s of the energy landscape withenergy density (cid:15) , the landscape in its vicinity is populated by rank-1 saddles, that constitute2 ciPost Physics Submission available escape states when the system is trapped in s . By extending to dynamics thetheoretical framework developed for the high-dimensional Kac-Rice method (see also [24]),we derive dynamical equations describing the evolution of the system conditioned to startfrom such unstable saddles as initial states. By analyzing these equations analytically andintegrating them numerically, we show that they admit two solutions which are associatedto the descents toward the two minima reachable from the saddle. In this way, we mapout the first geometrical properties of the Morse complex, i.e. we characterize all the localminima that are connected to the original reference one through rank-1 saddles (as illustratedin Fig. 1). We then resort to dynamical field theory and to the time-reversal property ofstochastic dynamics to construct the dynamical instantons for the two-point functions. Thetwo dynamical solutions discussed above are used as building blocks: the part of the dynamicalinstanton associated to the ascent of the system from the original minimum to the nearbysaddle is obtained through the time reversal of the relaxation path from the saddle down tothe minimum [18, 19]. We then combine this contribution with the one corresponding to thedescent from the saddle to the new minimum to finally obtain the shape of the dynamicalinstanton, see Fig. 4.In the following section a summary of the state-of-the-art and our main contributions ispresented, we will then expose in details our analysis. We focus on the energy landscape associated to the p -spin spherical model: E [ s ] = − (cid:88) i , ··· ,i p J i ··· i p s i · · · s i p , (1)defined at each point s = ( s , · · · , s N ) of an N -dimensional sphere, s · s = N . The couplings J i ··· i p are independent Gaussian random variables with zero average and variance (cid:104) J i ··· i p (cid:105) =1 / p ! N p − , and are symmetric under permutations of the indexes. The functional (1) hasbeen the subject of an extensive amount of research devoted to understanding its statisticalproperties, which started with the earlier investigations [25, 26, 27, 28, 29, 30] and culminatedin the most recent results [31, 32, 21, 22]. These works highlighted a peculiar organizationof the landscape stationary points in terms of their energy density (cid:15) = E /N and of theirstability: while at large value of the energy the landscape is dominated by saddles witha huge index (i.e., number of unstable directions), the local minima and low-index saddlesconcentrate at the bottom of the landscape, below a critical threshold value of the energydensity (cid:15) th . Their number N k ( (cid:15) ) ( k being the index) is exponentially large in N , its typicalvalue being governed by a positive complexity Σ k ( (cid:15) ) = lim N →∞ (cid:104) log N k ( (cid:15) ) (cid:105) /N that decreaseswith k . The high-dimensionality of configuration space entails that most of these low-energyminima and saddles are orthogonal to each others on the sphere, i.e. they normalized overlap q ( s, s (cid:48) ) = lim N →∞ s · s (cid:48) /N is typically equal to zero.In order to find the escape paths from a given minimum, one needs to perform a morethorough analysis. In particular, it is important to scan the landscape in the vicinity ofany of its stationary points. This information is accessible via large deviation techniques by3 ciPost Physics Submission computing the complexity of the stationary points constrained to be at fixed overlap q (cid:54) = 0from the reference stationary point [23, 21, 22].Figure 1: Pictorial representation of the landscape with a pair of local minima connected by a saddle s . The lines represent the dynamical evolution of the system conditioned to start from the saddle asan initial condition. In Ref. [21] we computed the complexity of the typical stationary points that are foundin the vicinity of a reference minimum, extracted uniformly from the ensemble of minimahaving a given energy density larger than the ground state (cid:15) gs and smaller than the threshold (cid:15) th . Henceforth we denote with s the reference minimum, and with s a stationary point atoverlap s · s = N q from it. We let (cid:15) , (cid:15) be the corresponding energy densities, and Σ( (cid:15) , q | (cid:15) )the complexity of the stationary points at energy (cid:15) . The results for a representative valueof energy (cid:15) of the reference minimum are summarized in Fig. 2. The stationary points thatare closer to the reference minimum (i.e., at larger overlap) typically appear at an overlapthat we denote with q M , and are at high energy density (equal to (cid:15) th in the case of Fig. 2).At each overlap smaller than q M we find an exponentially large number of stationary points(Σ > (cid:15) > (cid:15) ( q | (cid:15) ). The lower bound (cid:15) ( q | (cid:15) ) corresponds to theenergy of the deepest stationary points at overlap q : their complexity is exactly zero. Forany fixed (cid:15) smaller than (cid:15) th (see for instance the dashed arrow in Fig. 2), the closeststationary points with that energy are found at an overlap q m ( (cid:15) ), and are rank-1 saddles:their Hessian has an eigenvalue density with a positively supported bulk (like for minima),plus an isolated eigenvalue that is separated from the bulk, and negative. The eigenvectorassociated to that eigenvalue has a macroscopic projection along the direction connecting s and s in configuration space, indicating that the saddle s is unstable in a direction that‘points’ towards the minimum s . This remains true decreasing the overlap, up to a value q ms ( (cid:15) ) where a transition to minima occurs. This is the overlap at which the curve (cid:15) ms ( q | (cid:15) )intersects the given (cid:15) : the points at this overlap are marginally stable rank-1 saddles, with oneflat mode (the isolated eigenvalue is exactly equal to zero). For smaller overlaps the stationarypoints are minima; the closest ones are still correlated to the minimum s (dashed gray regionin Fig. 2), since their Hessian still exhibits an isolated eigenvalue, that is nevertheless positive.4 ciPost Physics Submission Eventually, for even smaller q the points become minima that are totally uncorrelated to s . At q = 0, we recover the unconstrained complexity of local minima. Therefore, all the stationarypoints enclosed in the violet region of Fig. 2 are typically saddles that are geometricallyconnected to the reference minimum in configuration space. Their complexity is shown in thesame figure. Among them, the deepest ones have parameters q ∗ ( (cid:15) ) , (cid:15) ∗ ( (cid:15) ) that correspond tothe intersection between the curves (cid:15) ms ( q | (cid:15) ) and (cid:15) ( q | (cid:15) ).Figure 2: Left.
The colored regions identify the range of energy densities (cid:15) of the stationary pointsfound at overlap q with a minimum of energy density (cid:15) = − . ρ ( λ ) of the Hessian matrices at the stationary points is sketchedat the bottom of the plot. Right.
Color plot of the complexity of the rank-1 saddles as a function oftheir energy and overlap q with the reference minimum of energy (cid:15) = − . All the saddles lying in the vicinity of the reference minimum s , and corresponding to theviolet region in Fig. 2, represent possible escape states for the system trapped in the localminimum. In this work we study where gradient descent dynamics starting from these saddlesland in the energy landscape. By developing a theoretical framework that combines the Kac-Rice method and dynamical mean-field theory, we obtain the equations characterizing theminima that are connected to the original reference minimum through the saddles, see Fig. 1:these are the states that the system can reach if it manages to escape from s through one of thesurrounding saddles. The connectivity of s in configuration space can thus be characterizedby studying the energy density (cid:15) ∞ and the overlap N x ∞ = s ∞ · s of the minima s ∞ reachedasymptotically, as a function of the saddle parameters q, (cid:15) .Fig. 3 shows the resulting distributions, for a representative value of (cid:15) as above. Interestingcorrelations emerge between minima and saddles: at fixed energy (cid:15) of the saddle, those athigher overlap (i.e., those closer to the reference minimum) are more optimal, as they allowto reach minima that lie deeper in the landscape, and at furthest distance from the referenceminimum. Upon changing the energy of the saddle, one discovers that there exists a trade-offbetween energy and overlap: the saddles that connect the reference minimum to the deeperones are not the same ones that connect it to the further ones, thus allowing to explore alarger portion of configuration space. In particular, the saddles at q ∗ , (cid:15) ∗ that correspond to theminimal energy barrier are optimal in terms of energy of s ∞ , but not in terms of its overlap5 ciPost Physics Submission x ∞ . Overall, we see that both the range in energy and in overlap of the connected minimais rather limited: escaping through these saddles, the system reaches minima that are highlycorrelated from the reference one. We comment on the implications of this on the dynamicsin Sec. 7, and refer to Sec. 4 for a more detailed analysis of the asymptotic solutions of thedynamical equations.Figure 3: Left.
Energy density (cid:15) ∞ of the minimum reached asymptotically by the dynamics startingfrom an index-1 saddle at energy (cid:15) and overlap q with the reference minimum having energy (cid:15) = − . Right.
Overlap x ∞ between the reference minimum and the one reached asymptotically bythe dynamics starting from the saddle at energy (cid:15) and overlap q with the reference minimum. As already stressed in the introduction, one of the main aim of this work is to obtain thedynamical instanton that corresponds to the activated process associated to the escape froma given minimum s towards a new minimum s ∞ , see Fig. 1. Instantons are in generalobtained as special extremal solutions of a large deviation functional [33, 15]. In the case ofmean-field spin glasses the corresponding mathematical object is a functional of the two-pointfunctions [34, 35]. Although, in principle one could look for dynamical solutions by extremizingthis functional and imposing suitable boundary conditions in time, in practice analyzing thecorresponding equations represents a formidable challenge. No numerical solution has beenobtained yet. On the analytic side, despite the results in [17, 18, 19, 20] the problem remainslargely open. The main obstacle is the lack of intuition on the kind of solution one is lookingfor — an information that is missing so far. The only case in which a dynamical instanton hasbeen fully worked out is in the study of finite-time metastable states where periodic boundarycondition in time are enforced [35]. This is quite a different situation with respect to the onewe are interested in here.In the following we show how to obtain the dynamical instanton corresponding to theactivated process sketched in Fig. 1. The resulting shape of the two-point correlation functionis shown in Fig. 4. It displays three time regimes: the first one corresponding to the ascentfrom s , the second one corresponding to the approach and the departure from the saddle,and the final one associated to the descent towards the new minimum. Since the basic objectsis a symmetric two-time functions, C ( t, t (cid:48) ), this leads to six different time-sectors and sixdifferent behaviors for the correlation function (depending on which of the three regimes thetimes t and t (cid:48) belong to). The explicit form of the dynamical instanton associated to the6 ciPost Physics Submission simple activated process in Fig. 1 will be instrumental in finding instantons associated tomore complex relaxation processes, in particular to equilibrium relaxation. We shall get backto this issue in the conclusion.Figure 4: Representation of the correlation function c ( t, t (cid:48) ) along the reconstructed (see Sec. 6)instantonic solution that links a reference minimum ( s ) at energy (cid:15) = − .
167 to a neighboringminimum ( s ∞ ) reached through a saddle ( s ) at energy (cid:15) = − . q = 0 .
75 with s . The plot shows correlation equal to one on the diagonal and plateaux on other three differentlevels, corresponding to the overlaps q = 0 .
75 between s and s , c ∞ = 0 .
957 between s and s ∞ , and x ∞ = 0 .
