Exponent relation for the multiplicity exponent

Prove that the multiplicity exponent of the dominant cluster-fusion event satisfies the relation $\omega=\psi+1/\nu$ in the random transverse-field Ising model across the spatial dimensions governed by an infinite-disorder fixed point.

Background

The threshold annealing time is controlled both by the excitation energy of the dominant fusion event and by its multiplicity, the number of original fields involved in that fusion. The paper introduces the multiplicity exponent ω\omega through the algebraic scaling of this quantity with system size, while ψ\psi is the tunneling exponent and ν\nu is the correlation-length exponent. A heuristic finite-size scaling argument yields the conjectured relation ω=ψ+1/ν\omega=\psi+1/\nu; it is exact in one dimension and numerically compatible with the available estimates in two and three dimensions, but is not established generally.

References

We conjecture that $\omega$ is not independent from the usual critical exponents, but can be expressed as a combination of $\psi$ and $\nu$.

— Quantum annealing of the random transverse-field Ising model in one, two, and three dimensions  (2610.11411 - Kovács et al., 8 Oct 2026) in Section 2.2, “Generalization to higher dimensions”