Equality of the one-step replica-symmetry-breaking and independence-ratio bounds

Prove that the 1-RSB cavity-method bound _d equals the asymptotic independence ratio _d for every integer degree d 20.

Background

The paper discusses several bounds for the independence ratio of random regular graphs. The Ding–Sly–Sun result identifies the asymptotic independence ratio with a sharper bound for sufficiently large degrees, while the cavity method yields a 1-RSB prediction _d. The text states that equality between the 1-RSB prediction and the true asymptotic independence ratio is conjectured for all degrees at least 20, and notes that this equality is still only expected rather than established in the paper.

References

More precisely, a 1-RSB (1-step replica symmetric breaking) formula _d was obtained through the cavity method, and it was conjectured that _d=_d for every d 20. (This is still widely expected to hold true.)

— Star decompositions and independent sets in random regular graphs  (2503.09458 - Harangi, 12 Mar 2025) in Section 3.1, subsection “Bounds on the independence ratio”