619 between s and s ∞ . In this section we derive the equations describing the system evolution with specific initialconditions, that correspond to being in a saddle at a fixed distance of a given local minimumof the energy landscape.We remark that dynamical equations with constrained initial conditions for the p -spin spher-ical model have been derived in simpler settings, see for instance Refs. [36, 37, 24]. Inparticular, Ref. [36] studies the exponential relaxation of the system initialized within oneof the metastable states that contribute to the Boltzmann measure in the so called dynam-ical phase, at temperatures between the static and the dynamic transition temperatures. InRef. [37] and in the more recent [24] the overlap between the initial condition of the dynamicsand a thermalized condition in the same temperature range is also enforced to take a fixed,non-zero tunable value.The approach we present below goes one step further since we condition on the initial con-7 ciPost Physics Submission dition s to be itself a stationary point, beside conditioning on its energy density and onthe overlap with the reference minimum s . From the technical point of view our approachcombines the Kac-Rice method developed to study critical points of high-dimensional roughlandscapes [38, 31] with dynamical field theory [33, 34]. Let s ( t ) denote the spin configuration at time t , and let E [ s t ] be the time-dependent energyfield evaluated at s ( t ): E [ s t ] = − (cid:88) i , ··· ,i p J i ··· i p s i ( t ) · · · s i p ( t ) . (2)The vector s ( t ) is obtained as a solution of the Langevin equation: ds i ( t ) dt = − δ E [ s t ] δs i ( t ) − z ( t ) s i ( t ) + ξ i ( t ) , (3)where ξ i ( t ) is white noise with correlations (cid:104) ξ i ( t ) ξ j ( t (cid:48) ) (cid:105) = αδ ij δ ( t − t (cid:48) ) , α ≥ , (4) z ( t ) is a Lagrange multiplier that enforces the spherical constraint s ( t ) · s ( t ) = N , α is equalto twice the temperature T , and δ E [ s t ] δs i ( t ) = − p (cid:88) i , ··· ,i p J ii ··· i p s i ( t ) · · · s i p ( t ) . (5)With this normalization, typically E [ s t ] ∼ √ N . We assume that the dynamics has a specifiedinitial condition s (0) = s , with corresponding energy: E = N (cid:15) ≡ E [ s ] = − (cid:88) i , ··· ,i p J i ··· i p ( s ) i · · · ( s ) i p . (6)The dynamical generating functional corresponding to the stochastic evolution (3) and ob-tained integrating over the noise reads: Z D ( s ) = (cid:90) s (0)= s D s t D ˆ s t exp (cid:40) N (cid:88) i =1 (cid:90) ∞ dt ˆ s i ( t ) (cid:20) α s i ( t ) − ds i ( t ) dt − z ( t ) s i ( t ) − δ E [ s t ] δs i ( t ) (cid:21)(cid:41) , (7)where ˆ s i ( t ) are auxiliary fields and we highlighted the dependence on the initial condition ofthe dynamical evolution. We aim at averaging the above dynamical functional over all possible initial conditions s thatare stationary points found in the vicinity of some local minimum s of the energy landscape,having energy: E = N (cid:15) ≡ E [ s ] = − (cid:88) i , ··· ,i p J i ··· i p ( s ) i · · · ( s ) i p . (8) This is obtained exponentiating the delta function imposing the dynamical constraint (3), and performingthe rotation i ˆ s j ( t ) → − ˆ s j ( t ). The Itˆo prescription is used, implying that the Jacobian is equal to one. ciPost Physics Submission We assume that the two stationary points are at overlap
N q = s · s , and define the followingaverage over the initial conditions: E q [ ⊗ ] = (cid:90) ds (cid:90) ds δ ( s · s − N q ) f (cid:15) ( s ) N ( (cid:15) ) f (cid:15) ( s ) N s ( (cid:15) , q | (cid:15) ) ⊗ , (9)where the integrals are over the sphere of radius √ N . The explicit form of the measure isgiven by: f (cid:15) ( s ) = N − (cid:89) α =1 δ (cid:32) N (cid:88) i =1 e αi [ s ] ∂ E [ s ] ∂s i (cid:33) δ ( E [ s ] − N (cid:15) ) | det H [ s ] | . (10)This measure encodes the spherical constraint, since e α [ s ] with α = 1 , · · · , N − s , and H [ s ] is the Hessianmatrix of the energy functional at the point s , which is also projected onto the tangent planeand has components: H αβ = e α [ s ] · ∂ E [ s ] ∂s · e β [ s ] − N (cid:18) ∂ E [ s ] ∂s · s (cid:19) δ αβ . (11)The normalization N s ( (cid:15) , q | (cid:15) ) denotes the total number of stationary points of energy density (cid:15) that are found at overlap q with a fixed stationary point s having energy density (cid:15) ,whereas N ( (cid:15) ) is the total number of stationary points of energy density (cid:15) . We thus definethe generating functional averaged over the initial conditions as: Z D = E q [ Z D ( s )] . (12)The interpretation of (9) is as follows: the measure (10) weights uniformly all stationary pointsat a given value of energy density (cid:15) ; for energies below the threshold energy, these stationarypoints are typically local minima, since the number of saddles of finite index is exponentiallysuppressed in the dimension N (with respect to the number of minima). Therefore, in thelarge- N limit the point s extracted with such measure will be a local minimum with a highprobability, that converges to one as N → ∞ . Similarly, the analysis of Ref. [21] reveals thatfor each choice of q, (cid:15) , the energy landscape is dominated by one specific type of stationarypoints that are either local minima or index-1 saddles, see Fig. 2. More precisely, any localminimum s is typically surrounded by an exponentially large (in the dimension N ) populationof stationary points distributed over a finite range of overlaps q ; among them, the ones thatare at larger overlap q (and thus closer to the minimum) are typically index-1 saddles. Thisimplies that if a stationary point is selected at random among those at large-enough overlap,with probability that converges to one in the large- N limit this point will be an index-1 saddle.By suitably changing the parameters q, (cid:15) at fixed energy (cid:15) , we can select the initial condition s to be either an index-1 saddle or a local minimum. Of course we are particularly interestedin the regime of parameters in which the initial condition is an unstable stationary point, i.e.a saddle. The generating functional (12) can be averaged over the quenched random couplings J i , ··· ,i p .We denote the corresponding average with (cid:104)Z D (cid:105) = (cid:68) (cid:90) (cid:90) ds ds δ ( s · s − N q ) f (cid:15) ( s ) N ( (cid:15) ) f (cid:15) ( s ) N s ( (cid:15) , q | (cid:15) ) Z D ( s ) (cid:69) . (13)9 ciPost Physics Submission Computing this average is a priori non-trivial because of the normalizations in the denomina-tor, which are explicit functions of the random couplings. The computation can be performedexploiting the identity x − = lim n → x n − . This naturally leads to a replica calculation,in which higher moments of the quantities N ( (cid:15) ) and N s ( (cid:15) , q | (cid:15) ) have to be determined.As it follows from the results of [21], however, such a replica calculation reproduces the re-sults obtained within the so called annealed approximation, which in this case corresponds toaveraging separately the numerators and the denominator of (13): (cid:104)Z D (cid:105) = (cid:68) (cid:82) (cid:82) ds ds δ ( s · s − N q ) f (cid:15) ( s ) f (cid:15) ( s ) Z D ( s ) (cid:69) (cid:104)N ( (cid:15) ) N s ( (cid:15) , q | (cid:15) ) (cid:105) . (14)We can therefore focus on the average of the numerator, which we denote with: I ( (cid:15) , q | (cid:15) ) = (cid:90) ds ds δ ( s · s − N q ) (cid:90) s (0)= s D s t D ˆ s t e V [ s t , ˆ s t ] J ( (cid:15) , q | (cid:15) ) (15)where J ( (cid:15) , q | (cid:15) ) = (cid:68) f (cid:15) ( s ) f (cid:15) ( s ) e − (cid:80) Ni =1 (cid:82) ∞ dt ˆ s i ( t ) δ E [ st ] δsi ( t ) (cid:69) (16)and V [ s t , ˆ s t ] = N (cid:88) i =1 (cid:90) ∞ dt ˆ s i ( t ) (cid:20) α s i ( t ) − ds i ( t ) dt − z ( t ) s i ( t ) (cid:21) . (17)To perform the average over the random couplings, we make use of the following trick:because the energy functional is homogeneous, it can be expressed (together with its gradient)in terms of the time-dependent symmetric matrix field M ij ( t ) defined as: M ij [ s t ] ≡ δ E [ s t ] δs i ( t ) δs j ( t ) = − p ( p − (cid:88) i , ··· ,i p J iji ··· i p s i ( t ) · · · s i p ( t ) . (18)Indeed, we can write: δ E [ s t ] δs i ( t ) = 1 p − N (cid:88) j =1 M ij [ s t ] s j ( t ) , E [ s t ] = 1 p ( p − N (cid:88) i,j =1 M ij [ s t ] s i ( t ) s j ( t ) . (19)The matrix field (18) is symmetric and random, with a Gaussian statistics induced by the cou-plings. The covariance of the field evaluated along two fixed different dynamical trajectories s a ( t ) , s b ( t (cid:48) ) is given by: (cid:104) M ij [ s ta ] M kl [ s t (cid:48) b ] (cid:105) = p ( p − N ( δ ik δ jl + δ il δ jk ) (cid:18) s a ( t ) · s b ( t (cid:48) ) N (cid:19) p − + p ( p − p − N ×× (cid:8) [ s b ( t (cid:48) )] i ( δ jl [ s a ( t )] k + δ jk [ s a ( t )] l ) + [ s b ( t (cid:48) )] j ( δ il [ s a ( t )] k + δ ik [ s a ( t )] l ) (cid:9) (cid:18) s a ( t ) · s b ( t (cid:48) ) N (cid:19) p − + p ( p − p − p − N [ s b ( t (cid:48) )] i [ s b ( t (cid:48) )] j [ s a ( t )] k [ s a ( t )] l (cid:18) s a ( t ) · s b ( t (cid:48) ) N (cid:19) p − . (20)10 ciPost Physics Submission As a consequence, the average over the random couplings can be equivalently re-writtenas an average over this matrix field. This is convenient as it allows us to account for the con-straints in the initial condition of the dynamics (encoded in the measure (10)) in a straight-forward way, given that for both s a with a = 1 , E [ s a ] = s a · M [ s t =0 a ] · s a p ( p −
1) =
N (cid:15) a , N (cid:88) i =1 e αi [ s a ] ∂ E [ s a ] ∂ ( s a ) i = e αi [ s a ] · M [ s t =0 a ] · s a p − . (21)We define the initial conditions of the matrix field M [ s a ( t = 0)] = m a . As we show inAppendix B, by averaging over the matrix field and implementing the constraints (21), wecan re-write (15) in the form: I ( (cid:15) , q | (cid:15) ) ∝ (cid:90) s · s = Nq ds ds (cid:90) s (0)= s D s t D ˆ s t e S [ s t , ˆ s t ] K [ s t , ˆ s t ] , (22)where the proportionality factors do not depend on the dynamical variables s t , ˆ s t , and thusdo not matter for the derivation of the dynamical equations. In this formula S [ s t , ˆ s t ] is thedynamical action, whereas the term K [ s t , ˆ s t ] is given by an integral over the initial conditionsof the matrix field (18). We describe the structure of the two terms in the following subsection,and refer to the Appendices for the detailed derivations. We have: S [ s t , ˆ s t ] = V [ s t , ˆ s t ] + S [ s t , ˆ s t ] − S B [ s t , ˆ s t ] , (23)where V [ s t , ˆ s t ] is given in (17). The action S encodes the dynamical evolution given by(5), and is generic. The term S B instead accounts for the peculiar initial conditions of thedynamics: it arises when imposing that the initial condition s is a stationary point of energydensity (cid:15) , at overlap q from a local minimum of energy (cid:15) . Both actions depend on thedynamical variables only through the two-point functions contained in the 2 × Q ( t, t (cid:48) )with components: Q ( t, t (cid:48) ) = 1 N (cid:18) s ( t ) · s ( t (cid:48) ) s ( t ) · ˆ s ( t (cid:48) ) s ( t (cid:48) ) · ˆ s ( t ) ˆ s ( t ) · ˆ s ( t (cid:48) ) (cid:19) ≡ (cid:18) c ( t, t (cid:48) ) r ( t, t (cid:48) ) r ( t (cid:48) , t ) d ( t, t (cid:48) ) (cid:19) . (24)In addition, we introduce the 2 × c ( t ) = 1 N (cid:18) s (0) · s ( t ) s (0) · s ( t ) s (0) · s ( t ) s (0) · s ( t ) (cid:19) , r ( t ) = 1 N (cid:18) ˆ s (0) · s ( t ) ˆ s (0) · s ( t )ˆ s (0) · s ( t ) ˆ s (0) · s ( t ) (cid:19) (25)and the vector: (cid:18) x ( t ) x ( t ) (cid:19) = (cid:18) c ( t ) r ( t ) (cid:19) . (26)With this notation, it holds: S [ s t , ˆ s t ] → S [ Q ] = N p (cid:90) ∞ dtdt (cid:48) (cid:8) [ c ( t, t (cid:48) )] p − d ( t, t (cid:48) ) + ( p − c ( t, t (cid:48) )] p − r ( t, t (cid:48) ) r ( t (cid:48) , t ) (cid:9) . (27)11 ciPost Physics Submission This term thus reproduces the dynamical action obtained when starting from random initialconditions [25]. All the non-trivial information on the initial condition of the dynamicalevolution is contained in S B . This term depends explicitly on the overlap q , as well as on theenergy densities (cid:15) , (cid:15) . It contains two contributions: S B [ s t , ˆ s t ] → S B [ Q, x ] = S (1) B [ Q, x ] + S (2) B [ Q, x ] , (28)which are derived in Appendix D. The first contribution S (1) B [ Q, x ] is generated by conditioning s a to be stationary points. We can write it as: S (2) B [ Q,x ]= N p q q − q p (cid:90) ∞ ds (cid:90) ∞ ds (cid:48) (cid:88) a,b =1 (cid:2) δ ab (1+ q p − ) − q p − (cid:3)(cid:2) c a ( s ) c b ( s (cid:48) ) (cid:3) p − X ab ( s,s (cid:48) ) , (29)with X ab ( s, s (cid:48) ) = c a ( s ) c b ( s (cid:48) ) (cid:18) d ( s, s (cid:48) ) − f [ r ( s ) , r ( s (cid:48) )]1 − q (cid:19) + r a ( s ) r b ( s (cid:48) ) (cid:18) c ( s, s (cid:48) ) − f [ c ( s ) , c ( s (cid:48) )]1 − q (cid:19) ++ ( p − (cid:0) r a ( s ) c b ( s (cid:48) ) + c a ( s (cid:48) ) r b ( s ) (cid:1) (cid:18) r ( s, s (cid:48) ) − f [ c ( s ) , r ( s (cid:48) )]1 − q (cid:19) (30)and where, for arbitrary 2 × x ab with a, b ∈ { , } , we haveintroduced the form: f ( x, y ) = x y + x y − q ( x y + x y ) . (31)The second contribution S (2) B [ Q, x ] follows from conditioning both on the gradient and on theenergy density of the points s a , and reads: S (2) B [ Q, x ] = 12 p ( p − (cid:88) i,j =1 V i [ Q, x ] A ij V j [ Q, x ] , (32)where V [ Q, x ] = (cid:90) ∞ dtp [ c ( t )] p − r ( t ) + 2 (cid:15) V [ Q, x ] = (cid:90) ∞ dtp [ c ( t )] p − r ( t ) + 2 (cid:15) V [ Q, x ] = (cid:90) ∞ dt [ c ( t )] p − r ( t ) − qr ( t ) (cid:112) − q + (cid:90) ∞ dt ( p − c ( t )] p − r ( t ) c ( t ) − qc ( t ) (cid:112) − q V [ Q, x ] = (cid:90) ∞ dt [ c ( t )] p − r ( t ) − qr ( t ) (cid:112) − q + (cid:90) ∞ dt ( p − c ( t )] p − r ( t ) c ( t ) − qc ( t ) (cid:112) − q . (33)and A is a 4 × K [ s t , ˆ s t ]. This term is obtained as an integral over the( N − × ( N −
