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 $\alpha ,\theta ,\kappa$ that control the gap, force, and overlap distributions by the relations $\alpha =a/b$, $\theta =(c-a)/(b-c)$, $\kappa =c+1$. Crucially, the relation $a+b=1$ yields the scaling relations $\alpha =1/(2+\theta )$ and $\kappa =2-2/(3+\theta )$ 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.

arXiv comment12 pages, no figures

References in corpus (9)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Front propagation into unstable states
• Jamming of Soft Particles: Geometry, Mechanics, Scaling and Isostaticity
• Effects of compression on the vibrational modes of marginally jammed solids
• Exact theory of dense amorphous hard spheres in high dimension. III. The full RSB solution
• Stability at Random Close Packing
• Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition
• Exact theory of dense amorphous hard spheres in high dimension. I. The free energy
• The distribution of forces affects vibrational properties in hard sphere glasses