Uniform zero-temperature asymptotics

Establish error estimates in the fifth-order and six-fifths asymptotics that are uniform as the inverse temperature b2 tends to infinity, or develop a corresponding plateau calculus for the zero-temperature Parisi problem, in order to determine the constant governing the six-fifths upper-tail behavior of the Sherrington–Kirkpatrick ground-state energy.

Background

The paper proves the six-fifths upper-deviation constant for the Sherrington–Kirkpatrick free energy at each fixed b2>1. Its asymptotic error terms are not shown to be uniform as b2 becomes large, so the argument does not identify the corresponding constant for the ground-state energy.

Aronow and Lopatto provide two-sided bounds with unspecified constants, while the paper identifies uniform control or a zero-temperature analogue of its plateau analysis as the missing ingredient.

References

The $o(1)$ terms are not known to be uniform as $\beta\to\infty$, so Theorem~\ref{thm:sk} gives no constant for the ground-state energy, whose six-fifths upper-tail behaviour is a physics prediction recorded in (1.9), who also prove that the ground-state rate is quadratic near its minimum in a field and superquadratic without one Theorem~1.3; Aronow and Lopatto prove two-sided bounds of this order with unspecified constants Theorem~1.6 and (1.14). The constant would require control of our $o(1)$ terms uniformly in large $\beta$, or a plateau calculus for the zero-temperature Parisi problem.

— Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures  (2610.06480 - Pei, 5 Oct 2026) in Section "Earlier work and open questions", subsection "Inputs and open questions", paragraph "Zero temperature"