On the Spectral Geometry and Small Time Mass of Anderson Models on Planar Domains

This paper establishes small-time asymptotics for the Anderson Hamiltonian and parabolic Anderson model on bounded planar domains using probabilistic methods, demonstrating that a single observation of the system's eigenvalues or mass can almost surely recover geometric properties of the domain (such as area, boundary length, or fractal dimension) and the noise variance.

The Gist

The Big Picture: Listening to a Noisy Drum

Imagine you have a drum (a flat, bounded shape like a circle or a square). In the physics of the real world, if you hit this drum, it vibrates at specific frequencies. These frequencies are like the "notes" the drum can play.

In mathematics, these notes are called eigenvalues. By listening to the notes a drum makes, you can often figure out the drum's shape, size, and even the length of its rim. This is the classic idea of "Can you hear the shape of a drum?"

The Twist:
Now, imagine the drum isn't perfect. Imagine the surface of the drum is covered in static noise (like the hiss you hear on an old radio). This is what the paper calls White Noise.

The authors are asking: If we add this chaotic static noise to our drum, can we still figure out the drum's shape, size, and the strength of the noise just by listening to the notes?

The Two Main Characters

The paper studies two ways this noisy drum behaves:

1. The Anderson Hamiltonian (The Drum's Notes): This is the list of frequencies (eigenvalues) the noisy drum produces.
2. The Parabolic Anderson Model (The Heat Diffusion): Imagine spreading a drop of ink (or heat) on the drum. How does it spread out over time? The "mass" is the total amount of ink remaining on the drum after a tiny bit of time.

The Discovery: The "Logarithmic" Secret

For a long time, mathematicians knew that if you listen to the drum for a very long time, the noise makes the sound chaotic and hard to predict. But this paper looks at what happens in extremely short time (or very high frequencies).

They discovered a hidden pattern. When you add the noise, the math changes in a very specific way. It's not just a messy jumble; the noise leaves a distinct "fingerprint" in the math.

The Analogy:
Think of the drum's notes as a song.

• Without noise: The song is a clear melody.
• With noise: The song is the same melody, but someone is whispering a specific code into the microphone.

The authors found that this "whisper" (the noise) adds a logarithmic term (a specific mathematical shape involving $\log t$) to the song. This is unusual because usually, adding noise just makes things messy. Here, the noise actually creates a new, predictable structure.

What Can We Recover? (The Three Superpowers)

Because of this special "whisper," the authors proved we can recover three things almost perfectly, even with just one observation of the noisy drum:

1. The Area and the Rim Length:

   • The Metaphor: Imagine you are blindfolded and can only hear the drum's highest notes. Usually, you can guess the size of the drum. The authors proved that even with the static noise, you can still calculate exactly how big the drum is (Area) and how long the edge is (Perimeter).
   • Why it's cool: Previous math said you could only get the size. This paper says you can get the size and the edge length, even with the noise.

2. The Fractal Edge:

   • The Metaphor: What if the drum's edge isn't smooth, but looks like a jagged coastline or a snowflake (a fractal)? These shapes have a weird "dimension" between 1 (a line) and 2 (a flat surface).
   • The Result: The authors showed that by watching how the "ink" spreads out in a tiny fraction of a second, you can calculate exactly how "jagged" the edge is (its Minkowski dimension), even if it's a fractal.

3. The Volume of the Noise:

   • The Metaphor: Imagine the static on the radio is very loud in one room and quiet in another. Can you tell how loud the static is just by listening to the drum?
   • The Result: Yes! The specific "logarithmic whisper" the noise adds tells you exactly how strong the noise is ($\kappa^2$). This is surprising because usually, if the noise is too messy, you can't measure its strength. But because this noise is "white noise" (extremely rough), it leaves a clear signature.

How Did They Do It? (The Probabilistic Detective Work)

Most mathematicians try to solve this using heavy calculus and analysis (like trying to solve a puzzle by looking at the pieces under a microscope).

The authors took a different path: Probability and Random Walks.

