Spectral expansion of LQG heat trace and KPZ scaling

Abstract

Let $h$ be a whole plane Gaussian free field, and let $\Omega$ be a bounded domain in two dimensions. We study the asymptotics as $t\to 0$ of the Liouville quantum gravity (LQG) heat trace, defined as the integral over $\Omega$ of the on-diagonal LQG heat kernel. Our main result is to show that the second term in the spectral expansion as $t\to 0$ of the expected heat trace is governed by a nontrivial exponent, given by the KPZ (Knizhnik--Polyakov--Zamolodchikov) relation. A similar but stronger (almost sure) result applies to the related notion of heat content. Along the way we obtain various results on the short-term behaviour of the heat kernel, notably solving a conjecture of \cite{BW} concerning its annealed asymptotics, and showing the finiteness of all moments of the properly rescaled heat kernel.