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arxiv: 2606.26958 · v1 · pith:JVD7NCK4new · submitted 2026-06-25 · 🧮 math.PR · math-ph· math.MP· math.SP

Spectral expansion of LQG heat trace and KPZ scaling

Nathana\"el Berestycki , Jakob Klein This is my paper

Pith reviewed 2026-06-26 04:08 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MPmath.SP
keywords Liouville quantum gravityheat traceKPZ relationGaussian free fieldspectral expansionheat kernel asymptoticsheat content
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The pith

The second term in the short-time spectral expansion of the expected LQG heat trace follows the KPZ exponent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the short-time asymptotics of the Liouville quantum gravity heat trace, defined as the integral of the on-diagonal heat kernel over a bounded domain with respect to the whole-plane Gaussian free field. It proves that the second term in the expansion of the expected heat trace as time approaches zero is controlled by a nontrivial exponent coming from the KPZ relation. A parallel almost-sure result holds for the heat content. This matters because it links the spectral geometry of random surfaces directly to conformal field theory scaling predictions. The work also settles a prior conjecture on the annealed short-time behavior of the heat kernel and establishes finiteness of all moments of the rescaled kernel.

Core claim

For the LQG heat trace on a bounded domain driven by the whole-plane Gaussian free field, the expected value admits an asymptotic expansion in which the second term as t to 0 is governed by the KPZ exponent; an almost-sure version holds for the associated heat content. The same analysis yields the conjectured annealed short-time asymptotics of the heat kernel together with finiteness of all moments after proper rescaling.

What carries the argument

The LQG heat trace (integral over the domain of the on-diagonal LQG heat kernel), whose short-time expansion is shown to obey KPZ scaling via the underlying Gaussian free field.

If this is right

  • The annealed short-time asymptotics of the LQG heat kernel match the conjectured form.
  • All moments of the properly rescaled LQG heat kernel remain finite.
  • The heat content admits an almost-sure short-time expansion controlled by the same KPZ exponent.
  • Spectral expansions of the LQG heat trace therefore inherit the scaling predicted by conformal field theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar KPZ exponents may control other spectral quantities such as eigenvalue distributions in the same LQG setting.
  • Moment finiteness opens the door to concentration or law-of-large-numbers statements for the random heat trace.
  • The approach could extend to time-dependent or non-whole-plane fields once domain regularity is verified.

Load-bearing premise

The whole-plane Gaussian free field on a bounded domain is enough for the short-time asymptotics of the LQG heat trace to be governed by the KPZ relation.

What would settle it

A numerical or analytic check that the coefficient of the second term in the expected heat trace expansion deviates from the KPZ-predicted exponent for a concrete bounded domain.

Figures

Figures reproduced from arXiv: 2606.26958 by Jakob Klein, Nathana\"el Berestycki.

Figure 1
Figure 1. Figure 1: Quantum cone, conditioned on Ex, in cylindrical coordinates. The main observation in the proof of Lemma 3.10 is that sC finds itself far away from both log(1/r) and log(1/r′ ), so that the effect of both conditions (at short and large scales) disappear asymptotically. Remark 3.11. For the final claim of this lemma it is crucial that the compact sets in the topology underlying the convergence in law ⇒ are n… view at source ↗
Figure 2
Figure 2. Figure 2: Example of a curve with different one-sided Minkowski dimensions on each side: [PITH_FULL_IMAGE:figures/full_fig_p056_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Idea of the proof of Lemma C.1. = ε + ε−1X /3+o(1) n=1  2ε log(n) − 2ε 2 c  + X∞ n=ε−1/3+o(1) 2 n3 log(n) − 4˜c n6 = ε + ε 1−1/3+o(1) + (ε −1/3 ) −(2−1)+o(1) = ε 2/3+o(1) , where o(1) vanishes when ε → 0. In conclusion, the inner Minkowski dimension is 2 − limε→0 log(Iε)/ log(ε −1 ) = 2 − 2/3 = 4/3. Outer neighborhood. Similar considerations apply, estimating the contribution of each rectangle. Now we mu… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper studies the short-time asymptotics (t→0) of the LQG heat trace, defined as the integral over a bounded domain Ω of the on-diagonal LQG heat kernel constructed from the whole-plane Gaussian free field h. The central claim is that the second term in the spectral expansion of the expected heat trace is governed by a nontrivial exponent given by the KPZ relation. A stronger almost-sure result is stated for the related heat content. Along the way, the manuscript resolves the BW conjecture on annealed heat-kernel asymptotics and establishes finiteness of all moments of the properly rescaled heat kernel.

Significance. If the derivations hold, the work supplies a rigorous link between the spectral expansion of the LQG heat trace and the KPZ scaling relation, while resolving an open conjecture of BW and furnishing moment bounds. These are concrete technical contributions to the study of Liouville quantum gravity and random planar geometry; the moment bounds in particular strengthen the analytic control available for LQG objects.

minor comments (2)
  1. The abstract refers to the conjecture of \cite{BW} without giving the full bibliographic entry; the bibliography should list the reference explicitly.
  2. Notation for the LQG heat kernel and heat trace is introduced in the abstract but not defined until later sections; a brief reminder of the precise integral definition in the introduction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recognition of the link to the KPZ relation, resolution of the BW conjecture, and the moment bounds. The recommendation of minor revision is noted. No major comments were listed in the report, so we have no specific points to address point-by-point at this stage. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity; KPZ exponent invoked as external input

full rationale

The paper's central claim is that the second term in the short-time spectral expansion of the expected LQG heat trace (and almost-sure heat content) is governed by the KPZ exponent. The abstract and claim structure treat the KPZ relation as a known external input rather than deriving or fitting it internally. No equations or steps reduce the claimed exponent to a self-defined quantity, a fitted parameter renamed as prediction, or a self-citation chain. The resolution of the BW conjecture and moment bounds are presented as separate results. The derivation is therefore self-contained against external benchmarks with no load-bearing reductions visible.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty.

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