1) matrices m a , which denote (up to a shift) the projection of M [ s a ( t = 0)] = m a ciPost Physics Submission on the tangent plane at s a . Its explicit form reads: K [ s t , ˆ s t ] = (cid:90) (cid:89) a =1 dm a e − (cid:80) N − α ≤ β =1 (cid:80) N − γ ≤ δ =1 (cid:80) a,b =1 m aαβ [Ω ∗ ] abαβ,γδ m bγδ (cid:89) a =1 | det( m a − Φ a [ s t , ˆ s t ] − p(cid:15) a ) | . (34)From this expression we see that the Hessian matrices m a are Gaussian distributed, withinverse covariances Ω ∗ = [Σ ∗ ] − that are given explicitly in Appendix C. The term Φ a [ s t , ˆ s t ]inside the determinant denotes a matrix whose components can be written as:Φ aαβ [ s t , ˆ s t ] = φ aαβ [ s t , ˆ s t ] − δ α,N − δ β,N − µ a ( q, (cid:15) , (cid:15) ) , (35)where we introduced the functions (cid:18) µ ( q, (cid:15) , (cid:15) ) µ ( q, (cid:15) , (cid:15) ) (cid:19) = 1 a ( q ) (cid:18) (cid:15) a ( q ) − (cid:15) a ( q ) (cid:15) a ( q ) − (cid:15) a ( q ) (cid:19) , (36)with: a ( q ) = p ( p − (cid:0) − q (cid:1) (cid:2) ( p − q p +2 − ( p − q p + q (cid:3) a ( q ) = p ( p − (cid:0) − q (cid:1) q p (cid:2) q p − ( p − q + ( p − q (cid:3) a ( q ) = q − p + q p +2 − (cid:0) ( p − q − p − pq + ( p − (cid:1) q p +2 . (37)Thus, these matrices are a sum of a rank-one projector and of a second matrix φ a whichdepends in principle on the dynamical variables s ( t ) , ˆ s ( t ), and can not be expressed com-pactly in terms of the order parameters (24) and (25). It might therefore seem that thedeterminants in (34) give a contribution to the action that depends explicitly on the wholetime evolution, and that therefore has to be taken into account when deriving the dynamicalequations. However, as it appears from the analysis performed in Appendix C, the compo-nents of φ a vanish when the dynamical average is restricted to trajectories that fulfill therequirement of causality. As a consequence, when the dynamical evolution is causal the ma-trices Φ a reduce to rank-1 projectors, that depend explicitly only on the parameters q, (cid:15) and (cid:15) that characterize the initial condition. This is consistent with the natural expectation thatthe terms appearing in the measure (9), that select the initial condition of the dynamics, arenot affected by the subsequent dynamical evolution of the system. Inspecting the distributionof the entries of the matrix m a and the explicit form of the functions µ a ( q, (cid:15) , (cid:15) ), one caneasily show that the integrand in K [ s t , ˆ s t ] reproduces exactly the flat measure over criticalpoints at overlap q with each others, see Appendix C for details. Therefore, accounting forthe causality of the dynamical evolution we recover K [ s t , ˆ s t ] causality −→ K ( q, (cid:15) , (cid:15) ) = (cid:104)N ( (cid:15) ) N s ( (cid:15) , q | (cid:15) ) (cid:105) , (38)which cancels precisely with the denominator in (14). As it follows from this simplification,all the information on the initial conditions s enters in the boundary terms of the dynamicalaction (23) only. These terms turn out to encode the statistical properties of the Hessian atthe initial condition s , as we show explicitly in Sec. 3.3.1. To finally obtain the dynamical equations, we focus on the remaining term: (cid:90) D s t D ˆ s t e V + S −S B = (cid:90) D Q D x A [ Q, x ] e S [ Q ] −S B [ Q,x ] , (39)13 ciPost Physics Submission where we introduced the order parameters (24) and (25), and: A [ Q, x ] = (cid:90) D s t D ˆ s t e V [ s t , ˆ s t ] δ (cid:16) N Q αβ ( t, t (cid:48) ) − s ( α )2 ( t ) · s ( β )2 ( t (cid:48) ) (cid:17) δ (cid:16) N x α ( t ) − N s ( α )1 (0) · s ( t ) (cid:17) , (40)where s (1) a ( t ) = s a ( t ), s (2) a ( t ) = ˆ s a ( t ) and the product over α, β is implicit. The integral overthe dynamical variables s ( t ) , ˆ s ( t ) is Gaussian, with kernel: M ( t, t (cid:48) ) = (cid:18) − ∂ t + z ( t )) δ ( t − t (cid:48) )( ∂ t + z ( t )) δ ( t − t (cid:48) ) − αδ ( t − t (cid:48) ) (cid:19) . (41)A standard calculation gives: A [ Q, x ] = (cid:90) d Λ αβ e − N a [Λ; Q,x ] (42)with: a [Λ; Q, x ] =tr log ( M + 2 i Λ) + (cid:90) dsds (cid:48) x ( s ) ( M + 2 i Λ) ( s, s (cid:48) ) x ( s (cid:48) ) − i (cid:88) α,β (cid:90) dsds (cid:48) Q αβ ( s, s (cid:48) )Λ αβ ( s, s (cid:48) )(43)Substituting (42) into (39) and taking the variation with respect to Λ and Q we get: M + 2 i Λ = (cid:0) Q − xx T (cid:1) − , N δδQ [ S − S B ] + i Λ = 0 , (44)Combining these equations with the one obtained taking the variation of the action withrespect to x we obtain the coupled equations: M ⊗ (cid:0) Q − xx T (cid:1) − N δδQ [ S − S B ] ⊗ (cid:0) Q − xx T (cid:1) = ,M ⊗ x = 2 N δδQ [ S − S B ] ⊗ x + 1 N δδx [ S − S B ] (45)where we used the notation ( A ⊗ B ) αβ ( t, t (cid:48) ) = (cid:80) γ (cid:82) ds A αγ ( t, s ) B γ,β ( s, t (cid:48) ). A lengthy (butstraightforward) calculation of the functional variations of the action leads to the dynamicalequations reported below (see [34] for details of the derivation in the simplified case in whichthe boundary terms are absent). We stress that the equations are given under the assumptionthat the resulting typical dynamical trajectories are causal, meaning that we assume that thesaddle-point solution satisfies: d ( t, t (cid:48) ) = 0 and r ( t, t (cid:48) ) = 0 for t < t (cid:48) , (46)which implies in particular r (0 , t ) = 0 for any t >
0. The remaining equations are for thecorrelation function c ( t, t (cid:48) ), the response function r ( t, t (cid:48) ) for t > t (cid:48) , and the overlap x ( t ) ≡ x ( t ) = c ( t ) with the minimum s . We report them in the following, and refer to Appendix Efor the explicit expression of the constants involved.14 ciPost Physics Submission3.2.1 Dynamical equation for the overlap with the nearby minimum The equation for x ( t ) reads:[ ∂ t + z ( t )] x ( t ) = p ( p − (cid:90) t dsr ( t, s ) c p − ( t, s ) x ( s ) − p ( p − q q − q p (cid:90) t dsr ( t, s ) (cid:26) c p − ( t ) c p − ( s ) − q p − (cid:0) c p − ( t ) x p − ( s ) + x p − ( t ) c p − ( s ) x ( s ) (cid:1)(cid:27) − p ( p − q q − q p (cid:90) t dsr ( t, s ) (cid:26) x p − ( t ) x p − ( s ) − q p − (cid:0) x p − ( t ) c p − ( s ) + c p − ( t ) x p − ( s ) c ( s ) (cid:1)(cid:27) + G (cid:15),q [ c ( t ) , x ( t )] , (47)and G (cid:15),q depends linearly on the energies, and reads: G (cid:15),q [ c ( t ) , x ( t )] = (cid:88) a =1 (cid:15) a (cid:8) G a ( q ) c p − ( t ) + G a ( q ) x p − ( t ) + G a ( q ) c p − ( t ) x ( t ) + G a ( q ) x p − ( t ) c ( t ) (cid:9) (48)and the constants G ai ( q ) are functions of x (0) = q , and are reported in Appendix E. The equation for the correlation c ( t, t (cid:48) ) reads:[ ∂ t + z ( t )] c ( t, t (cid:48) ) = αr ( t (cid:48) , t ) + p ( p − (cid:90) t dsr ( t, s )[ c ( t, s )] p − c ( t (cid:48) , s ) + p (cid:90) t (cid:48) ds [ c ( t, s )] p − r ( t (cid:48) , s ) − p ( p − q q − q p c ( t (cid:48) ) (cid:90) t dsr ( t, s ) (cid:26) c p − ( t ) c p − ( s ) − q p − (cid:0) c p − ( t ) x p − ( s ) + x p − ( t ) x ( s ) c p − ( s ) (cid:1)(cid:27) − p ( p − q q − q p x ( t (cid:48) ) (cid:90) t dsr ( t, s ) (cid:26) x p − ( t ) x p − ( s ) − q p − (cid:0) x p − ( t ) c p − ( s ) + c p − ( t ) x p − ( s ) c ( s ) (cid:1)(cid:27) − p q q − q p (cid:90) t (cid:48) dsr ( t (cid:48) , s ) (cid:8) [ x ( t ) x ( s )] p − + [ c ( t ) c ( s )] p − − q p − (cid:0) [ x ( t ) c ( s )] p − + [ c ( t ) x ( s )] p − (cid:1)(cid:9) + F (cid:15),q [ c, x ] (49)where the energy-dependent part is a linear combination of (cid:15) , (cid:15) given by: F (cid:15),q [ c, x ] = (cid:88) a =1 (cid:15) a (cid:110) F a ( q ) c p − ( t ) c ( t (cid:48) ) + F a ( q ) c ( t (cid:48) ) x p − ( t ) + F a ( q ) c ( t (cid:48) ) c p − ( t ) x ( t )+ F a ( q ) x ( t (cid:48) ) x p − ( t ) + F a ( q ) x ( t (cid:48) ) x p − ( t ) c ( t ) + F a ( q ) x ( t (cid:48) ) c p − ( t ) (cid:111) , (50)15 ciPost Physics Submission and the constants are given in Appendix E. Setting t = t (cid:48) we obtain the equation for themultiplier z ( t ) enforcing the spherical constraint during the dynamics : z ( t ) = α p (cid:90) t dsr ( t, s ) [ c ( t, s )] p − − p ( p − q q − q p c ( t ) (cid:90) t dsr ( t, s ) (cid:26) c p − ( t ) c p − ( s ) − q p − (cid:0) c p − ( t ) x p − ( s ) + x p − ( t ) x ( s ) c p − ( s ) (cid:1)(cid:27) − p ( p − q q − q p x ( t ) (cid:90) t dsr ( t, s ) (cid:26) x p − ( t ) x p − ( s ) − q p − (cid:0) x p − ( t ) c p − ( s ) + c p − ( t ) x p − ( s ) c ( s ) (cid:1)(cid:27) − p q q − q p (cid:90) t dsr ( t, s ) (cid:8) [ x ( t ) x ( s )] p − + [ c ( t ) c ( s )] p − − q p − (cid:0) [ x ( t ) c ( s )] p − + [ c ( t ) x ( s )] p − (cid:1)(cid:9) + F (cid:15),q [ c, x ] (cid:12)(cid:12)(cid:12) t = t (cid:48) , (52)and F at equal times reduces to: F (cid:15),q [ c, x ] (cid:12)(cid:12)(cid:12) t = t (cid:48) = (cid:88) a =1 (cid:15) a (cid:110) F a c p ( t ) + ( F a + F a ) x p − ( t ) c ( t ) + ( F a + F a ) c p − ( t ) x ( t ) + F a x p ( t ) (cid:111) . (53)When t = 0, setting x ( t = 0) = q and c ( t = 0) = 1 we get: z (0) = α (cid:88) a =1 (cid:15) a (cid:110) F a + ( F a + F a ) q p − + ( F a + F a ) q + F a q p (cid:111) = α − p(cid:15) , (54)which for α = 0 reduces to the correct value of the Lagrange multiplier enforcing the sphericalconstraint at a stationary point of energy N (cid:15) . The equation for the response r ( t, t (cid:48) ) is formally unaltered by the coupling to the initialconditions, and reads:[ ∂ t + z ( t )] r ( t, t (cid:48) ) = δ ( t − t (cid:48) ) + p ( p − (cid:90) ∞ dsr ( t, s ) r ( s, t (cid:48) )[ c ( t, s )] p − . (55)In this equation the information on the initial conditions is implicitly encoded in the Lagrangemultiplier z ( t ). We now consider two interesting limits of the above equations, which we remind are derivedunder the assumption that the initial condition s ( t = 0) = s is a stationary point of energy This equation is obtained starting from the identity: (cid:2) ∂ t c ( t, t (cid:48) ) + ∂ t (cid:48) c ( t, t (cid:48) ) (cid:3) (cid:12)(cid:12)(cid:12) t,t (cid:48) = s = 0; (51)In particular, the factor 1 / α comes from the fact that only one of these two derivatives gives anon-zero contribution multiplying α , while all the other terms are doubled. ciPost Physics Submission density (cid:15) , at overlap q with a local minimum s of energy density (cid:15) .The first case we focus on consists in the limit α = 2 T →
0, when the noise in the Langevinequation vanishes and the dynamics reduces to gradient descent starting from a stationarypoint. As we shall see, and as expected, if this point is a minimum then the system remainsstuck there, otherwise if this point is a saddle a dynamical instability takes place.The second case corresponds to the limit q →
0, when the initial condition decouples from s , and one samples uniformly all stationary points at a given energy. For this reason we willrefer to it as “microcanonical initial conditions”. This limit is useful to check our equationssince it can be connected to the one analyzed in [36]. In the noiseless limit α →
