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Spectral expansion of LQG heat trace and KPZ scaling

This paper establishes that the short-time asymptotics of the expected Liouville quantum gravity heat trace on a bounded domain are governed by a nontrivial exponent determined by the KPZ relation, while also resolving a conjecture regarding annealed heat kernel asymptotics and proving the finiteness of its moments.

Original authors: Nathanaël Berestycki, Jakob Klein

Published 2026-06-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nathanaël Berestycki, Jakob Klein

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing in a room, and you want to understand the shape of the room by listening to how sound echoes inside it. In the world of mathematics, this is similar to studying the "heat trace." If you heat up a room and watch how the heat spreads, the way it dissipates tells you about the room's size, its walls, and even the texture of its boundaries.

This paper, written by mathematicians Nathanaël Berestycki and Jakob Klein, explores this concept in a very strange, wobbly, and random universe called Liouville Quantum Gravity (LQG).

Here is a breakdown of their discovery using everyday analogies:

1. The Setting: A Room Made of Jello

In normal geometry (like a standard room), the walls are straight and the floor is flat. If you drop a drop of ink (heat) on the floor, it spreads out in a predictable circle.

In this paper, the "room" is a Liouville Quantum Gravity surface. Imagine the floor isn't flat; it's made of jello that is constantly vibrating and rippling. Sometimes the floor is incredibly bumpy and thick (like a mountain), and sometimes it's incredibly thin and stretched out. This "jello" is generated by something called a Gaussian Free Field, which is essentially a random, wiggly landscape.

2. The Experiment: The Heat Trace

The authors are interested in a specific measurement called the Heat Trace.

The Analogy: Imagine you have a magical thermometer that measures the total amount of heat in the room at a specific moment in time ( $t$ ).
The Question: As time gets very close to zero (the very first split second after you turn on the heat), how does the total heat behave?

In a normal, flat room, the math is simple:

The Big Term: The total heat is mostly determined by the size (area) of the room.
The Small Term: The next most important thing is the length of the walls. The heat near the walls behaves differently than heat in the middle.

3. The Discovery: The "Wobbly" Walls

The authors wanted to see if this simple rule (Area + Wall Length) works in their wobbly, jello-like quantum room.

What they knew: They already knew the first part (the Area) still mattered, but the number was different because the "jello" changes the effective size of the room.
The Big Surprise: They looked at the second part (the walls). In a normal room, the wall effect scales with the square root of time ( $1/\sqrt{t}$ ). But in this quantum jello room, the wall effect is weird.

They found that the "wall effect" doesn't follow the normal rules. Instead, it follows a mysterious exponent (a power) related to the KPZ relation.

The Metaphor: Imagine the walls of the room aren't just a line; they are a fractal coastline. If you zoom in, the coastline gets longer and longer. In this quantum world, the "length" of the wall depends on how you measure it. The authors proved that the way the heat interacts with these fractal walls is governed by a specific, non-trivial number derived from the KPZ equation (a famous formula in physics that relates flat geometry to this wobbly quantum geometry).

4. The "In-Out" Trick

How did they figure this out? They used a clever trick called the "In-Out Decomposition."

The Analogy: Imagine you are tracking a random walker (a person wandering aimlessly) in a park.
- The "In" part: You count how many times the walker stays inside the park boundaries.
- The "Out" part: You count how many times the walker steps outside the park.
The Insight: The total heat trace is the "In" part minus the "Out" part.
- The "In" part is easy to calculate (it's just the volume of the room).
- The "Out" part is the hard part. It represents the heat that "leaks" out because the walker hit the wall.
- The authors realized that the "Out" part is exactly where the strange KPZ exponent comes from. The probability of the walker hitting the wall in this wobbly universe is governed by the fractal nature of the boundary.

5. The "Heat Content" (The Temperature of the Room)

They also studied something called Heat Content.