• The Metaphor: Imagine a drunk person walking randomly on the drum (a Brownian motion).
• The Trick: They asked: "How often does this drunk person cross their own path?" or "How often do two drunk people walking on the drum bump into each other?"
• The Connection: They found that the "noise" in the drum is mathematically linked to how often these random walkers intersect.
• The "Log" Connection: In 2D space, the expected number of times a random walker crosses its own path grows like a logarithm ($\log t$). This is exactly where the mysterious "logarithmic term" in their main formula comes from. The noise isn't just chaos; it's the statistical result of these random paths crossing over and over again.

Summary

This paper is a triumph of Spectral Geometry (listening to shapes). It proves that even when you add a layer of extreme, chaotic noise to a geometric shape, the shape doesn't lose its identity. Instead, the noise adds a specific, calculable "fingerprint" (the logarithmic term).

By understanding this fingerprint, we can:

1. Measure the size and edge of the shape.
2. Measure the "jaggedness" of a fractal edge.
3. Measure the intensity of the noise itself.

It's like being able to look at a blurry, static-filled photo and perfectly reconstruct the original image, the size of the frame, and the strength of the static, all at once.

Technical Summary

1. Problem Statement

The paper investigates the Anderson Hamiltonian (AH) and the Parabolic Anderson Model (PAM) on a bounded planar domain $D \subset \mathbb{R}^2$ perturbed by white noise $\xi$. The central question is how the presence of the noise term $\kappa \xi$ (where $\kappa > 0$) affects the spectral properties and short-time behavior of the system compared to the noise-free case ($\kappa = 0$).

Specifically, the authors analyze:

• The Anderson Hamiltonian: $H_\kappa = -\frac{1}{2}\Delta + \kappa \xi$, defined with Dirichlet boundary conditions.
• The Parabolic Anderson Model: The solution $u_\kappa(t, x)$ to the heat equation $\partial_t u = (\frac{1}{2}\Delta - \kappa \xi)u$.
• Key Quantities:
  • The Exponential Trace (Heat Trace): $T_\kappa(t) = \text{Tr}(e^{-tH_\kappa}) = \sum e^{-t\lambda_n(H_\kappa)}$.
  • The Mass (Heat Content): $M_\kappa(t) = \int_D u_\kappa(t, x) dx = \sum e^{-t\lambda_n(H_\kappa)} \langle \psi_n(H_\kappa), \mathbf{1}_D \rangle^2$.

The goal is to determine the small-time asymptotics ($t \to 0$) of the expectation and variance of these quantities, specifically looking for terms beyond the leading order that reveal geometric information about $D$ or the noise strength $\kappa$.

2. Methodology

The authors adopt a purely probabilistic approach, contrasting with previous works that relied heavily on analytic machinery like regularity structures or paracontrolled calculus. Their proof strategy consists of three main steps:

Step 1: Construction via Renormalization

They assume the existence of $H_\kappa$ and $u_\kappa$ as renormalized limits of smooth approximations $H_{\kappa, \epsilon}$ and $u_{\kappa, \epsilon}$.

• The noise $\xi$ is approximated by $\xi_\epsilon = \xi * p_{\epsilon/2}$ (convolution with a Gaussian kernel).
• A diverging renormalization constant $c_{\kappa, \epsilon} = \frac{\kappa^2}{2\pi} \log(1/\epsilon)$ is introduced to compensate for the singularity of the noise as $\epsilon \to 0$.
• The limit is taken in the sense of convergence of eigenvalues and eigenfunctions (Assumption 4.9).

Step 2: Feynman-Kac Formulas and Intersection Local Times

Using the smooth approximations, the authors derive Feynman-Kac formulas for the moments of $T_{\kappa, \epsilon}(t)$ and $M_{\kappa, \epsilon}(t)$.

• The expectations involve Brownian bridges ($B^{x,y}_t$) and Brownian motions ($B^x$).
• The exponential of the noise integral is converted into terms involving Approximate Self-Intersection Local Times (SILTs) and Mutual Intersection Local Times (MILTs).
• Crucially, they show that the renormalized moments converge to limits involving the centered SILT ($\gamma_t$) and the MILT ($\alpha_t$).
• The logarithmic terms in the asymptotics emerge from the expected values of these approximate local times (specifically Lemma 5.9), which scale as $\log(1/\epsilon) + \log t$.