0, the dynamical equations must admit a solution in which thesystem does not move away from the initial condition, given that the latter is a stationarypoint. We refer to this as the “static” solution. It is easy to check using the explicit form ofthe constants given in Appendix E that x ( t ) = q and c ( t, t (cid:48) ) = 1 solve the above equations inthis limit. Indeed, plugging this ansatz into (47) we get: z ( t ) q = z q = G E ,q [1 , q ] = (cid:88) a =1 (cid:15) a (cid:8) G a ( q ) + G a ( q ) q p − + G a ( q ) q + G a ( q ) q p − (cid:9) = − pq(cid:15) , (56)which rightly gives the value of the zero-time multiplier z (0) = z = − p(cid:15) . The same identityis obtained from (49). The equation (55) for the response becomes:( ∂ t + z ) r ( t, t (cid:48) ) = δ ( t − t (cid:48) ) + p ( p − (cid:90) ∞ dxr ( t, x ) r ( x, t (cid:48) ) . (57)Assuming time-translation invariance, this is equivalent to( ∂ τ + z ) R ( τ ) = δ ( τ ) + p ( p − (cid:90) τ dxR ( τ − x ) R ( x ) . (58)The Laplace transform of this equation is simply:[ ω + z ] ˆ R ( ω ) = 1 + p ( p − (cid:104) ˆ R ( ω ) (cid:105) , (59)where we used that the Laplace transform of the derivative is ω ˆ R ( ω ) − R (0 − ), and R (0 − ) = 0.The equation admits the shifted GOE resolvent as a solution, i.e.,ˆ R ( ω ) = G σ ( ω + z ) , z = − p(cid:15) , σ = σ p ( p − . (60)where the function G is given in (113). The inverse Laplace transform is proportional to aBessel function, R ( τ ) = e − z τ στ I (2 στ ) . (61)This result coincides with the one of stochastic dynamics in purely quadratic landscapes[39, 40], as in the noiseless limit the non-linear part of the potential is not explored.17 ciPost Physics Submission The initial condition s has a Hessian whose statistics depends on the parameters q, (cid:15) and (cid:15) , as recalled in Appendix A. Its eigenvalue density is almost entirely positive definite(and GOE-like), with the exception of possibly one negative eigenvalue that appears forcertain values of the parameters (given by the condition (111)). When the initial condition s is a saddle with one single negative eigenvalue, the “static” solution must be dynamicallyunstable, since there exist a direction in configuration space in which the landscape hasnegative curvature, allowing the system to escape from the stationary point, see Fig. 1. Forfixed q , this happens whenever the initial condition s is chosen to have energy (cid:15) ∈ [ (cid:15) ms , (cid:15) ],see Fig. 2. In order to check this instability from the dynamical equations, we consider thelinearization of Eq. (47) around the static solution x ( t ) = q . Setting x ( t ) = q + δx ( t ), we get: ddt δx ( t ) = O [ δx ] ( t ) (62)where the operator O acts as: O [ δx ] ( t ) = (cid:18) − z + ˜ G (cid:15),q − p ( p − p − q p − (1 − q ) q − q p (cid:90) t dsR ( t − s ) (cid:19) δx ( t )+ p ( p − (cid:18) − p q p − (1 − q ) q − q p (cid:19) (cid:90) t dsR ( t − s ) δx ( s ) , (63)and R ( s ) is the response in the stationary point with energy density (cid:15) , z = − p(cid:15) and ˜ G (cid:15),q reads: ˜ G (cid:15),q = (cid:88) a =1 (cid:15) a (cid:2) G a ( p − q p − + G a + G a ( p − q p − (cid:3) = (cid:15) a ( q ) − (cid:15) a ( q ) a ( q ) (64)with the a i ( q ) given in (37). The static solution becomes unstable when the linear operator O has eigenvalues that becomes positive. We assume that δx ( s ) is slowly varying, which iscorrect close to the transition where the instability is small. As a consequence, we can extractit from the integration in (63). Taking t → ∞ we get: O [ δx ] ( t ∞ ) = λ ∞ δx ( t ∞ ) , (65)with λ ∞ = − z + ˜ G (cid:15),q + p ( p − (cid:20) − ( p − q p − (1 − q ) q − q p (cid:21) (cid:90) ∞ dsR ( s ) (66)Using that the integral is the Laplace transform evaluated at zero, which is related to theGOE resolvent G σ with σ = p ( p − / (cid:90) ∞ dsR ( s ) = ˆ R (0) = G σ ( z ) , (67)see Eq. (60), one finds that λ ∞ = − z + ˜ G (cid:15),q + p ( p − (cid:20) − ( p − q p − (1 − q ) q − q p (cid:21) G σ ( z ) . (68)18 ciPost Physics Submission Using (64) we get that the instability condition λ ∞ = 0 reads: p(cid:15) + (cid:15) a ( q ) − (cid:15) a ( q ) a ( q ) + p ( p − (cid:20) − ( p − q p − (1 − q ) q − q p (cid:21) G σ ( − p(cid:15) ) √ . (69)This equation is precisely equivalent to the one corresponding to the isolated eigenvalue of theHessian at s being equal to zero. Indeed, as we recall in Appendix A the isolated eigenvalueof the Hessian is given by: λ ( q, (cid:15) , (cid:15) ) = λ min ( q, (cid:15) , (cid:15) ) − √ p(cid:15) , (70)where λ min solves the equation λ − µ ( q, (cid:15), (cid:15) ) − ∆ ( q ) G σ ( λ ) = 0 . (71)with µ ( q, (cid:15), (cid:15) ) = − √ (cid:15) a ( q ) − √ (cid:15) a ( q ) a ( q ) , ∆ ( q ) ≡ p ( p − (cid:20) − ( p − q p − (1 − q ) q − q p (cid:21) . (72)Multiplying (69) by √ G σ √ ( z ) = −√ G σ (cid:0) −√ z (cid:1) we obtain √ p(cid:15) − µ ( q, (cid:15) , (cid:15) ) − ∆ ( q ) G σ ( √ p(cid:15) ) = 0 , (73)which corresponds to λ min = √ p(cid:15) and thus λ ( q, (cid:15) , (cid:15) ) = 0. Therefore, the dynamicalsolution c ( t, t (cid:48) ) = 1 and x ( t ) = q becomes unstable exactly at the values of parameters atwhich s transitions from being a minimum to being a saddle, as expected. We now consider the case in which q →
0, where the initial condition of the dynamics s decorrelates from the minimum s . In this limit, the only non-vanishing G ak ( q ) constant is G ( q ) → − p , while the non-vanishing F ak ( q ) constants are F ( q ) , F ( q ) → − p . The equationfor x ( t ) reduces to:[ ∂ t + z ( t )] x ( t ) = p ( p − (cid:90) t dsr ( t, s ) (cid:2) c p − ( t, s ) x ( s ) − x p − ( t ) x p − ( s ) (cid:3) − p (cid:15) x p − ( t ) , (74)which is homogeneous and thus admits the solution x ( t ) ≡ x (0) = q = 0. The equationfor the correlation when x ( t ) = 0 for any t reduces to:[ ∂ t + z ( t )] c ( t, t (cid:48) ) = αr ( t (cid:48) , t ) + p ( p − (cid:90) t dsr ( t, s )[ c ( t, s )] p − c ( t (cid:48) , s ) + p (cid:90) t (cid:48) ds [ c ( t, s )] p − r ( t (cid:48) , s ) − p ( p − c ( t (cid:48) ) (cid:90) t dsr ( t, s ) c p − ( t ) c p − ( s ) − p (cid:90) t (cid:48) dsr ( t (cid:48) , s )[ c ( t ) c ( s )] p − − p(cid:15) c p − ( t ) c ( t (cid:48) ) , (75)while the Lagrange multiplier reads: z ( t ) = α − p(cid:15) c p ( t ) + p (cid:90) t dsr ( t, s ) c p − ( t, s ) − p c p − ( t ) (cid:90) t dsc p − ( s ) r ( t, s ) . (76)19 ciPost Physics Submission These equations give the evolution of the correlation function for a dynamics conditionedto start from a typical stationary point of energy density (cid:15) , which therefore will be a localminimum for (cid:15) < (cid:15) th . The first two terms in the second line of (75) and the last term in(76) are generated by conditioning on the stationarity of the initial condition: setting them tozero, we get the dynamical equations conditioned to start from a point extracted with uniformmeasure from the manifold at a given energy density (cid:15) .This is a case that has been already considered in the literature: it is the microcanonicalequivalent of the one analyzed in [36], where the initial condition is extracted with a Bolzmannmeasure at a temperature T (cid:48) between the statical and the dynamical transition temperatures.It provides a useful check of our method, which is different from the one followed in [36]. Infact, we recover the same dynamical equations, in particular the same boundary terms . The aim of this section is to study where the system falls after escaping from the saddleillustrated in Fig.1. One (trivial) possibility is to come back to the original reference minimum.The other possibility—the interesting one—is that the system lands in a different basin. Inorder to analyze this case, we consider the large time limit in which the system equilibrateswithin the basin. This allows us to obtain closed equations describing the properties of thebasin, or more precisely the minimum since we consider the small-noise case.
When the initial condition s is an unstable saddle, in presence of weak thermal fluctuations( α = 2 T (cid:54) = 0) the system eventually escapes from the saddle (even though this might requireextremely large times). In this section we study the asymptotic solutions of the dynamicalequations representing the dynamics within the basin that has been reached. We thereforeassume that after a finite time t eq a stationary limit is reached (see [37] for a similar compu-tation), meaning that: x ( t ) → x ∞ , c ( t ) → c ∞ , z ( t ) → z ∞ , r ( t, t (cid:48) ) → R ( t − t (cid:48) ) , c ( t, t (cid:48) ) → C ( t − t (cid:48) ) . (78)Moreover, we assume that in the asymptotic limit the dynamics equilibrates into some localminimum of the energy landscape, and that the fluctuation-dissipation relation holds at largetimes: R ( τ ) = − β ∂ τ C ( τ ) , with C ( τ ) → A ∞ . (79)In the limit of zero temperature, if the dynamics ends up asymptotically in a minimum then C ( τ ) → C (0) = 1). To capture the dynamical evolution it is necessary to For a comparison, one needs to keep in mind that the Lagrange multiplier µ ( t ) in [36] and the z ( t ) in thiswork are related by z ( t ) = µ ( t ) + p ( p − (cid:90) t dsr ( t, s ) c p − ( t, s ) + p T (cid:48) c p − ( t, . (77) ciPost Physics Submission introduce the scaling variable φ ( τ ) = β (1 − C ( τ )), with φ ∞ = β (1 − A ∞ ). Assuming that theinitial transient decouples from the long-time dynamics:lim t →∞ (cid:90) t eq dsR ( t, s ) {· · ·} = 0 , (80)the equation for z ( t ) becomes for t, t (cid:48) → ∞ : z ∞ = α pβ − A p ∞ ) + ˜ F , (81)with: ˜ F = − p q q − q p β (1 − A ∞ ) (cid:2) c p − ∞ − q p − c p − ∞ Q p − ∞ + Q p − ∞ (cid:3) + (cid:88) a =1 (cid:15) a (cid:2) F a c p ∞ + ( F a + F a ) Q p − ∞ c ∞ + ( F a + F a ) c p − ∞ Q ∞ + F a Q p ∞ (cid:3) (82)and β (1 − A p ∞ ) ≈ pβ (1 − A ∞ ). The equation for the correlation with these assumptions is ∂ t C ( t − t (cid:48) ) + z ∞ C ( t − t (cid:48) ) = pβ C ( t − t (cid:48) ) − A p ∞ ) − pβ (cid:90) tt (cid:48) ds C p − ( t − s ) ∂ s C ( s − t (cid:48) ) + ˜ F , (83)and using (82) we get: ∂ τ C ( τ ) − z ∞ (1 − C ( τ )) = − α − pβ − C ( τ )) − pβ (cid:90) τ ds C p − ( s ) ∂ s C ( τ − s ) . (84)Setting φ ( x ) = β (1 − C ( x )) and C p − ( x ) ≈ − ( p − φ ( x ) β − , β∂ x C ( x ) = − ∂ x φ ( x ) (85)we finally obtain: ∂ τ φ ( τ ) + z ∞ φ ( τ ) = α T + p ( p − (cid:90) τ ds φ ( τ − s ) ∂ s φ ( s ) , (86)which has a finite limit when T → α = 2 T ; integratingthe equation for the response from t (cid:48) to t reproduces (84). In the limit τ → ∞ we get: z ∞ φ ∞ = 1 + p ( p − φ ∞ ( φ ∞ − φ ) (87)and φ = 0. The fluctuation-dissipation relation implies that φ ∞ coincides with the static sus-ceptibility in the minimum reached asymptotically by the dynamics. The equation above is in-deed consistent with this interpretation since φ ∞ satisfies the same equation of G σ ( z ∞ ), where G σ is the GOE resolvent which is directly related to the static susceptibility, see Eq. (113),as well as (60). 21 ciPost Physics Submission Additional relations between the parameters x ∞ , c ∞ , z ∞ and φ ∞ are obtained from the t → ∞ limit of the equations for x ( t ) , z ( t ) and c ( t ), which gives the following coupled equations: z ∞ c ∞ = p ( p − φ ∞ (cid:20) c ∞ − q q − q p (cid:0) c p − ∞ − q p − x p − ∞ c p − ∞ − q p c p − ∞ x p − ∞ + qx p − ∞ (cid:1)(cid:21) + (cid:88) a =1 (cid:15) a (cid:2) c p − ∞ ( F a + qF a ) + x p − ∞ ( F a + qF a ) + c p − ∞ x ∞ F a + qF a x p − ∞ c ∞ (cid:3) (88)and z ∞ = p φ ∞ − p q q − q p φ ∞ (cid:0) c p − ∞ − q p − x p − ∞ c p − ∞ + x p − ∞ (cid:1) + (cid:88) a =1 (cid:15) a (cid:2) F a c p ∞ + x p − ∞ c ∞ ( F a + F a ) + c p − ∞ x ∞ ( F a + F a ) + F a x p ∞ (cid:3) (89)and z ∞ x ∞ = p ( p − φ ∞ (cid:20) x ∞ − q q − q p (cid:0) x p − ∞ − q p − x p − ∞ c p − ∞ − q p c p − ∞ x p − ∞ + qc p − ∞ (cid:1)(cid:21) + (cid:88) a =1 (cid:15) a (cid:2) G a c p − ∞ + G a x p − ∞ + G a c p − ∞ x ∞ + G a x p − ∞ c ∞ (cid:3) . (90)The solution of these four coupled equations gives information on the minima reachedasymptotically by the dynamics, as a function of the parameters q and (cid:15) a that specify theinitial conditions. In particular, the energy of the minimum reached asymptotically by thedynamics can be read out from z ∞ .As we anticipated, we expect two kinds of solutions for these equation when the initial condi-tion s is an unstable saddle. One solution should correspond to the trajectory that escapesfrom the saddle and goes back to the original minimum s . In fact, we do find that this setof equations admits the solution x ∞ = 1 and c ∞ = q , and substituting these values into (88),(89) and (90) we obtain the identity z ∞ = − p(cid:15) . This indicates that the two stationary points s and s are not only geometrically connected (meaning that the unstable direction of thesaddle s is oriented towards the minimum s in configuration space) but also dynamicallyconnected, since there exists a solution of the dynamical equations that corresponds to therelaxation from the saddle to the reference minimum, see also Sec. 5.The other (less-trivial) solution of the above system of equations instead corresponds to thesystem relaxing to another local minimum s ∞ that is connected to the reference one throughthe rank-1 saddle s . We focus on this second solution in the following. We now want to discuss the correlations between the pairs of minima connected by the index-1 saddles. In order to do so, we solve the asymptotic Eqs. (88), (89) and (90) for theparameters c ∞ , x ∞ and z ∞ . Given z ∞ , the response φ ∞ is then readily obtained solving thequadratic equation (87). We choose a representative energy of the reference minimum, equalto (cid:15) = − .