The Analogy: Imagine the room starts at freezing cold ( $0^\circ$ ). You suddenly heat the walls to a warm temperature ( $1^\circ$ ). How much total heat is inside the room after a tiny amount of time?
The Result: They proved that the amount of heat absorbed from the walls follows the same strange KPZ scaling rule. This was crucial because it helped them prove the result for the Heat Trace. It's like saying, "If we know how fast the walls warm up the room, we can figure out how the heat trace behaves."

6. Solving a Mystery

The paper also solves a conjecture from a previous study (by Berestycki and Werner, 2023).

The Mystery: What happens to the heat kernel (the probability of a walker being exactly where they started) if you look at it from the perspective of a "typical" point in this quantum jello?
The Solution: They proved that if you zoom in close enough and look at the heat kernel, it behaves in a very specific, stable way. It turns out that all the moments (averages of different powers) of this heat kernel are finite. This means the "spikes" in the heat distribution aren't wild enough to break the math.

Summary

In simple terms, this paper takes a complex, random, and bumpy universe (Liouville Quantum Gravity) and asks: "How does heat behave here?"

They found that while the main behavior depends on the area, the secondary behavior (the boundary effects) is governed by a strange, fractal-like rule known as the KPZ relation. They proved this by breaking the problem into "what stays inside" and "what leaks out," showing that the "leakage" is determined by the fractal dimension of the walls in this quantum world.

They didn't just guess this; they built a rigorous mathematical bridge connecting the random geometry of the walls to the behavior of heat, confirming that the "wobbly" nature of the universe changes the rules of geometry in a very specific, predictable way.

Technical Summary: Spectral expansion of LQG heat trace and KPZ scaling

Problem Statement
This paper investigates the short-time asymptotics ( $t \to 0$ ) of the Liouville Quantum Gravity (LQG) heat trace, $H_\Omega(t)$ , defined as the integral of the on-diagonal heat kernel over a bounded domain $\Omega \subset \mathbb{C}$ with respect to the Liouville area measure $\mu$ . While the leading-order behavior of the heat trace was previously established in [BW23] as proportional to the LQG area (yielding a Weyl law with spectral dimension $d_s=2$ ), the behavior of the second term in the spectral expansion remained an open question. In classical Riemannian geometry, this second term is governed by the boundary length and scales as $t^{-1/2}$ . The authors aim to determine the scaling exponent of this second term in the LQG setting, specifically investigating whether it follows a non-trivial exponent governed by the Knizhnik–Polyakov–Zamolodchikov (KPZ) relation, and to establish almost sure convergence results for related quantities like the heat content.

Methodology
The authors employ a combination of probabilistic analysis, Gaussian Free Field (GFF) theory, and stochastic processes. Key methodological components include:

In-Out Decomposition: The core analytical tool is the decomposition of the heat kernel $p^\Omega_t(x,x)$ into a "bulk" term (the heat kernel on the full plane, $p^C_t(x,x)$ ) and a "boundary" term (trajectories leaving $\Omega$ before time $t$ , $p^{C\setminus\Omega}_t(x,x)$ ). This allows the authors to separate the volume contribution from the boundary contribution.
Rooted Measures and Girsanov Transform: To analyze the heat kernel under the annealed measure (averaging over the GFF), the authors utilize the Peyrière rooted measure $P^*_x$ . This involves a Girsanov transform that introduces a logarithmic singularity $\gamma G(x, \cdot)$ to the field, effectively changing the "thickness" of the point $x$ .
Scaling and Quantum Cones: The analysis relies heavily on the scale-invariance properties of the $\gamma$ -quantum cone. The authors prove that the heat kernel on the quantum cone satisfies an exact scaling relation $p^C_t(0,0) \stackrel{d}{=} t^{-1} p^C_1(0,0)$ . This allows them to relate local behavior near a point to the global geometry of the quantum cone.
Moment Estimates and Concentration: The proof involves delicate first and second moment computations for the heat content. To handle the divergence of moments in the $L^2$ -phase (when $\gamma \ge \sqrt{2}$ ), the authors introduce a truncation argument based on "good events" where the field has specific thickness properties (circle averages) at scales related to the distance to the boundary.
0-1 Law and Decoupling: To upgrade convergence in probability to the desired almost sure (or high probability) results, the authors partition the boundary into many small, approximately independent regions. They use a Paley-Zygmund inequality combined with a 0-1 law type argument to boost the probability of the lower bound.

