Universality and mechanism of full-RSB jamming criticality in non-convex continuous constraint satisfaction problems

The spherical perceptron with negative stability parameter, solved by Franz and Parisi via full replica symmetry breaking (full-RSB), is the simplest analytically tractable model of jamming: at its satisfiability threshold it is isostatic and marginally stable, and its small-gap and small-force distributions carry the same scaling functions and exponents ($\gamma\approx0.41269$, $\theta\approx0.42311$) as hard spheres in the limit of infinite spatial dimension. The research program posed in that work has as its overarching question whether this criticality is universal across a wide class of non-convex continuous random constraint satisfaction problems (CSPs) with inequality constraints, and what physically causes it. Concretely: (i) the conjectured universality remains unproven, and the specific deformations proposed as tests — restricting the allowed gaps to finite intervals, diluting the perceptron connectivity, correlating the constraint positions — are still unsolved; (ii) the association noted in that work between fragmentation of the solution space (continuous RSB) and the singular gap and force statistics still lacks an interpretation, and no structural criterion for it has been identified. Later literature has resolved side threads (vibrational spectrum, quantum specific heat, quantum jamming dynamics, cavity-type dynamics) and broadened the evidence for universality, but the central conjecture and its mechanism remain open.

Background

Jamming is the transition by which a disordered collection of hard, frictionless particles loses the ability to rearrange as its density increases: the packing becomes mechanically rigid while remaining disordered. At jamming, such packings are typically isostatic — the number of mechanical contacts equals the number of degrees of freedom, the marginal minimum needed for rigidity. A hallmark of the transition, seen in simulations in all spatial dimensions, is criticality: the distribution of small gaps $r$ between nearly touching particles and of small contact forces $f$ follow power laws, $g(r)\sim r^{-\gamma}$ and $P(f)\sim f^{\theta}$, with exponents that appear essentially dimension-independent. The mean-field theory of this criticality comes from the replica analysis of hard spheres in the limit of infinite spatial dimension: the glass phase is described by a full replica symmetry breaking (full-RSB) solution — the hierarchical Parisi solution, in which the order parameter is a function of overlap rather than a number, encoding a fragmentation of the space of configurations into clusters within clusters — and from its scaling regime Charbonneau, Kurchan, Parisi, Urbani and Zamponi extracted $\gamma\approx0.41269$ and $\theta\approx0.42311$, tied by the scaling relation $\gamma=1/(2+\theta)$. Franz and Parisi then identified the minimal exactly solvable model with the same criticality: the spherical perceptron. In machine learning the perceptron is the simplest binary classifier; here it is the problem of finding a vector $x$ with $\sum_i x_i^2=N$ satisfying $M=\alpha N$ random inequality constraints $\xi^\mu\cdot x/\sqrt{N}\ge\sigma$ with independent Gaussian entries $\xi_i^\mu$. In a packing picture, $x$ is the position of a single particle surrounded by $M$ random spherical obstacles, and the constraints forbid overlap; $\sigma$ plays the role of an obstacle size. The gap variables $r^\mu=\xi^\mu\cdot x/\sqrt{N}-\sigma\ge0$ measure the distance from contact with obstacle $\mu$ and are the analogue of the interparticle gaps $r$ in a packing. For $\sigma>0$ the feasible set is convex and jamming is hypostatic (fewer contacts than degrees of freedom) and non-critical. For $\sigma<0$ the problem is non-convex: the set of satisfying configurations (whose logarithm is the entropy, computable by the Gardner-volume replica construction — the replica computation of the volume that the set of solutions occupies in configuration space) is fragmented, and its replica solution is of full-RSB type. Franz and Parisi showed that at the satisfiability (SAT/UNSAT) threshold — the analogue of the jamming point — the non-convex perceptron is isostatic and marginally stable (marginally stable meaning that the system sits at the edge of mechanical instability: removing a single contact unlocks a soft deformation mode), and that the scaling analysis of the full-RSB equations yields exactly the hard-sphere scaling functions and exponents. Two further results frame the open question. First, a random linear-programming variant of the model is always replica symmetric; it jams isostatically but with completely regular gap and force distributions, so isostaticity alone does not produce singular statistics — fragmentation of the solution space (RSB) appears crucial. Second, Müller and Wyart argued on physical grounds that isostaticity implies infinite-range correlations, so that a jammed packing under stress responds as a system with long-range effective forces, and that this marginal stability explains the force pseudogap — the depletion of contacts carrying very small forces, $P(f)\sim f^{\theta}$ at small $f$; the source suggests, but does not derive, that the same mechanism underlies the perceptron scaling. On this basis the source conjectures that universal jamming behaviour occurs in a large class of non-convex continuous random constraint satisfaction problems (CSPs) with inequality constraints (random optimization problems on continuous variables, of which the non-convex perceptron is the simplest instance), and proposes concrete tests: correlating the obstacle positions, restricting the allowed gap variables to finite intervals, and diluting the perceptron so that only finitely many $x_i$ enter each constraint. It also notes that its numerics used moderate sizes ($N\le400$) and an off-equilibrium minimization protocol without equilibration, and lists cavity-method (a single-site iterative solution technique, equivalent to the replica treatment, that computes the statistics of solutions directly), vibrational-spectrum, and quantum specific-heat studies as feasible future work — extensions that have since been carried out (Refs. 4–6, 12), leaving the universality conjecture and the interpretation of the association between fragmentation (RSB) and the singular gap and force statistics as the surviving open core.

Problem Statement

The question, taken from the conclusions of Franz and Parisi's 'The simplest model of jamming', is whether the critical jamming behaviour found there — isostaticity, marginal stability, and the full-RSB scaling functions and exponents $\gamma\approx0.41269$, $\theta\approx0.42311$ with $\gamma=1/(2+\theta)$ — is universal for non-convex continuous random constraint satisfaction problems with inequality constraints, and why. Two linked aspects follow the source. (1) Universality: determine whether the same scaling functions and exponents persist under the deformations the source proposes as tests — restricting the allowed gap variables to finite intervals; diluting the perceptron so that only a finite number of the $x_i$ enter each constraint; correlating the positions of the obstacles — and more generally across other non-convex continuous CSPs with inequality constraints, or whether instead genuinely different jamming criticality (a distinct universality class) occurs within this setting. The three named deformations are independent sufficient sub-targets: establishing that the same scaling functions and exponents persist under any one of them is by itself a valid partial resolution of aspect (1), and the three named tests together are settled only when all three are. The general clause is the wider target, its class being all non-convex continuous random CSPs with inequality constraints that undergo a sharp SAT/UNSAT (jamming) transition in the large-$N$ limit, an umbrella that contains the three named deformations; an affirmative resolution of aspect (1) requires the general statement rather than finitely many examples, whereas its negative resolution requires only a single member of this class exhibiting genuinely distinct criticality. A negative result on one of the three named deformations therefore resolves aspect (1) outright, while an affirmative result on one is partial. (2) Interpretation: explain the association, noted in the source and still lacking an interpretation, between fragmentation of the space of configurations (continuous RSB, with a scaling solution approaching $q\approx1$ at the jamming line) and the singular small-gap and small-force statistics at jamming; any account must be consistent with the known counterexample of the replica-symmetric, isostatic random linear program, whose gap and force statistics are completely regular. Aspects (1) and (2) jointly address whether the perceptron criticality is the universal criticality of jamming and what causes it.

References

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