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Peer-reviewedCC BYJournal of Statistical Mechanics Theory and Experiment

A proof of an identity for the critical exponents of jamming

  • Giorgio Parisi A1,2,3
  • Francesco Zamponi3
  1. 1Beihang University
  2. 2Rome Unit
  3. 3Sapienza Università di Roma
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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 )

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Cite

BibTeXRIS
BibTeX
@article{parisi2026proof,
  title = {{A proof of an identity for the critical exponents of jamming}},
  author = {Giorgio Parisi A and Francesco Zamponi},
  journal = {Journal of Statistical Mechanics Theory and Experiment},
  year = {2026},
  month = jul,
  doi = {10.1088/1742-5468/ae7bd7},
  url = {https://doi.org/10.1088/1742-5468/ae7bd7},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - A proof of an identity for the critical exponents of jamming
AU  - Giorgio Parisi A
AU  - Francesco Zamponi
PY  - 2026
DA  - 2026/07/01
JO  - Journal of Statistical Mechanics Theory and Experiment
DO  - 10.1088/1742-5468/ae7bd7
UR  - https://doi.org/10.1088/1742-5468/ae7bd7
AB  - 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 )
SN  - 1742-5468
ER  -