Key Contributions and Results

Spectral Expansion of the Heat Trace: The main result (Theorem 1.2) establishes that the second term in the expansion of the expected heat trace $\mathbb{E}[H_\Omega(t)]$ scales as $t^{-1+\Delta}$ , where $\Delta$ is the KPZ scaling exponent associated with the inner Minkowski dimension of the boundary $\partial \Omega$ . Specifically, if the inner Euclidean scaling exponent of the boundary is $x$ (where $\text{Leb}(\Omega_\epsilon) \sim \epsilon^{2x}$ ), then $\Delta$ is the unique solution to the KPZ equation $x = \frac{\gamma^2}{4}\Delta + (1-\frac{\gamma^2}{4})\Delta^2$ .
$\mathbb{E}[H_\Omega(t)] = \frac{c_\gamma \mathbb{E}[\mu(\Omega)]}{t} - t^{-1+\Delta+o(1)}$
This confirms that the boundary effects in LQG are governed by the KPZ relation, deviating from the classical $t^{-1/2}$ scaling unless $\Delta = 1/2$ .
Heat Content Asymptotics: The paper proves (Theorem 1.3) that the heat content $K_\Omega(t)$ , which measures the integrated temperature in $\Omega$ , satisfies $K_\Omega(t) = t^{\Delta+o(1)}$ in probability. This result is stronger than the heat trace result as it holds without taking the expectation over the field (convergence in probability). The proof requires specific assumptions on the boundary (Assumptions 1.2 and 1.3) regarding the uniformity of the Minkowski dimension.
Heat Kernel Moments and Conjecture Resolution: The authors resolve a conjecture from [BW23] regarding the annealed convergence of the heat kernel (Theorem 1.4). They show that under the rooted measure $P^*_x$ , the rescaled heat kernel $t p^\Omega_t(x,x)$ converges in law to a random variable $X$ representing the heat kernel at time 1 on the $\gamma$ -quantum cone. Furthermore, they prove that all moments of this limiting random variable are finite.
Quantitative Bulk Convergence: The paper provides quantitative bounds showing that the difference between the expected bulk heat trace and the leading Weyl law term decays faster than any polynomial (FTP) as $t \to 0$ .

Significance and Claims
The paper claims to provide a rigorous spectral geometric characterization of LQG surfaces, linking the spectral invariants (heat trace expansion) directly to the geometric KPZ scaling. The authors emphasize that their results confirm numerical observations from [BW23] regarding the anomalous exponent of the second term.

The significance of the work lies in:

Connecting Spectral Geometry and KPZ: It demonstrates that the KPZ relation, typically used to map Euclidean dimensions to LQG dimensions, naturally governs the spectral expansion of the heat trace, specifically the boundary correction term.
Solving Open Problems: It resolves the conjecture on the annealed asymptotics of the heat kernel and establishes the finiteness of all moments for the rescaled heat kernel, which was previously unknown.
Methodological Advances: The development of the "in-out" decomposition in the context of LQG and the use of rooted measures to analyze local heat kernel behavior provide new tools for studying spectral geometry on random surfaces.

The authors note that while they prove the result for the expected heat trace, the almost sure behavior for the heat trace itself remains an open question, though they successfully prove the almost sure (in probability) behavior for the heat content. They also conjecture that for "quantum discs" (where the boundary is an SLE curve), the exponent $\Delta$ should be exactly $1/2$ , recovering the classical Riemannian scaling, but state they are currently unable to prove this due to technical difficulties.