A proof of an identity for the critical exponents of jamming

Abstract: Within the full replica-symmetry-breaking (fullRSB) solution of dense hard spheres in infinite dimension, Charbonneau, Kurchan, Parisi, Urbani, and Zamponi (CKPUZ; J.Stat.Mech.P10009, 2014) introduced three critical exponents $a$, $b$, $c$ governing the matching region of the fullRSB profile near the jamming transition. These exponents satisfy two scaling relations. The first, $b=(1+c)/2$, was established analytically by the diffusion-drift balance in the scaling ansatz. The second, $a+b=1$, was observed numerically to arbitrary precision but could not be proven. The exponents $a,b,c$ of the scaling fullRSB ansatz are related to the physical exponents $α, θ, κ$ that control the gap, force, and overlap distributions by the relations $α=a/b$, $θ=(c-a)/(b-c)$, $κ=c+1$. Crucially, the relation $a+b=1$ yields the scaling relations $α=1/(2+θ)$ and $κ=2-2/(3+θ)$ predicted on independent grounds by the mechanical-marginal-stability arguments of Wyart and collaborators. Here, we give an analytic proof of the identity $a+b=1$ from the scaling fullRSB equations. The proof was obtained through interaction with Claude (Sonnet 4.6 and Opus 4.7) and verified by us.

• The paper presents an analytic proof that the scaling relation a + b = 1 holds at the jamming transition in hard sphere models.
• It employs integration-by-parts identities and maximum principle techniques within the fullRSB framework to rigorously establish the result.
• The findings bridge phase-space marginality with mechanical stability, confirming the predicted critical exponent relationships in disordered systems.

Rigorous Proof of a Scaling-Exponent Identity at the Jamming Transition

Introduction

The jamming transition in amorphous packings, such as hard or soft spheres, constitutes a central theme in the theory of disordered systems. The mean-field, infinite-dimensional solution to this problem—established through replica symmetry breaking (RSB) techniques—has revealed a critical manifold described by power-law behaviors of several structural and mechanical observables. Charbonneau, Kurchan, Parisi, Urbani, and Zamponi (CKPUZ) previously identified three scaling exponents, $a, b, c$, that characterize the matching regime of the fullRSB solution near the jamming density. These exponents control the asymptotic form of the fullRSB profile and are related to physically measurable exponents $(\alpha, \theta, \kappa)$ governing the distributions of inter-particle gaps, forces, and the Edwards-Anderson (EA) parameter, respectively.

Two scaling relations between $a, b, c$ emerge in the CKPUZ theory. The first, $b = (1 + c)/2$, was derived analytically. The second, $a + b = 1$, was supported by high-precision numerics but lacked a formal analytic proof. The fulfillment of $a + b = 1$ implies, in combination with $b = (1 + c)/2$, that the physical exponents satisfy

$$\alpha = \frac{1}{2 + \theta}, \qquad \kappa = 2 - \frac{2}{3 + \theta}$$

which are exactly the relations postulated by Wyart et al. based on mechanical marginal stability arguments. The present work gives a first-principles analytic proof of $a + b = 1$ within the scaling fullRSB equations for hard spheres in $d \to \infty$ (2606.03300).

Mathematical Framework

FullRSB Equations and the Scaling Regime

The analysis is based on the continuum fullRSB equations for the (rescaled) mean-square displacement $(\alpha, \theta, \kappa)$0 and probability field $(\alpha, \theta, \kappa)$1 as $(\alpha, \theta, \kappa)$2, close to the jamming transition. The matching region, where the scaling ansatz applies, is characterized by the critical scaling:

$(\alpha, \theta, \kappa)$3

with $(\alpha, \theta, \kappa)$4, and where the exponents $(\alpha, \theta, \kappa)$5, $(\alpha, \theta, \kappa)$6, $(\alpha, \theta, \kappa)$7 are determined by consistency with the fullRSB solution and marginal stability. The crucial point is the reduction, in the scaling limit, to a pair of coupled ODEs: a nonlinear equation for an auxiliary function $(\alpha, \theta, \kappa)$8 and a linear eigenvalue problem for $(\alpha, \theta, \kappa)$9.

Formal Statement of the Proof Strategy

Unlike prior approaches, the proof does not seek deeper direct relationships between the ODE solutions $a, b, c$0 and $a, b, c$1. Instead, it relies on a weak-form identity: integrating the $a, b, c$2 equation against an appropriate test function, constructed from the solution $a, b, c$3. This strategy yields the key algebraic identity:

$a, b, c$4

where $a, b, c$5, $a, b, c$6, $a, b, c$7.

Crucially, marginal stability imposes $a, b, c$8, which, together with the positivity of $a, b, c$9 on the physical branch ($b = (1 + c)/2$0 throughout), forces $b = (1 + c)/2$1.

Analytic Control and Selection of the Physical Branch

Fisher-KPP Structure and Maximum Principles

An obstacle to the proof’s rigor is ensuring that the physical (no-node) branch is selected from the multiplicity of solutions to the ODE for $b = (1 + c)/2$2. The authors demonstrate that the evolution equation for a two-variable extension $b = (1 + c)/2$3, closely related to $b = (1 + c)/2$4, takes the form of a Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher-KPP) type reaction-diffusion equation. The initial condition is a Boolean indicator $b = (1 + c)/2$5, and maximum principles from parabolic PDE theory preserve the bound $b = (1 + c)/2$6 for all $b = (1 + c)/2$7. Passing to the scaling limit, one obtains $b = (1 + c)/2$8, guaranteeing $b = (1 + c)/2$9 for the physical branch.

Uniqueness and the Role of Marginal Stability

While the proof hinges on a specific scaling ansatz and the existence of a no-node solution, the uniqueness and rigorous derivation of the matching regime from the underlying fullRSB equations are assumed, as full mathematical control of these aspects remains open.

Implications and Theoretical Significance

Equivalence of Mechanical and Phase-Space Marginality

The analytic proof that $a + b = 1$0 yields the scaling relations for the physical exponents previously conjectured to follow from mechanical marginal stability, thus tying the phase-space concept of marginality in spin-glass theory to the mechanical stability of jammed packings. The critical exponents $a + b = 1$1 follow relations predicted from different lines of argument in amorphous solids.

Mathematical Technique

The approach highlights the effectiveness of integrating linear differential constraints against functions derived from the nonlinear scaling solution, and leverages maximum-principle arguments for nonlinear parabolic PDEs to establish the positivity required to complete the algebraic proof.

Open Problems

The work outlines several remaining theoretical questions:

• The existence and uniqueness of the fullRSB profile in the hard-sphere context, by analogy with the SK model.
• A rigorous asymptotic theory connecting the fullRSB solution to the scaling regime.
• The potential for a converse derivation (mechanical marginality $a + b = 1$2 fullRSB criticality).

Conclusion

This paper provides the first analytic derivation of the scaling relation $a + b = 1$3 for the exponents controlling the jamming transition in the mean-field hard sphere model. Together with numerical evidence and the previously established relation $a + b = 1$4, this result rigorously links the replica-theoretic scaling picture of jamming with the mechanical marginal stability framework. The proof employs a combination of integration-by-parts identities for the ODE system and maximum principles for reaction–diffusion equations, offering methodological advances with potential utility in similar settings. This bridge between phase-space and mechanical stability deepens the theoretical understanding of criticality at jamming and motivates further rigorous study in the mathematics of structural glasses and related disordered systems.