167 as in Fig. 2 (recall that (cid:15) gs ≈ − .
172 and (cid:15) th ≈ − . ciPost Physics Submission the constrained complexity [21], we know that index-1 saddles are the dominant stationarypoints in a range of energies and overlaps corresponding to the violet region in the figure: forany (cid:15) ∈ [ (cid:15) ∗ , (cid:15) th ] we find that the typical stationary points are saddles if q ∈ [ q ms ( (cid:15) ) , q m ( (cid:15) )].For the chosen (cid:15) , the deepest energy of these saddles is (cid:15) ∗ ≈ − . q ∗ ≈ . q m ( (cid:15) th ) = q M ≈ . (cid:15) , q the static solution of the dynamical equations(corresponding to c ∞ = 1 and x ∞ = q ) is unstable, and another solution of the asymptoticequations is found, with c ∞ <
1. We denote with (cid:15) ∞ the energy density of the minimum thatis reached asymptotically by the dynamics. Fig. 3 shows the values of this asymptotic energydensity as a function of the energy of the saddle (cid:15) and of its overlap q with the referenceminimum, as well as the values of the asymptotic overlaps x ∞ with the reference minimum.The following features are observed: • At fixed energy (cid:15) of the saddle, the asymptotic energy (cid:15) ∞ decreases with q , meaningthat the saddles that are closer to the reference minimum connect the latter to minimathat lie deeper in the landscape; the same holds true for the overlap x ∞ for sufficientlysmall values of (cid:15) (see the caption of Fig. 5 for more details). Therefore, among theFigure 5: Left.
The energy (cid:15) ∞ of the minima reached asymptotically by the dynamics decreaseswith q , at fixed energy (cid:15) of the saddle. Middle and Right.
The behavior of the asymptotic overlap x ∞ with q depends on (cid:15) : for (cid:15) sufficiently small, x ∞ decreases with q , i.e., the closer is the saddle tothe reference e minimum, the farther is the one reached asymptotically; for energies (cid:15) closer to thethreshold, the behavior of x ∞ is non-monotonic. saddles at the same depth in the landscape, the ones that are closer to the minima leadto a more efficient exploration of configuration space, as they allow to explore fartherregions and to reach deeper minima. We recall that increasing q corresponds to selectingsaddles that are less numerous (have lower complexity) and that are in general steeperalong the direction connecting to the reference minimum (as they have a smaller isolatedeigenvalue). • At fixed overlap q with the reference minimum, deeper saddles connect the latter withlocal minima with smaller energy. In particular, the deepest minimum that can bereached through this family of index-1 saddles is connected to the reference one throughthe lowest saddle of energy (cid:15) ∗ . However, this is not the farthest point that can bereached through this family of index-1 saddles.In Fig. 6 we focus on the closest saddles to the minimum for each (cid:15) (i.e., on those atoverlap q M ( (cid:15) ) with the minimum, having zero complexity), and plot the asymptotic23 ciPost Physics Submission Figure 6:
Asymptotic overlap (
Left. ) and energy (
Right. ) of the minima reached from the zero-complexity saddles at energy (cid:15) that are closer to the reference minimum ( i.e. , that are at overlap q M ( (cid:15) )). The dashed line are linear fits. energy and overlaps reached from these saddles, which show an almost linear dependenceon (cid:15) . We see that moving along the curve corresponding to zero complexity of thesaddles, the ones having lower energy lead to lower energy minima, but that are atlarger overlap with the original minimum. Thus, there is a competition between energyand overlap of the asymptotic states: the saddles leading to lowest energies are not thoseleading to the farthest stationary points.More generally, the asymptotic analysis shows that the minima that are reached throughthis family of saddles have a distribution in energy concentrated around values that are muchhigher than (cid:15) (the energy of the reference minimum), and are rather close to the energy (cid:15) of the saddles. Moreover, the asymptotic correlation with the initial condition (the saddle)remains quite close to one, as we show in Fig. 7. This suggest that the minima reachedasymptotically are close to the saddles in configuration space. Moreover, we find that theyare correlated to the reference minimum . Indeed, the corresponding parameters ( x ∞ , (cid:15) ∞ )lie in a region of configuration space that is dominated by minima having an Hessian thatfeels the presence of the reference minimum through a single (positive) isolated eigenvalue,see Fig. 7 and the comparison with Fig. 2. The purpose of this section is to present a full numerical solution of the equations (47), (49),(52), (55). We shall show that after escaping from the selected saddle the system displays arelaxation dynamics towards the connected minima, thus validating the assumptions behindthe asymptotic solution obtained in Sec. 4 (in particular we exclude the existence of agingdynamics and trapping in spurious minima). The numerical solution of the free-fall dynamicsfrom the saddle will be instrumental in reconstructing the shape of the dynamical instantonin the next section, see Fig. 8 In this discussion we restrict to initial conditions lying in a region of configuration space where the com-plexity of stationary points is non-negative, i.e. , to q < q M ( (cid:15) ). For q > q M ( (cid:15) ), non-trivial solutions ofthe asymptotic dynamical equations can still be found; however, for q large enough, they lie in a region ofconfiguration space where stationary points of energy (cid:15) ∞ are exponentially rare (their complexity is negative). ciPost Physics Submission Figure 7:
Left.
Asymptotic correlation function, giving the overlap between the minimum reachedasymptotically by the dynamics and the saddle chosen as initial condition. The flatter is the negativedirection of the saddle ( i.e. , the closer is (cid:15) to (cid:15) ms ), the closer is the minimum reached asymptotically. Right.
The red points represent the parameters of the saddles chosen as initial conditions for thedynamics, while the black ones are the parameters of the minima reached asymptotically from thesaddles with the same symbol. The inset is a zoom of these points. The saddles that have a flatterunstable direction (those at smaller q ) lead to closer local minima. All minima reached asymptoticallylie in the region of configuration space that is dominated by minima correlated to the reference one,having one positive isolated eigenvalue (dashed gray area). As already discussed in the previous section, when the initial condition is on the saddle thesystem remains stuck there even though this is an unstable point. The reason is that thisunique unstable direction is one out of N , so in the large N limit the system does not escapefrom the saddle in any finite time. By linearizing the dynamics around the unstable saddle iseasy to establish that the escape time equals ln ( N/α ) / | λ | , i.e. it increases logarithmicallywith N ( λ is the negative eigenvalue of the Hessian corresponding to the unstable direction).In the following, since we are interested in the free-fall dynamics, we bypass this slow processby introducing a small perturbation aligned, or counter-aligned, with the unique unstabledirection of the saddle. We implement this perturbation in the form of an impulse, a kick,of infinitesimal amplitude and duration in the direction of s which has a finite projection onthe unstable direction [21, 22], i.e. along the vector s − s . However since the componentalong s is compensated anyway by the spherical constraint we simplify and consider a kick inthe direction s − qs (see Fig. 8) perpendicular to s . This leads to the modified dynamicalequations: ∂ t s i ( t ) = − δ E [ s t ] δs i ( t ) − z ( t ) s i ( t ) + ξ i ( t ) + εδ ( t )[ s − qs ] i , (91)with initial condition s ( t = 0) = s chosen as usual. For ε > < s . In the second case the convergence to the otherminimum s ∞ is favored. The equations for x ( t ), c ( t, t (cid:48) ) and z ( t ) change in a very simple waythat can be read from (91) and only affects the contributions coming from the initial condition25 ciPost Physics Submission P I P up P do " > " < s s s s s s s s s Figure 8:
Schematic representation of steps for numerical integration and instanton reconstruction.Kicks with amplitudes of opposite signs allow numerical integration of dynamical paths from the saddletowards the original minimum and from the saddle away from the original minimum. The second pathis P do . The time reversal of the first path is P up . The dynamical instanton path P I is obtained byjoining P up and P do . G (cid:15),q [ c, x ] and F (cid:15),q [ c, x ] in the following way: G ε(cid:15),q [ c, x ] = G (cid:15),q [ c, x ] + εδ ( t )[1 − q ] F ε(cid:15),q [ c, x ] = F (cid:15),q [ c, x ] + εδ ( t )[ x ( t (cid:48) ) − qc ( t (cid:48) )] . (92)The equation for r ( t, t (cid:48) ) that is not explicitly affected by initial conditions would changeuniquely through the Lagrange multiplier z ( t ), which has itself a null contribution δ ( t )[ x ( t ) − qc ( t )] = 0 from this kick by construction. The simplest form for the new system equationscan then be rewritten using (52), (55),[ ∂ t + z ( t )] c ( t, t (cid:48) ) = αr ( t (cid:48) , t ) + p ( p − (cid:90) t dsr ( t, s )[ c ( t, s )] p − c ( t (cid:48) , s ) + p (cid:90) t (cid:48) ds [ c ( t, s )] p − r ( t (cid:48) , s ) − p ( p − q q − q p c ( t (cid:48) ) (cid:90) t dsr ( t, s ) (cid:26) c p − ( t ) c p − ( s ) − q p − (cid:0) c p − ( t ) x p − ( s ) + x p − ( t ) x ( s ) c p − ( s ) (cid:1)(cid:27) − p ( p − q q − q p x ( t (cid:48) ) (cid:90) t dsr ( t, s ) (cid:26) x p − ( t ) x p − ( s ) − q p − (cid:0) x p − ( t ) c p − ( s ) + c p − ( t ) x p − ( s ) c ( s ) (cid:1)(cid:27) − p q q − q p (cid:90) t (cid:48) dsr ( t (cid:48) , s ) (cid:8) [ x ( t ) x ( s )] p − + [ c ( t ) c ( s )] p − − q p − (cid:0) [ x ( t ) c ( s )] p − + [ c ( t ) x ( s )] p − (cid:1)(cid:9) + F (cid:15),q [ c, x ] + εδ ( t )[ x ( t (cid:48) ) − qc ( t (cid:48) )] , (93)26 ciPost Physics Submission and[ ∂ t + z ( t )] x ( t ) = p ( p − (cid:90) t dsr ( t, s ) c p − ( t, s ) x ( s ) − p ( p − q q − q p (cid:90) t dsr ( t, s ) (cid:26) c p − ( t ) c p − ( s ) − q p − (cid:0) c p − ( t ) x p − ( s ) + x p − ( t ) c p − ( s ) x ( s ) (cid:1)(cid:27) − p ( p − q q − q p (cid:90) t dsr ( t, s ) (cid:26) x p − ( t ) x p − ( s ) − q p − (cid:0) x p − ( t ) c p − ( s ) + c p − ( t ) x p − ( s ) c ( s ) (cid:1)(cid:27) + G (cid:15),q [ c, x ] + εδ ( t )[1 − q ] . (94)From the last new equation it becomes evident that, if for ε = 0 x ( t ) = q ∀ t , setting ε > <
0) leads to an initial increase (decrease) of x ( t ) from q and therefore a consequentrelaxation towards (away from) s , as pictorially represented in Fig. 8. The algorithm used to integrate the dynamical equations is a modification of the code devel-oped for the Cugliandolo-Kurchan equations on a fixed time-grid used in [41, 42] and availableat https://github.com/sphinxteam/spiked_matrix-tensor .We introduced two modifications to it. The first one consists in adding the terms of the equa-tions derived in Sec. 3 that enforce the initial condition of the dynamics. The second one is dueto the presence of the kick. As it emerges from Eqs. (93) and (94), while introducing the effectof the kick for one time quantity is straightforward, two point functions should incorporate atany t (cid:48) > t = 0. However the standard numerical approach (see anexample of source code at https://github.com/sphinxteam/spiked_matrix-tensor ) ob-tains the two point correlation function c ( t, t (cid:48) ) with t > t (cid:48) from the integration of { c ( t − dt, s ) } with s ∈ [0 , t − dt ] (example in yellow in the Fig. 9). In this scheme the singular contributioncoming from the impulse at t = 0 would be only included in c ( dt, c ( dt, t ) with dt < t through a modifiedintegration routine on { c (0 , s ) } with s ∈ [0 , t ] and adding the contribution from the kick. Theresult, by symmetry, gives c ( t, dt ) (in red in Fig. 9) to be used in the subsequent integrationstep for c ( t + dt, t (cid:48) ). We now present the full numerical solution with initial condition on the saddle s . Theresults shown in this section refer to a reference minimum s at energy (cid:15) = − .
167 andinitial condition on a saddle s at overlap q = 0 .
75 from s and at energy (cid:15) = − . α = 0, i.e. zero temperature.A first check of our numerical scheme is that without the kick the numerically integrateddynamics is stuck on the saddle, which is indeed what we find, as anticipated in Sec. 3.3.1.We then implement the kick as explained above and find the results reported in Fig. 10 interms of the overlap x ( t ) with the original minimum s and the energy (cid:15) ( t ), for a positiveand negative kick of amplitude ε = 10 − . We observe that the dynamics on both sides of thesaddle lead to a finite time relaxation towards the two neighboring minima. We validate theprediction for the long time energies and correlation obtained in Sec. 4 under the TTI (Time27 ciPost Physics Submission
123 3’45t-dt tt’=t dtkick t+dt
Figure 9:
Integration scheme of two point functions proceeds imposing their values on the diagonal(green circle) to be 1 for correlation and 0 for response, and obtaining c ( t, t (cid:48) ) and r ( t, t (cid:48) ) (yellow circles)from the integration of the functions at t − dt (shaded yellow area). Including the initial kick in theintegration for c ( t, dt ) (red dot) deserves a particular treatment. It is obtained by symmetry from c ( dt, t ) (red circle) which is the result of integration of the functions c (0 , t (cid:48) ) (shaded red area) plus thecontribution from the kick. ε > ε <
50 100 200 500 10001.00.90.8q0.7 x ∞ t x ( t ) ε > ε <
50 100 200 500 1000 - ϵ ϵ ∞ - - ϵ - t ϵ ( t ) Figure 10:
Results of the numerical integration of dynamical relaxation with kick of positive (negative)amplitude are represented in green (red).
Left.
Overlaps x ( t ) between the reference minimum s and the configuration along the dynamics. It is x ( t = 0) = 0 .
75 = q . At large time it approachesasymptotic values 1 and x ∞ = 0 . Right.
Energies along the relaxationpaths start from the energy of the saddle (cid:15) = − . (cid:15) = − .
