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 (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 )