Full Replica Symmetry Breaking Research Articles

Replica-Symmetry Breaking Transitions in the Large Deviations of the Ground-State of a Spherical Spin-Glass

We derive, within the replica formalism, a generalisation of the Crisanti–Sommers formula to describe the large deviation function (LDF) L(e)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{L}(e)$$\\end{document} for the speed-N atypical fluctuations of the intensive ground-state energy e of a generic spherical spin-glass in the presence of a random external magnetic field of variance Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. We then analyse our exact formula for the LDF in much detail for the Replica symmetric, single step Replica Symmetry Breaking (1-RSB) and Full Replica Symmetry Breaking (FRSB) situations. Our main qualitative conclusion is that the level of RSB governing the LDF may be different from that for the typical ground-state. We find that while the deepest ground-states are always controlled by a LDF of replica symmetric form, beyond a finite threshold e≥et\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e\\ge e_{t}$$\\end{document} a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document} spherical spin-glass (Fyodorov and Le Doussal in J Stat Phys 154:466, 2014) and the rigorous results by Dembo and Zeitouni (J Stat Phys 159:1306, 2015) which we are able to reproduce invoking an 1-RSB pattern. Finally at an even larger critical energy ec≥et\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e_{c}\\ge e_{t}$$\\end{document}, acting as a “wall”, the LDF diverges logarithmically, which we interpret as a change in the large deviation speed from N to a faster growth. In addition, we show that in the limit Γ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\rightarrow 0$$\\end{document} the LDF takes non-trivial scaling forms (i) L(e)∼G((e-ec)/Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{L}(e) \\sim G((e-e_c)/\\Gamma )$$\\end{document} in the vicinity of the wall (ii) L(e)∼ΓηνF((e-etyp)/Γν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{L}(e) \\sim \\Gamma ^{\\eta \ u } F((e-e_{\ extrm{typ}})/\\Gamma ^{\ u })$$\\end{document} in the vicinity of the typical energy, characterised by two new exponents η≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta \\ge 1$$\\end{document} and ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of typical fluctuations, identified as e-etyp∼N-1/η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e-e_{\ extrm{typ}} \\sim N^{-1/\\eta }$$\\end{document} and Γ∼N-1/(ην)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\sim N^{-1/(\\eta \ u )}$$\\end{document}.

Replica symmetry broken states of some glass models.

We have studied in detail the M-p balanced spin-glass model, especially the case p=4. These types of model have relevance to structural glasses. The models possess two kinds of broken replica states; those with one-step replica symmetry breaking (1RSB) and those with full replica symmetry breaking (FRSB). To determine which arises requires studying the Landau expansion to quintic order. There are nine quintic-order coefficients, and five quartic-order coefficients, whose values we determine for this model. We show that it is only for 2≤M<2.4714⋯ that the transition at mean-field level is to a state with FRSB, while for larger M values there is either a continuous transition to a state with 1RSB (when M≤3) or a discontinuous transition for M>3. The Gardner transition from a 1RSB state at low temperatures to a state with FRSB also requires the Landau expansion to be taken to quintic order. Our result for the form of FRSB in the Gardner phase is similar to that found when 2≤M<2.4714⋯, but differs from that given in the early paper of Gross etal. [Phys. Rev. Lett. 55, 304 (1985)0031-900710.1103/PhysRevLett.55.304]. Finally we discuss the effects of fluctuations on our mean-field solutions using the scheme of Höller and Read [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] and argue that such fluctuations will remove both the continuous 1RSB transition and discontinuous 1RSB transitions when 8>d≥6 leaving just the FRSB continuous transition. We suggest values for M and p which might be used in simulations to confirm whether fluctuation corrections do indeed remove the 1RSB transitions.

How to count in hierarchical landscapes: A full solution to mean-field complexity.

We derive the general solution for counting the stationary points of mean-field complex landscapes. It incorporates Parisi's solution for the ground state, as it should. Using this solution, we count the stationary points of two models: one with multistep replica symmetry breaking and one with full replica symmetry breaking.