167 and (cid:15) ∞ = − . Translation Invariance) hypothesis, see the perfect correspondence in Fig. 10 with the longtime limit of numerical integration for the corresponding quantities. We have also verifiedexplicitly that TTI holds for correlation and response asymptotically (only the latter has anon-trivial TTI dynamics since α = 0). 28 ciPost Physics Submission In this section we focus on the dynamical instanton, which corresponds to the activatedprocess that allows the system to escape from the minimum s to the new minimum s ∞ by crossing the barrier associated to s . In order to obtain the dynamical instanton, wecombine the results on free-fall dynamics derived above with time-reversal transformations.In fact, the theory of activated process at low temperature developed in theoretical physicsand mathematics (referred to Freidlin and Wentzell in probability theory) established that anactivated process can be decomposes in two parts: first an upward trajectory to the saddle,which is the time-reversal of the free-fall descent (in our case from s to s ), and then thefree-fall descent from s to the new minimum. In the following we recall the time-reversalfield transformations that will allow us to reconstruct the dynamical instanton. The time reversal c R ( t, t (cid:48) ) , r R ( t, t (cid:48) ) of the correlation c ( t, t (cid:48) ) and the response function r ( t, t (cid:48) )for t > t (cid:48) follows from the relation between the time reversal fields s R ( t ) , ˆ s R ( t ) and the originalfield s ( t ) and auxiliary field ˆ s ( t ). Let us recall them [33, 43] in a simplified setting where Z = (cid:90) D s t D ˆ s t e S [ s, ˆ s ; τ ] , (95)with an action S [ s, ˆ s ; τ ] = (cid:90) τ dt ˆ s ( t ) (cid:20) α s ( t ) − ds ( t ) dt − δ R [ s t ] δs ( t ) (cid:21) (96)and with R [ s t ] = E [ s t ] + z ( t ) s ( t ) /
2. The single path time-reversal is as follows s R ( t ) = s ( τ − t ) (97)ˆ s R ( t ) = ˆ s ( τ − t ) + 2 α ds ( τ − t ) dt . This choice is self-explanatory for s R ( t ). The non trivial transformation of the auxiliary fieldis obtained instead by imposing the invariance under time inversion of the action in Eq. (96),except from the production of boundary terms at s (0) = s I = s R ( τ ) and s ( τ ) = s F = s R (0)that assure detailed balance all along the dynamical path: P [ s ( τ ) | s I ] = (cid:90) D ˆ s t e S [ s, ˆ s ; τ ] = P [ s R ( τ ) | s F ]exp (cid:20) − α ( R ( s F ) − R ( s I )) (cid:21) . (98)The transformations under time reversal for correlation and response functions, as defined inEq. (24), are therefore inherited from the single field transformations as follows c R ( t, t (cid:48) ) = lim N →∞ s R ( t ) · s R ( t (cid:48) ) N = lim N →∞ s ( τ − t ) · s ( τ − t (cid:48) ) N = c ( τ − t, τ − t (cid:48) ) (99) r R ( t, t (cid:48) ) = lim N →∞ s R ( t ) · ˆ s R ( t (cid:48) ) N = lim N →∞ s ( τ − t ) · (ˆ s ( τ − t (cid:48) ) + α ds ( τ − t (cid:48) ) dt (cid:48) ) N = r ( τ − t, τ − t (cid:48) ) + 2 α ddt (cid:48) c ( τ − t, τ − t (cid:48) ) (100)29 ciPost Physics Submission d R ( t, t (cid:48) ) = lim N →∞ ˆ s R ( t ) · ˆ s R ( t (cid:48) ) N = lim N →∞ (ˆ s ( τ − t ) + α ds ( τ − t ) dt ) · (ˆ s ( τ − t (cid:48) ) + α ds ( τ − t (cid:48) ) dt (cid:48) ) N = 2 α (cid:20) ddt r ( τ − t, τ − t (cid:48) ) + ddt (cid:48) r ( τ − t (cid:48) , τ − t ) (cid:21) + 4 α d dtdt (cid:48) c ( τ − t, τ − t (cid:48) ) (101)as d ( t, t (cid:48) ) = lim N →∞ ˆ s ( t ) · ˆ s ( t (cid:48) ) /N = 0. As schematically shown in Fig. 8, since we know by direct numerical integration the correlationand response function along the free-fall dynamics s → s , we can obtain their time-reversedcounterparts using the relations above. We shall denote the corresponding correlation function c up ( t, t (cid:48) ) and the associated dynamical path P up . In order to construct the dynamical instan-ton, the time-reversed path thus obtained is merged with the forward dynamical path P do from the saddle to the new minimum s ∞ . Accordingly, the correlation functions c do ( t, t (cid:48) ) forthis process is obtained by direct numerical integration along the free-fall dynamics s → s ∞ .The probability rate of such dynamical instanton equals at leading order e − E − E ) /α , with E and E the energy of the saddle and the original minimum, respectively, as it follows fromthe results recalled in the previous section. Since the difference in energy between the saddleand the original minimum is extensive, this implies that the activated process associated tothe dynamical instanton typically takes place on a time-scale that diverges exponentially with N .We wish to describe the reconstructed dynamical instanton in terms of a global two timecorrelation function c ( t, t (cid:48) ) defined on the entire time span t ∈ [0 , τ f ] and t (cid:48) ∈ [0 , τ f ] where τ f = τ up + τ do , and τ up , τ do are the time span of the dynamical paths respectively towards( P up ) and from ( P do ) the saddle. Finally τ s = τ up is the time at which the saddle is visited.However, a reconstruction based on a junction at time τ s of these two distinct dynamicalpaths lacks the off-diagonal sectors where t ∈ [ τ s , τ f ] and t (cid:48) ∈ [0 , τ s ], and viceversa. To fillthis gap we propose an approximated interpolation of the correlation function c ( t, t (cid:48) ) in thesedynamical sectors based on the following decomposition for t > τ s and t (cid:48) < τ s s ( t ) = s c do ( t, τ S ) + v (cid:113) − c ( t, τ S ) , (102) s ( t (cid:48) ) = s c up ( τ S , t (cid:48) ) + v (cid:48) (cid:113) − c ( τ S , t (cid:48) ) , (103)with v and v (cid:48) two vectors on the sphere, perpendicular to s . For t and t (cid:48) approaching τ s both vectors correspond to the saddle s . The above decomposition corresponds to fixing theprojection of the dynamical variables s ( t ) , s ( t (cid:48) ) along the direction of the saddle to its typicalvalue, which is given by the solution of the dynamical equations. The projection along theorthogonal direction is then automatically fixed by the spherical constraint. The directions v and v (cid:48) are in principle varying with time during the dynamical evolution, and so is theiroverlap. We neglect this time dependence and set:lim N →∞ v · v (cid:48) N = x ∞ − qc ∞ (cid:112) − q (cid:112) − c ∞ . (104)This condition ensures that the boundary conditions are verified: at t = τ f , t (cid:48) = 0, where itis expected s ( τ f ) = s ∞ , s (0) = s , we have that their scalar product is x ∞ as it should, since30 ciPost Physics Submission c do ( τ f , τ s ) = c ∞ , c up ( τ S ,
0) = q .The resulting expression for the correlation function for t ∈ [ τ s , τ f ] and t (cid:48) ∈ [0 , τ s ] then reads c ( t, t (cid:48) ) = c do ( t, τ s ) c up ( τ S , t (cid:48) ) + x ∞ − qc ∞ (cid:112) − q (cid:112) − c ∞ (cid:112) − c do ( t, τ s ) (cid:113) − c up ( τ s , t (cid:48) ) . (105)Finally we get c ( t (cid:48) , t ) = c ( t, t (cid:48) ) by symmetry. We are now in position to completely reconstructthe dynamical instanton corresponding to barrier crossing in mean-field glassy systems. Itsshape is shown in Fig. 4. The main outcome of this work is the identification for mean-field models of glasses of thesimplest activated processes, which correspond to the escape from a given minimum throughsaddles of index one. By combining the Kac-Rice method and dynamical field theory, wehave constructed explicitly the dynamical instanton associated to the jump over the barrier,and characterized the new minima that the system can reach after the jump. This representsa first step towards a general classification and analysis of dynamical instantons in roughhigh-dimensional energy landscapes. In particular, the dynamical equations derived in thiswork allow us to describe escapes from local minima passing through a particular family ofrank-1 saddles, those that are closer to the reference minimum in configuration space. Thereason for this is that these rank-1 saddles are the typical stationary points (i.e., those thatare exponentially more numerous than any other type of stationary points) found at high-overlap with the minimum. We have found that the minima reached dynamically throughthese saddles are correlated to the reference one, being quite close to it in configurationsspace; moreover, they are at higher energy. Other saddles geometrically connected to thereference minimum exist at higher distance (smaller value of the overlap) but are atypical[22], i.e. they are still exponentially numerous in N , but their number is subleading withrespect to that of local minima, which are instead the typical critical points for that value ofthe overlap. Initializing the dynamics in one of these saddles requires to condition explicitlyon the properties of the Hessian of the initial condition, thus generating additional terms inthe dynamical equations. Deriving the corresponding dynamical equations and characterizingtheir asymptotic solutions is potentially interesting, since these saddles might connect thereference minimum to other local minima that are less correlated with the reference one,being at smaller overlap with it or having lower energy. These saddles can provide more directescape paths, whereas the ones analysed in this work are more likely to give rise to back andforth motions with frequent returns to the original minimum. We leave this interesting openproblem to future work. More broadly, it is worth examining the extremization equation ofthe large deviation dynamical functional by leveraging on the special solution we constructed.Generalizing such solution (numerically or analytically) provides a new way to obtain thedynamical instantons which correspond to more complex activated processes, and in particularthe ones leading to thermal relaxation. 31 ciPost Physics Submission Acknowledgements
We acknowledge Stefano Sarao Mannelli and Pierfrancesco Urbani for sharing the originalversion of the code, available at https://github.com/sphinxteam/spiked_matrix-tensor .We also thank J. Kurchan and G. Tarjus for interesting discussions.
Funding information
This work is supported by the Simons Foundation collaborationCracking the Glass Problem (No. 454935 to G. Biroli). V.Ros acknowledges funding by theLabEx ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*
A Statistics of the Hessian at critical points
In this Appendix we recall the statistics of the Hessian matrices of the functional (1), evaluatedat stationary points s that are at fixed overlap q from a reference minimum s . This statisticshas been computed in [21] (see also Lemma 13 in [32] and [22]), and we refer to that work forthe details of the derivation. For a fixed realization of the random field, the Hessian matrix H [ s ] at an arbitrary point s on the sphere is given by (11): the first contribution is simplythe projection of the matrix of second derivatives of E [ s ] into the tangent plane at s , whilethe second term comes from enforcing the spherical constraint. Conditioning on E [ s ] = N (cid:15) ,we see that the Hessian can be re-written as: H αβ = (cid:18) e α [ s ] · δ E [ s ] δs · e β [ s ] − p (cid:15) δ αβ (cid:19) , (106)where the vectors e α [ s ] form a basis of the tangent plane at s . Following the notation in[21, 22] we focus on the rescaled matrix: M αβ [ s ] = √ N (cid:18) e α [ s ] · δ E [ s ] δs · e β [ s ] (cid:19) (107)and describe the statistics of its entries averaged over the random couplings J i , ··· ,i p , onceconditioning the point s to be a stationary point at overlap q from another stationary point s with energy density (cid:15) . To do so, we choose the basis vectors e α [ s ] in such a way thatonly the vector e N − [ s ] has a non-zero projection on s , e N − [ s ] = s − qs (cid:112) N [1 − q ] , (108)while the remaining e α for α ≤ N − s and s . The statistics of the conditioned matrix M is invariant with respect to the particularchoice of these N − M αβ with α, β ≤ N − σ = p ( p − M α N − with α (cid:54) = N − ( q ) ≡ σ (cid:18) − ( p − − q ) q p − − q p − (cid:19) . (109)32 ciPost Physics Submission Finally, the diagonal element M N − N − has yet another variance (that we do not report sinceit is not relevant in the following), and a non-zero average equal to: (cid:104)M N − N − (cid:105) = √ N µ ( q, (cid:15) , (cid:15) ) ≡ √ N [ (cid:15) a ( q ) − (cid:15) a ( q )] a (2) , (110)with the constants a i ( q ) already defined in (37) in the main text. Therefore, M is a GOE ma-trix modified by finite-rank additive and multiplicative perturbations that alter the statisticsof the entries in the last line and column, that single out the direction connecting s and s inconfiguration space. The bulk of the eigenvalue density of M is given by a semicircle, and itis insensitive to the modified statistics of the elements outside the ( N − × ( N −
