Equality of the 1-RSB and true independence-ratio thresholds

Prove that the 1-step replica-symmetry-breaking independence-ratio bound equals the asymptotic independence ratio of the random d-regular graph for every integer degree d ≥ 20.

Background

The paper discusses several thresholds for the independence ratio of random regular graphs. The 1-RSB cavity-method formula provides a non-rigorous bound, while the asymptotic independence ratio is the limiting independence ratio of the random d-regular graph.

The text states that equality between these quantities is conjectured for every d ≥ 20 and remains widely expected, but the paper does not resolve this conjecture. The notation is partially corrupted in the supplied source; the statement refers to the 1-RSB bound and the asymptotic independence-ratio threshold introduced in that subsection.

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 $d \geq 20$. (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”