Algorithmic Pure States for the Negative Spherical Perceptron

We consider the spherical perceptron with Gaussian disorder. This is the set S of points \(\varvec{\sigma }\in \mathbb {R}^N\) on the sphere of radius \(\sqrt{N}\) satisfying \(\langle \varvec{g}_a , \varvec{\sigma }\rangle \ge \kappa \sqrt{N}\) for all \(1 \le a \le M\), where \((\varvec{g}_a)_{a=1}^M\) are independent standard gaussian vectors and \(\kappa \in \mathbb {R}\) is fixed. Various characteristics of S such as its measure and the largest M for which it is non-empty, were computed heuristically in statistical physics in the asymptotic regime \(N \rightarrow \infty \), \(M/N \rightarrow \alpha \). The case \(\kappa <0\) is of special interest as S is conjectured to exhibit a hierarchical tree-like geometry known as full replica-symmetry breaking (\(\text {FRSB}\)) close to the satisfiability threshold \(\alpha _{\text {SAT}}(\kappa )\), whose characteristics are captured by a Parisi variational principle akin to the one appearing in the Sherrington–Kirkpatrick model. In this paper we design an efficient algorithm which, given oracle access to the solution of the Parisi variational principle, exploits this conjectured \(\text {FRSB}\) structure for \(\kappa <0\) and outputs a vector \(\hat{\varvec{\sigma }}\) satisfying \(\langle \varvec{g}_a , \hat{\varvec{\sigma }}\rangle \ge \kappa \sqrt{N}\) for all \(1\le a \le M\) and lying on a sphere of non-trivial radius \(\sqrt{\bar{q}N}\), where \(\bar{q}\in (0,1)\) is the right-end of the support of the associated Parisi measure. We expect \(\hat{\varvec{\sigma }}\) to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that \(\bar{q}\rightarrow 1\) as \(\alpha \rightarrow \alpha _{\text {SAT}}(\kappa )\), so that \(\big \langle \varvec{g}_a, \hat{\varvec{\sigma }} \big \rangle / \vert \hat{\varvec{\sigma }} \vert \ge \kappa -o(1)\) near criticality.

Crisanti–Sommers Formula and Simultaneous Symmetry Breaking in Multi-species Spherical Spin Glasses

There is a rich history of expressing the limiting free energy of mean-field spin glasses as a variational formula over probability measures on $[0,1]$, where the measure represents the similarity (or "overlap") of two independently sampled spin configurations. At high temperatures, the formula's minimum is achieved at a measure which is a point mass, meaning sample configurations are asymptotically orthogonal up to a magnetic field correction. At low temperatures, though, a very different behavior emerges known as replica symmetry breaking (RSB). The deep wells in the energy landscape create more rigid structure, and the optimal overlap measure is no longer a point mass. The exact size of its support remains in many cases an open problem. Here we consider these themes for multi-species spherical spin glasses. Following a companion work in which we establish the Parisi variational formula, here we present this formula's Crisanti-Sommers representation. In the process, we gain new access to a problem unique to the multi-species setting. Namely, if RSB occurs for one species, does it necessarily occur for other species as well? We provide sufficient conditions for the answer to be yes. For instance, we show that if two species share any quadratic interaction, then RSB for one implies RSB for the other. Moreover, the level of symmetry breaking must be identical, even in cases of full RSB. In the presence of an external field, any type of interaction suffices.

Marginal stability of soft anharmonic mean field spin glasses

We investigate the properties of the glass phase of a recently introduced spin glass model of soft spins subjected to an anharmonic quartic local potential, which serves as a model of low temperature molecular or soft glasses. We solve the model using mean field theory and show that, at low temperatures, it is described by full replica symmetry breaking. As a consequence, at zero temperature the glass phase is marginally stable. We show that in this case, marginal stability comes from a combination of both soft linear excitations—appearing in a gapless spectrum of the Hessian of linear excitations—and pseudogapped non-linear excitations—corresponding to nearly degenerate two level systems. Therefore, this model is a natural candidate to describe what happens in soft glasses, where quasi localized soft modes in the density of states appear together with non-linear modes triggering avalanches and conjectured to be essential to describe the universal low temperature anomalies of glasses.

Dynamical mean-field theory and aging dynamics

Dynamical mean-field theory (DMFT) replaces the many-body dynamical problem with one for a single degree of freedom in a thermal bath whose features are determined self-consistently. By focusing on models with soft disordered p-spin interactions, we show how to incorporate the mean-field theory of aging within DMFT. We study cases with only one slow time-scale, corresponding statically to the one-step replica symmetry breaking phase, and cases with an infinite number of slow time-scales, corresponding statically to the full replica symmetry breaking (FRSB) phase. For the former, we show that the effective temperature of the slow degrees of freedom is fixed by requiring critical dynamical behavior on short time-scales, i.e. marginality. For the latter, we find that aging on an infinite number of slow time-scales is governed by a stochastic equation where the clock for dynamical evolution is fixed by the change of the effective temperature, hence obtaining a dynamical derivation of the stochastic equation at the basis of the FRSB phase. Our results extend the realm of the mean-field theory of aging to all situations where DMFT holds.

