Short-time asymptotics of the central heat trace

Determine the short-time asymptotic behavior of the central heat trace, namely the trace of the heat kernel acting on the space of square-integrable central functions, for sequences of compact classical groups.

Background

The paper studies the central heat trace, which differs from the usual heat trace on L²(G_N) because it acts on the subspace of square-integrable conjugation-invariant functions. The main results establish a quantitative large-N expansion for this quantity for compact classical groups, but they do not address its short-time behavior.

The authors explicitly identify the short-time asymptotics as unresolved, in contrast with the known short-time expansion for the ordinary heat trace on L²(G_N).

References

For instance, as far as we are aware, its short-time asymptotics are unknown, and its large-$N$ asymptotics have been mainly studied by theoretical physicists at the level of formal power series (see Section~\ref{sec:string}).

— The central heat trace on large compact classical groups  (2511.08288 - Lemoine et al., 11 Nov 2025) in Introduction, subsection “Large-N expansion of the trace”