2) invariantblock. As argued in [21, 22], the perturbations to the GOE statistics can nevertheless generatea sub-leading correction to this density, in the form of a single isolated eigenvalue λ min ( q, (cid:15), (cid:15) )that lies outside the support of the semicircle. This eigenvalue exists whenever [22] µ < − σ (cid:20) σ (cid:48) ) σ (cid:21) where σ (cid:48) ( q ) = (cid:112) σ − ∆ ( q ) , (111)and it solves the equation λ − µ ( q, (cid:15), (cid:15) ) − ∆ ( q ) G σ ( λ ) = 0 , (112)with G σ ( x ) = 12 σ (cid:16) x − sign( x ) (cid:112) x − σ (cid:17) . (113)The solution to this equation can be compactly written as: λ min ( q, (cid:15) , (cid:15) ) = G − σ ( G σ (cid:48) ( µ )) = 1 G σ (cid:48) ( µ ) + σ G σ (cid:48) ( µ ) with G − σ ( x ) = 1 x + σ x. (114)When (111) holds and when the smallest eigenvalue of the matrix √ H , λ ( q, (cid:15) , (cid:15) ) ≡ λ min ( q, (cid:15) , (cid:15) ) − √ p(cid:15) (115)is negative, the point s is a rank-1 saddle. The eigenvector associated to this eigenvalue hasa macroscopic projection along the direction in configuration space connecting the saddle s to the minimum s (see [22] for the explicit calculation of the magnitude of this projection).This is what happens for parameters that correspond to the violet region in figure 2. B Derivation of Eq. 22
In this Appendix, we derive Eq. 22. We introduce the shorthand notation M [ s ta ] ≡ M a ( t )and enforce the initial conditions as: (cid:90) N (cid:89) i ≤ j =1 dm ij dm ij (cid:90) N (cid:89) i ≤ j =1 dλ ij dλ ij e iλ ij ( M ij (0) − m ij ) e iλ ij ( M ij (0) − m ij ) . (116) The sign in front of the square root of G σ ( x ) guarantees that the resolvent is positive for x >
0, and decaysto zero as | x | → ∞ . ciPost Physics Submission We can therefore re-write the average (16) as J = (cid:90) (cid:89) a =1 N (cid:89) i ≤ j =1 (cid:104) dm aij dλ aij e − iλ aij m aij (cid:105) F [ m a , s a ] (cid:90) D M a e − (cid:80) a =1 N (cid:80) i ≤ j (cid:82) ∞ dt M aij ( t ) [ δ a, O ij ( t ) − iλ aij δ ( t ) ](117)where D M a denotes the joint Gaussian measure: D M a = D [ M aij ( t )]exp − (cid:88) a =1 (cid:88) i ≤ j (cid:88) k ≤ l (cid:90) ∞ dt dt (cid:48) M aij ( t )[Σ − ] abij,kl ( t, t (cid:48) ) M bkl ( t (cid:48) ) , (118)and given that M ij ( t ) is symmetric we have restricted the covariance matrix to i ≤ j and k ≤ l : Σ abij,kl ( t, t (cid:48) ) ≡ χ i ≤ j χ k ≤ l (cid:104) M aij ( t ) M bkl ( t (cid:48) ) (cid:105) , (119)where χ is an indicator function. The matrix at the exponent in (117) reads O ij ( t ) = 1 p − { [ˆ s ( t )] i [ s ( t )] j + [ˆ s ( t )] j [ s ( t )] i − δ ij [ˆ s ( t )] i [ s ( t )] i } , (120)and: F [ m a , s a ] = (cid:89) a =1 N − (cid:89) α =1 δ (cid:18) e α [ s a ] · m a · s a p − (cid:19) δ (cid:18) s a · m a · s a p ( p − − N (cid:15) a (cid:19) (cid:12)(cid:12)(cid:12) det (cid:18) m a − p E [ s a ] N (cid:19) (cid:12)(cid:12)(cid:12) , (121)where m a is the projection of the matrix m a onto the tangent plane at s a , and is theidentity matrix. Notice that the fields in (120) are exactly at equal time: this will be relevantfor the discussion in Appendix C. The integration over the matrix field and over the auxiliaryvariables λ aij gives for (15) the following expression: I ( (cid:15) , q | (cid:15) ) ∝ (cid:90) s · s = Nq ds ds (cid:90) (cid:89) a =1 N (cid:89) i ≤ j =1 dm aij F [ m a , s a ] (cid:90) s (0)= s D s t D ˆ s t e V + V , (122)with an action V = 12 (cid:88) i ≤ j (cid:88) k ≤ l (cid:26)(cid:90) ∞ dtdt (cid:48) O ij ( t )Σ ij,kl ( t, t (cid:48) ) O kl ( t (cid:48) ) − (cid:0) Ξ aij + m aij (cid:1) Ω abij,kl (Ξ bkl + m bkl ) (cid:27) , (123)where Ω = [Σ(0 , − , Ξ aij = (cid:90) ∞ dt Σ a ij,kl (0 , t ) O kl ( t ) . (124)The proportionality is due to the fact that we are neglecting the functional determinantarising from the integration over the matrix field, as well as the determinant resulting fromthe Gaussian integration over λ ij . These terms can be disregarded as they do not dependexplicitly on the spin variables, and therefore will not matter when deriving the dynamicalequations from the optimization of the dynamical action. The expression for V is givenin (17). 34 ciPost Physics Submission We now focus on the integration over the initial conditions m aij . In order to implementthe constraints in (121), it is convenient to express the components of the matrices m a in thebases e α [ s a ] in which the constraints are given, which span the tangent planes to the sphereat s a . To this aim, we introduce the rescaled unit vectors σ a = s a / √ N for a = 1 ,
2. Weintroduce a first set of unit vectors B = { e , · · · , e N − , w N − , w N } such that: w N = σ , w N − = σ − qσ (cid:112) − q , (125)and the remaining e α for α ≤ N − s and s . Analogously, we introduce a second set B = { e , · · · , e N − , v N − , v N } such that: v N = σ , v N − = σ − qσ (cid:112) − q . (126)These two sets are related by: (cid:18) v N − v N (cid:19) = (cid:32) − q (cid:112) − q (cid:112) − q q (cid:33) (cid:18) w N − w N (cid:19) . (127)The vectors e α [ s ] in (121) spanning the tangent plane at s can be chosen to be equal to B \{ w N } , while the vectors e α [ s ] can be identified with B \ { v N } . It is convenient to determinethe covariances (119) between the matrix elements M aαβ expressed in the corresponding bases B a . For K = N −
2, let us collect the matrix elements M aαβ into the following vectors: (cid:126)M = ( M , M , · · · , M KK , M KK , M , M , · · · , M K − K , M K − K ) (cid:126)M / = ( M N − , M N , M N − , M N , · · · , M KN − , M KN , M KN − , M KN ) (cid:126)M = ( M N − N − , M NN , M N − N , M N − N − , M NN , M N − N ) . (128)It is easy to check that at t = 0 = t (cid:48) the covariance matrix Σ ≡ Σ(0 , ,
0) = Σ /
00 0 Σ −→ Ω = [Σ(0 , − = Ω /
00 0 Ω . (129)Let us determine the explicit form of Σ(0 , ] abαβ,γδ = (cid:104) M aαβ M bγδ (cid:105) = δ αγ δ βδ p ( p − N ( σ a · σ b ) p − (1 + δ αβ ) for α, β, γ, δ ≤ K = N − , (130)indicating that the first ( N − × ( N − M a have acoupled GOE statistics: each M aαβ is correlated only with itself and with the correspondingentry M bαβ of the other matrix. For what concerns the correlations between the componentsin (cid:126)M / , it can be easily shown that (cid:104) M aαx M bγy (cid:105) ∝ δ αγ for α, γ ≤ N − x, y ∈ { N − , N } .35 ciPost Physics Submission The blocks in the covariance matrix have the same form for each α : (cid:18) Σ N − N − Σ N − N Σ NN − Σ NN (cid:19) ≡ (cid:104) M αN − M αN − (cid:105) (cid:104) M αN − M αN − (cid:105) (cid:104) M αN − M αN (cid:105) (cid:104) M αN − M αN (cid:105)(cid:104) M αN − M αN − (cid:105) (cid:104) M αN − M αN − (cid:105) (cid:104) M αN − M αN (cid:105) (cid:104) M αN − M αN (cid:105)(cid:104) M αN M αN − (cid:105) (cid:104) M αN M αN − (cid:105) (cid:104) M αN M αN (cid:105) (cid:104) M αN M αN (cid:105)(cid:104) M αN M αN − (cid:105) (cid:104) M αN M αN − (cid:105) (cid:104) M αN M αN (cid:105) (cid:104) M αN M αN (cid:105) = p ( p − N q p − (cid:0) − pq + q + p − (cid:1) p − q p − (cid:112) − q q p − (cid:0) − pq + q + p − (cid:1) p − q p − (cid:112) − q
00 ( p − q p − (cid:112) − q p − p − q p − ( p − q p − (cid:112) − q p − q p − p − (131)where we introduced the compact notation Σ NN for the 2 × abαN,αN , which are equal for any α ≤ N −
2, and similarly for the other blocks. Notice thatthis reduces to a diagonal matrix for q →
0, when the initial condition s of the dynamicsis orthogonal (and thus uncorrelated) to the minimum s . Finally, the correlations of thecomponents of (cid:126)M form a 6 × = p ( p − N Σ N − N − ,N − N − Σ N − N − ,N − N Σ N − N − ,NN Σ N − N,N − N − Σ N − N,N − N Σ N − N,NN Σ NN,N − N − Σ NN,N − N Σ NN,NN , (132)where each block is a 2 × abxy,zξ = (cid:104) M axy M bzξ (cid:105) and x, y, z, ξ ∈{ N − , N } . The various block read:Σ N − N − ,N − N − = (cid:18) aa (cid:19) , Σ N − N − ,NN = (cid:18) bb (cid:19) , Σ N − N − ,N − N = (cid:18) cc (cid:19) Σ NN,NN = p ( p − (cid:18) q p q p (cid:19) , Σ NN,N − N = (cid:18) dd (cid:19) , Σ N − N,N − N = ( p − (cid:18) ff (cid:19) (133)with a = q p − (cid:0) ( p − pq − p − p − q + p − p + 6 (cid:1) b = ( p − pq p − (cid:0) − q (cid:1) c = − ( p − q p − (cid:112) − q (cid:0) p (cid:0) q − (cid:1) + 2 (cid:1) d = p ( p − q p − (cid:112) − q f = − q p − [1 − p (1 − q )] . (134)This general structure allows to decompose the sum in (123) in the following way: (cid:88) α ≤ β (cid:88) γ ≤ δ (cid:88) a,b =1 (cid:0) Ξ aαβ + m aαβ (cid:1) Ω abαβ,γδ (Ξ bγδ + m bγδ ) = U + U / + U , (135) The case p = 3 has to be treated with more care, as in this case the off-diagonal matrix elements shouldbe set to zero from the onset. ciPost Physics Submission where: U = N − (cid:88) α ≤ β =1 N − (cid:88) γ ≤ δ =1 2 (cid:88) a,b =1 (cid:0) Ξ aαβ + m aαβ (cid:1) [Ω ] abαβ,γδ (Ξ bγδ + m bγδ ) U / = N − (cid:88) α =1 N (cid:88) x,y = N − (cid:88) a,b =1 (Ξ aαx + m aαx ) [Ω ] abαx,αy (Ξ bαy + m bαy ) U = N (cid:88) x,y,z,ξ = N − (cid:88) a,b =1 (cid:0) Ξ axy + m axy (cid:1) [Ω ] abxy,zξ (Ξ bzξ + m bzξ ) . (136)The constraints in (121) correspond to setting m aαN = 0 for α < N , and m aNN = p ( p − (cid:15) a .Notice that the term U / couples the matrix elements m aαN , that have to be put to zero,with the elements m aαN − , on which the integration is free. Similarly, the integration on theelements m aN − N − in U is free, while the elements m aN − N and m aNN are constrained totake a given value. To decouple the constrained matrix elements from the unconstrained ones,we make use of Gaussian conditioning . Introducing the vector notation Ξ αβ = (Ξ αβ , Ξ αβ ) T and imposing m aαN = 0, we obtain: U / −→ N − (cid:88) α =1 Ξ TαN [Σ NN ] − Ξ αN + ( m αN − + Ξ ∗ α ) T [Σ ∗ / ] − ( m αN − + Ξ ∗ α ) , (139)The second term in the sum (139) depend on some shifted 2-dimensional vectors Ξ ∗ α and ona modified 2 × ∗ / given by:Ξ ∗ α = Ξ αN − − Σ N − N Σ − NN Ξ αN , Σ ∗ / = Σ N − N − − Σ N − N Σ − NN Σ N N − . (140)We find: Σ ∗ / = p ( p − N (cid:32) − ( p − − q ) q p − − q p − − q p − ( p − q +( p − q q p +3 − q − p − q p − ( p − q +( p − q q p +3 − q − p − ( p − − q ) q p − − q p − (cid:33) . (141)With an analogous reasoning, setting (cid:15) = ( (cid:15) , (cid:15) ) T , we see that once the conditioning on m aN − N = 0 and m aNN = p ( p − (cid:15) a are implemented the sum U takes the form: U → (cid:18) Ξ N − N Ξ NN + p ( p − (cid:15) (cid:19) T [Σ { , } ] − (cid:18) Ξ N − N Ξ NN + p ( p − (cid:15) (cid:19) +( m N − N − +Ξ ∗∗ ) T [Σ ∗ ] − ( m N − N − +Ξ ∗∗ ) . (142) We make use of the following identity holding for two generic vectors x , x : (cid:88) ij =1 ( x i − x i ) T [Σ − ] ij ( x j − x j ) = ( x − x ) T [Σ ] − ( x − x ) + ( x − x ∗ ( x )) T [Σ ∗ ] − ( x − x ∗ ( x )) , (137)where Σ is a generic correlation matrix with blocks Σ ij and:Σ ∗ = Σ − Σ Σ − Σ , x ∗ ( x ) = x + Σ Σ − ( x − x ) (138) ciPost Physics Submission In this case Σ { , } is a shorthand notation for the 4 × { , } = (cid:18) Σ N − N,N − N Σ N − N,NN Σ NN,N − N Σ NN,NN (cid:19) = p ( p − N × p − − p ) q p − (cid:0) p (cid:0) q − (cid:1) +1 (cid:1) p − pq p − (cid:112) − q (1 − p ) q p − (cid:0) p (cid:0) q − (cid:1) +1 (cid:1) p − p − pq p − (cid:112) − q