Jamming and replica symmetry breaking of weakly disordered crystals

We discuss the physics of crystals with small polydispersity near the jamming transition point. For this purpose, we introduce an effective single-particle model taking into account the nearest neighbor structure of crystals. The model can be solved analytically by using the replica method in the limit of large dimensions. In the absence of polydispersity, the replica symmetric solution is stable until the jamming transition point, which leads to the standard scaling of perfect crystals. On the contrary, for finite polydispersity, the model undergoes the full replica symmetry breaking (RSB) transition before the jamming transition point. In the RSB phase, the model exhibits the same scaling as amorphous solids near the jamming transition point. These results are fully consistent with the recent numerical simulations of crystals with polydispersity. The simplicity of the model also allows us to derive the scaling behavior of the vibrational density of states that can be tested in future experiments and numerical simulations.

One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses.

The de Almeida-Thouless (AT) line is the phase boundary in the temperature-magnetic field plane of an Ising spin glass at which a continuous (i.e., second-order) transition from a paramagnet to a replica-symmetry-breaking (RSB) phase occurs, according to mean-field theory. Here, using field-theoretic perturbative renormalization group methods on the Bray-Roberts reduced Landau-Ginzburg-type theory for a short-range Ising spin glass in space of dimension d, we show that at nonzero magnetic field the nature of the corresponding transition is modified as follows: (a) For d-6 small and positive, with increasing field on the AT line, first, the ordered phase just below the transition becomes the so-called one-step RSB, instead of the full RSB that occurs in mean-field theory; the transition on the AT line remains continuous with a diverging correlation length. Then at a higher field, a tricritical point separates the latter transition from a quasi-first-order one, that is one at which the correlation length does not diverge, and there is a jump in part of the order parameter, but no latent heat. The location of the tricritical point tends to zero as d→6^{+}. (b) For d≤6, we argue that the quasi-first-order transition could persist down to arbitrarily small nonzero fields, with a transition to full RSB still expected at lower temperature. Whenever the quasi-first-order transition occurs, it is at a higher temperature than the AT transition would be for the same field, preempting it as the temperature is lowered. These results may explain the reported absence of a diverging correlation length in the presence of a magnetic field in low-dimensional spin glasses in some simulations and in high-temperature series expansions. We also draw attention to the similarity of the "dynamically frozen" state, which occurs at temperatures just above the quasi-first-order transition, and the "metastate-average state" of the one-step RSB phase, and discuss the issue of the number of pure states in either.

Possible instability of one-step replica symmetry breaking in p-spin Ising models outside mean-field theory.

The fully connected Ising p-spin model has for p>2 a discontinuous phase transition from the paramagnetic phase to a stable state with one-step replica symmetry breaking (1RSB). However, simulations in three dimension do not look like these mean-field results and have features more like those which would arise with full replica symmetry breaking (FRSB). To help understand how this might come about we have studied in the fully connected p-spin model the state of two-step replica symmetry breaking (2RSB). It has a free energy degenerate with that of 1RSB, but the weight of the additional peak in P(q) vanishes. We expect that the state with full replica symmetry breaking (FRSB) is also degenerate with that of 1RSB. We suggest that finite-size effects will give a nonvanishing weight to the FRSB features, as also will fluctuations about the mean-field solution. Our conclusion is that outside the fully connected model in the thermodynamic limit, FRSB is to be expected rather than 1RSB.

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature mu . We are interested in the mean spectral density rho (lambda ) of the Hessian matrix {{mathcal {K}}} at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for rho (lambda ) for a fixed L^d in the N rightarrow infty limit extending d=0 results of our previous work (Fyodorov et al. in Ann Phys 397:1–64, 2018). A particular attention is devoted to analyzing the limit of extended lattice systems by letting Lrightarrow infty . In all cases we show that for a confinement curvature mu exceeding a critical value mu _c, the so-called “Larkin mass”, the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at mu rightarrow mu _c. For mu <mu _c the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as (mu _c-mu )^4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called “marginal cases” in d=1,2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.

New analysis of the free energy cost of interfaces in spin glasses

In this work we want to enhance the calculation performed by Franz, Parisi and Virasoro (FPV) (Franz et al 1992 J. Physique I 2 1869–80; Franz et al 1994 J. Physique I 4 1657–67) to estimate the free energy cost of interfaces in spin glasses and evaluate the lower critical dimension at which replica symmetry is restored. In particular we evaluate the free energy cost for a general class of effective Hamiltonians showing full replica symmetry breaking, and study the dependence of this cost on the order parameter and on the temperature. We confirm the findings of the FPV papers for the scaling of the free energy, recovering a value for the lower critical dimension of . In addition to their results we find a non-trivial dependence of the free energy density cost on the order parameter and the temperature. Apart from the case of a restricted class of effective Hamiltonians this dependence cannot be expressed in terms of functions with a clear physical interpretation, as is the case in hierarchical models (Franz et al 2009 J. Stat. Mech. P02002). In addition we connect the results on the lower critical dimension with recent simulations (Maiorano and Parisi 2018 Proc. Natl Acad. Sci. 115 5129–34).