Step 3: Asymptotic Analysis of Local Times

The final step involves analyzing the small-time behavior of the expectations and variances of the intersection local times.

• They utilize scaling properties of Brownian motion/bridges and uniform integrability estimates (Lemma 5.7, 5.15).
• By expanding the exponential terms (e.g., $e^{\kappa^2 \gamma_t}$) and utilizing the fact that the variance of the trace is bounded ($O(1)$), they isolate the deterministic logarithmic corrections.

3. Key Contributions and Results

Main Theorem (Theorem 1.3)

The paper establishes the precise small-time asymptotics for the expectation and variance of the trace and mass. As $t \to 0$:

1. Trace Expectation:
   $$\mathbb{E}[T_\kappa(t)] = T_0(t) + \frac{\kappa^2 A(D)}{4\pi^2} \log t + o(\log t)$$
   $$\text{Var}[T_\kappa(t)] = O(1)$$
2. Mass Expectation:
   $$\mathbb{E}[M_\kappa(t)] = M_0(t) + \frac{\kappa^2 A(D)}{2\pi} t \log t + o(t \log t)$$
   $$\text{Var}[M_\kappa(t)] = O(t^2)$$

Here, $T_0(t)$ and $M_0(t)$ are the classical heat trace and heat content for the Laplacian. The term $A(D)$ is the area of the domain.

Significance of the Result:

• The noise introduces a deterministic logarithmic correction to the leading order terms.
• Unlike the case where $\kappa=0$ (where the expansion is in powers of $t$), the presence of white noise creates a $\log t$ term in the trace and $t \log t$ in the mass.
• The variance remains bounded (or small), implying that a single observation of the spectrum or mass is sufficient to recover these parameters almost surely.

Applications (Corollaries)

The authors demonstrate that these asymptotics allow for the recovery of geometric and noise parameters from a single realization:

1. Spectral Geometry (Corollary 2.6): If the boundary $\partial D$ is sufficiently regular, one can recover both the Area $A(D)$ and the Boundary Length $L(\partial D)$ almost surely from the eigenvalues of $H_\kappa$. This extends Mouzard's Weyl law results to the noisy case.
2. Fractal Geometry (Corollary 2.8): If $D$ is simply connected and $\partial D$ is a fractal with Minkowski dimension $d_M \in [1, 2)$, the dimension $d_M$ can be recovered almost surely from the small-time asymptotics of the mass $M_\kappa(t)$.
3. Variance Recovery (Corollary 2.10): The noise strength $\kappa^2$ can be recovered almost surely from the eigenvalues of $H_\kappa$. This is surprising because for smoother noise potentials, $\kappa$ typically cannot be recovered from the spectrum alone; the singularity of white noise makes it detectable via the logarithmic term.

4. Significance and Novelty

• Probabilistic vs. Analytic: While previous works on the Anderson Hamiltonian in 2D relied on deep analytic tools (Regularity Structures, Paracontrolled Calculus), this paper provides a probabilistic proof relying on the intersection properties of Brownian paths. This offers a new perspective and potentially simpler access to these results.
• Refinement of Weyl's Law: The paper provides a substantial refinement of the Weyl law for the Anderson Hamiltonian. It shows that while the leading order term ($t^{-1}$) is unchanged by the noise, the second-order term (the logarithmic correction) encodes the noise intensity and the area of the domain.
• Detectability of Noise: A major theoretical insight is that the singularity of white noise (as opposed to regular noise) manifests as a logarithmic term in the heat trace. This allows for the "hearing" of the noise variance $\kappa^2$ and the noise's effect on the geometry, which is impossible with regular potentials where corrections are typically power-law.
• Robustness: The results hold for arbitrary planar domains (with appropriate regularity assumptions for the boundary in specific applications), extending previous results that were often limited to smooth manifolds or specific boundary conditions.

5. Conclusion

The paper successfully quantifies the impact of white noise on the spectral geometry of planar domains. By linking the renormalization of the Anderson Hamiltonian to the asymptotics of Brownian intersection local times, the authors prove that the small-time behavior of the heat trace and mass contains a deterministic logarithmic signature of the noise. This signature is strong enough to allow for the almost sure recovery of the domain's area, boundary length, fractal dimension, and the noise variance from a single spectral observation.