00 ( p − pq p − (cid:112) − q p ( p −
1) ( p − pq p ( p − pq p − (cid:112) − q p − pq p p ( p − , (143)and: Ξ ∗∗ = Ξ N − N − − (Σ N − N − ,N − N Σ N − N − ,NN )[Σ { , } ] − (cid:18) Ξ N − N Ξ NN + p ( p − (cid:15) (cid:19) , Σ ∗ = Σ N − N − ,N − N − − (Σ N − N − ,N − N Σ N − N − ,NN )[Σ { , } ] − (cid:18) Σ N − N,N − N − Σ NN,N − N − (cid:19) . (144)By defining: S = 12 (cid:88) i ≤ j (cid:88) k ≤ l (cid:90) ∞ dtdt (cid:48) O ij ( t )Σ ij,kl ( t, t (cid:48) ) O kl ( t (cid:48) ) (145)and S B = 12 (cid:34) N − (cid:88) α =1 Ξ TαN [Σ NN ] − Ξ αN + (cid:18) Ξ N − N Ξ NN + p ( p − (cid:15) (cid:19) T [Σ { , } ] − (cid:18) Ξ N − N Ξ NN + p ( p − (cid:15) (cid:19)(cid:35) , (146)we see that the exponent V in (123) equals to V = S − S B − N − (cid:88) α ≤ β =1 N − (cid:88) γ ≤ δ =1 2 (cid:88) a,b =1 (cid:0) Ξ aαβ + m aαβ (cid:1) [Ω ] abαβ,γδ (Ξ bγδ + m bγδ ) − (cid:34) N − (cid:88) α =1 ( m αN − + Ξ ∗ α ) T [Σ ∗ ] − ( m αN − + Ξ ∗ α ) + ( m N − N − + Ξ ∗∗ ) T [Σ ∗ ] − ( m N − N − + Ξ ∗∗ ) (cid:35) . (147)Substituting this expression into (122) we obtain I ( (cid:15) , q | (cid:15) ) ∝ (cid:90) s · s = Nq ds ds (cid:90) (cid:89) a =1 (cid:90) s (0)= s D s t D ˆ s t e V + S −S B K [ s t , ˆ s t ] , (148)which coincides with Eq. (22) with the identification (23). The term K [ s t , ˆ s t ] contains all theterms depending on the components m aαβ : its structure is described in detail in the followingAppendix. 38 ciPost Physics Submission C The integral over the Hessian matrices
After shifting the integration variables m aαβ and implementing the constraints, we see thatthe term K [ s t , ˆ s t ] in (22) can be compactly written as K [ s t , ˆ s t ] = (cid:90) (cid:89) a =1 dm a e − (cid:80) N − α ≤ β =1 (cid:80) N − γ ≤ δ =1 (cid:80) a,b =1 m aαβ [Ω ∗ ] abαβ,γδ m bγδ (cid:89) a =1 | det( m a − Φ a [ s t , ˆ s t ] − p(cid:15) a ) | , (149)Where the components of the ( N − × ( N −
1) matrices m a are given in the particularbases B a introduced in Appendix B. It follows from (149) that the entries of m a are Gaussianvariables, with covariance matrix having the following structure:[Σ ∗ ] abαβ,γδ = δ αγ δ βδ (cid:16) χ α,β ≤ N − (1 + δ αβ )[Σ ∗ ] ab + χ α ≤ N − δ β N − [Σ ∗ / ] ab + δ α N − δ αβ [Σ ∗ ] ab (cid:17) (150)where the Σ ∗ i are 2 × ∗ = p ( p − N (cid:18) q p − q p − (cid:19) , Σ ∗ / = p ( p − N (cid:32) − ( p − − q ) q p − − q p − − q p − ( p − q +( p − q q p +3 − q − p − q p − ( p − q +( p − q q p +3 − q − p − ( p − − q ) q p − − q p − (cid:33) . (151)Each of two matrices m a is therefore made of an ( N − × ( N −
2) block of entries havinga GOE statistics that is basis invariant; every entry m aαβ in this block is correlated only withitself, and with the analogous entry m bαβ of the other matrix. This remains true also for theentries belonging to the last line and column of the matrices m a : their correlations, however,are different; moreover, even their variance depends explicitly on the overlap q .We now come to the shifts Φ a [ s t , ˆ s t ] in (149). It follows from the derivation in Appendix Bthat these are ( N − × ( N −
1) symmetric matrices with components:Φ aαβ [ s t , ˆ s t ] = χ α,β ≤ N − Ξ aαβ + χ α ≤ N − δ β,N − [Ξ ∗ α ] a + δ α,N − δ β,N − [Ξ ∗∗ ] a . (152)A simple calculation gives:Ξ aαβ = p ( p − N (cid:90) ∞ dt [ c a (0 , t )] p − ([ˆ s ( t )] α [ s ( t )] β + [ s ( t )] α [ˆ s ( t )] β )+ p ( p − p − N (cid:90) ∞ dt [ c a (0 , t )] p − r a (0 , t )[ s ( t )] α [ s ( t )] β , (153)where we used the notation c ab ( t (cid:48) , t ) = s a ( t (cid:48) ) · s b ( t ) /N and r ab ( t (cid:48) , t ) = s a ( t (cid:48) ) · ˆ s b ( t ) /N . Werecall that the components of Ξ αβ are given in the basis B , and those of Ξ αβ in the basis B .Performing the necessary algebra we find:[Ξ ∗ α ] a = c a Ξ aαN − + c a Ξ aαN , [Ξ ∗∗ ] a = d a Ξ aN − N − + d a Ξ aN − N + d a Ξ aNN − (cid:18) Σ N − N − ,N − N Σ N − N − ,NN (cid:19) T [Σ { , } ] − (cid:18) p ( p − (cid:15) (cid:19) , (154) This is due to the fact that we are expressing the components of each matrix in a basis in which only the( N − s and s , see Appendix B. ciPost Physics Submission where and c ax , d axy are constants (depending on the overlap parameter q ). Therefore the ma-trices Φ a for a = 1 , (cid:18) Φ αβ Φ αβ (cid:19) = L (cid:16)(cid:110) Ξ α (cid:48) β (cid:48) (cid:111)(cid:17) L (cid:16)(cid:110) Ξ α (cid:48) β (cid:48) (cid:111)(cid:17) − δ α,N − δ β,N − (cid:18) Σ N − N − ,N − N Σ N − N − ,NN (cid:19) T [Σ { , } ] − (cid:18) p ( p − (cid:15) (cid:19) , (155)with L a linear functions of their arguments. The second term takes the explicit form: (cid:18) Σ N − N − ,N − N Σ N − N − ,NN (cid:19) T [Σ { , } ] − (cid:18) p ( p − (cid:15) (cid:19) = (cid:18) µ ( q, (cid:15) , (cid:15) ) µ ( q, (cid:15) , (cid:15) ) (cid:19) = 1 a ( q ) (cid:18) (cid:15) a ( q ) − (cid:15) a ( q ) (cid:15) a ( q ) − (cid:15) a ( q ) (cid:19) (156)with the same functions as in (37). This implies thatΦ aαβ = φ aαβ [ s t , ˆ s t ] − δ α,N − δ β,N − µ a ( q, (cid:15) , (cid:15) ) , (157)as stated in Eq. 35, where φ aαβ [ s t , ˆ s t ] = L a (cid:16)(cid:110) Ξ aα (cid:48) β (cid:48) (cid:111)(cid:17) is a linear combination of the inte-grals (153).Equipped with these explicit expression, we can discuss the role of causality in the sim-plification of this term. The integrals (153) involve either products of the spin variable s ( t )and of the response field ˆ s ( t ) evaluated exactly at the same time, or terms proportional tothe response function r a (0 , t ). When the dynamical evolution is causal, these terms will typ-ically be equal to zero: therefore, when the average over dynamical trajectories is restrictedto causal ones, we can set φ = 0. This simplifies considerably the shifts Φ a , that reduce tosimple rank-1 projectors. Exploiting this crucial observation, we finally obtain: K [ s t , ˆ s t ] causality −→ (cid:90) (cid:89) a =1 dm a e − (cid:80) N − α ≤ β =1 (cid:80) N − γ ≤ δ =1 (cid:80) a,b =1 ( m aαβ − δ α N − δ αβ µ a ) [Ω ∗ ] abαβ,γδ ( m bγδ − δ γ N − δ γδ µ b ) ×× (cid:89) a =1 | det( m a − p(cid:15) a ) | . (158)By direct comparison with the the results recalled in Appendix A, we see that the matrix m in(158) reproduces exactly the statistics as the conditional Hessian matrices at a stationary point s at fixed overlap q from a reference minimum s , as expected. More precisely, √ N m = M with M defined in (107). The symmetric statement holds for m . This allows us to concludethat (38) holds true. D Derivation of the boundary terms in the action
In this Appendix we derive the boundary terms in (28). The first term is given by S (1) B = 12 N − (cid:88) α =1 Ξ TαN [Σ NN ] − Ξ αN . (159)40 ciPost Physics Submission This term arises from conditioning the points s and s to be stationary points: in fact, itemerges from the constraint m aα N = 0, which corresponds to setting the gradients to zero.From (133) we find that:[Σ NN ] − = 2 Np ( p − q ( q − q p ) (cid:18) − q p − − q p − (cid:19) . (160)Moreover, with the notation introduced in (25) we find: N − (cid:88) α =1 Ξ aαN Ξ bαN = p ( p − (cid:90) ∞ dsds (cid:48) (cid:110)(cid:2) c a ( s ) c b ( s (cid:48) ) (cid:3) p − d ( s,s (cid:48) )+ (cid:2) c a ( s ) c b ( s (cid:48) ) (cid:3) p − r a ( s ) r b ( s (cid:48) ) c ( s,s (cid:48) ) (cid:111) + p ( p − (cid:90) ∞ dsds (cid:48) ( p − (cid:2) c a ( s ) c b ( s (cid:48) ) (cid:3) p − (cid:8) r a ( s ) c b ( s (cid:48) ) r ( s,s (cid:48) )+ c a ( s ) r b ( s (cid:48) ) r ( s (cid:48) ,s ) (cid:9) , (161)where c ( s,s (cid:48) )= c ( s,s (cid:48) ) − − q (cid:2) c ( s ) c ( s (cid:48) ) − qc ( s ) c ( s (cid:48) ) − qc ( s ) c ( s (cid:48) )+ c ( s ) c ( s (cid:48) ) (cid:3) d ( s,s (cid:48) )= d ( s,s (cid:48) ) − − q (cid:2) r ( s ) r ( s (cid:48) ) − qr ( s ) r ( s (cid:48) ) − qr ( s ) r ( s (cid:48) )+ r ( s ) r ( s (cid:48) ) (cid:3) r ( s,s (cid:48) )= r ( s,s (cid:48) ) − − q (cid:2) c ( s ) r (0 ,s (cid:48) ) − qc (0 ,s ) r (0 ,s (cid:48) ) − qc ( s ) r (0 ,s (cid:48) )+ c (0 ,s ) r (0 ,s (cid:48) ) (cid:3) . (162)Combining everything we get the expression (29). The second contribution to the boundaryterms is given by: S (2) B = 12 (cid:18) Ξ N − N Ξ NN + p ( p − (cid:15) (cid:19) T [Σ { , } ] − (cid:18) Ξ N − N Ξ NN + p ( p − (cid:15) (cid:19) . (163)This arises from conditioning on both the gradient and the energy density of the points s a .In this case no summation over the indices has to be performed, and the expression (32) isobtained setting A = [Σ { , } ] − . Explicitly we find: A = A ( q ) (cid:18) A (1) A (2) A (2) A (3) (cid:19) (164)with A ( q ) = 1 p ( p − q p − (( p − q − p − pq + ( p − ) q p + q ] (165)and A (1) = (cid:18) q − q p (cid:0) p (cid:0) ( p − q +(3 − p ) q + p − (cid:1) +1 (cid:1) q p (cid:0) p (cid:0) q − (cid:1) +1 (cid:1) − q p +4 q p (cid:0) p (cid:0) q − (cid:1) +1 (cid:1) − q p +4 q − q p (cid:0) p (cid:0) q − (cid:1)(cid:0) ( p − q − p +2 (cid:1) +1 (cid:1) (cid:19) A (2) = (cid:32) ( p − pq p +1 (cid:0) − q (cid:1) / − pq p +1 (cid:112) − q (cid:0) q − q p (cid:1) − pq p +1 (cid:112) − q (cid:0) q − q p (cid:1) ( p − pq p +1 (cid:0) − q (cid:1) / (cid:33) A (3) = (cid:18) pq − pq p +2 (cid:0) − pq + q + p (cid:1) − pq p +2 (cid:0) q p − pq + p − (cid:1) − pq p +2 (cid:0) q p − pq + p − (cid:1) pq − pq p +2 (cid:0) − pq + q + p (cid:1) (cid:19) . (166)41 ciPost Physics Submission E Constants appearing in dynamical equations
Let us introduce: D ( q ) = q p − (cid:0) ( p − q − p − pq + ( p − (cid:1) q p + q . (167)The constants appearing in the equation for the overlap x ( t ) read: D ( q ) G ( q ) = pq p +1 (cid:0) ( p − q p − ( p − q + q (cid:1) D ( q ) G ( q ) = − pq (cid:2)(cid:0) ( p − p (cid:0) q − (cid:1) + q − (cid:1) q p + q (cid:3) D ( q ) G ( q ) = − p (cid:0) q − q p (cid:0)(cid:0) − p + p + 1 (cid:1) q + ( p − q + ( p − (cid:1)(cid:1) D ( q ) G ( q ) = (2 − p ) pq p +4 + p (cid:0) p − q − (cid:1) q p D ( q ) G ( q ) = ( p − pq p +2 (cid:0) q − q p (cid:1) D ( q ) G ( q ) = ( p − p (cid:0) q − (cid:1) q p +2 D ( q ) G ( q ) = ( p − p (cid:0) q − (cid:1) q p +1 D ( q ) G ( q ) = ( p − pq p +1 (cid:0) q − q p (cid:1) , (168)while those appearing in the equation for the correlation are given by: D ( q ) F ( q ) = ( p − pq p (cid:0) q p − q (cid:1) D ( q ) F ( q ) = − p (cid:0)(cid:0) ( p − p (cid:0) q − (cid:1) − (cid:1) q p + q (cid:1) D ( q ) F ( q ) = ( p − p (cid:0) q − (cid:1) q p +1 ( p − p (cid:0) q − (cid:1) q p +1 D ( q ) F ( q ) = pq p +1 (cid:0) q − q p (cid:1) D ( q ) F ( q ) = ( p − p (cid:0) q − (cid:1) q p +1 D ( q ) F ( q ) = ( p − p (cid:0) q − (cid:1) q p +1 D ( q ) F ( q ) = − p (cid:0)(cid:0) ( p − p (cid:0) q − (cid:1) − (cid:1) q p + q (cid:1) D ( q ) F ( q ) = ( p − pq p (cid:0) q p − q (cid:1) D ( q ) F ( q ) = ( p − p (cid:0) q − (cid:1) q p +1 D ( q ) F ( q ) = ( p − pq p +1 (cid:0) q − q p (cid:1) D ( q ) F ( q ) = pq p +1 (cid:0) q − q p (cid:1) D ( q ) F ( q ) = ( p − p (cid:0) q − (cid:1) q p +1 (169) References [1] L. Berthier and G. Biroli,
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