The Full Replica Symmetry Breaking in the Ising Spin Glass on Random Regular Graph

In this paper, we extend the full replica symmetry breaking scheme to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the limits of the Parisi-M\'ezard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and the free energy. We face up the problem only from a technical point of view: the physical meaning of this approach and the quantitative evaluation of the solution of the self-consistency equations will be discussed in next works.

On the energy landscape of spherical spin glasses

We investigate the energy landscape of the spherical mixed even p-spin model near its maximum energy. We relate the distance between pairs of near maxima to the support of the Parisi measure at zero temperature. We then provide an algebraic relation that characterizes one-step replica symmetric breaking Parisi measures. For these measures, we show that any two nonparallel spin configurations around the maximum energy are asymptotically orthogonal to each other. In sharp contrast, we study models with full replica symmetry breaking and show that all possible values of the asymptotic distance are attained near the maximum energy.

Finite Size Corrections to the Parisi Overlap Function in the GREM

We investigate the effects of finite size corrections on the overlap probabilities in the Generalized Random Energy Model (GREM) in two situations where replica symmetry is broken in the thermodynamic limit. Our calculations do not use replicas, but shed some light on what the replica method should give for finite size corrections. In the gradual freezing situation, which is known to exhibit full replica symmetry breaking, we show that the finite size corrections lead to a modification of the simple relations between the sample averages of the overlaps $ Y_k $ between $ k $ configurations predicted by replica theory. This can be interpreted as fluctuations in the replica block size with a \emph{negative} variance. The mechanism is similar to the one we found recently in the random energy model [1]. We also consider a simultaneous freezing situation, which is known to exhibit one step replica symmetry breaking. We show that finite size corrections lead to full replica symmetry breaking and give a more complete derivation of the results presented in [2] for the directed polymer on a tree.

Full replica symmetry breaking in p-spin-glass-like systems

It is shown that continuously changing the effective number of interacting particles in p-spin-glass-like model allows to describe the transition from the full replica symmetry breaking glass solution to stable first replica symmetry breaking glass solution in the case of non-reflective symmetry diagonal operators used instead of Ising spins. As an example, axial quadrupole moments in place of Ising spins are considered and the boundary value $p_{c_{1}}\cong 2.5$ is found.

Parisi Formula, Disorder Chaos and Fluctuation for the Ground State Energy in the Spherical Mixed p-Spin Models

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

Following the evolution of glassy states under external perturbations: the full replica symmetry breaking solution

.The state-following technique allows the study of metastable glassy states under external perturbations. Here we show how this construction can be used to study the behavior of glassy states of Hard Spheres in infinite dimensions under compression or shear strain. In Rainone et al (2015 Phys. Rev. Lett. 114 015701) it has been shown that in both cases, when the external perturbation is sufficiently strong, glassy states undergo a second-order transition, called the Gardner transition, whereupon a hierarchical structure of marginal micro-states manifests within the original glass state. The purpose of this work is to study the solution of the state-following construction in this marginal phase. We show that upon compression, close to the jamming transition, the metastable states are described by a scaling solution characterized by a set of non-trivial critical exponents that agree with the results of Charbonneau et al (2014 Nat. Commun. 5 3725), and we compute the value of the jamming density for various glassy states. Moreover we show that under the action of the shear strain, beyond the Gardner point, the metastable states can be followed in the marginal phase and we detect an overshoot in the stress-strain curve in agreement with numerical and experimental observations. Finally we further characterize the Gardner transition point by computing both the susceptibility and the exponent parameter λ that characterize the critical slowing down of the dynamics within a glassy state close to the transition.

Explicit generation of the branching tree of states in spin glasses

We present a numerical method to generate explicit realizations of the tree of states in mean-field spin glasses. The resulting study illuminates the physical meaning of the full replica symmetry breaking solution and provides detailed information on the structure of the spin-glass phase. A cavity approach ensures that the method is self-consistent and permits the evaluation of sophisticated observables, such as correlation functions. We include an example application to the study of finite-size effects in single-sample overlap probability distributions, a topic that has attracted considerable interest recently.

Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution

.In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai and Wolynes is realized in a strong sense in the mean-field limit. We also suggest how the computation could be generalized in an approximate way to finite-dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results.