On the moments of the mass of shrinking balls under the Critical $2d$ Stochastic Heat Flow

This paper investigates the intermittency of the Critical 2d Stochastic Heat Flow by determining that the ratio of the $h$-th moment of the mass assigned to shrinking balls of radius $\epsilon$ to their Lebesgue volume scales asymptotically as $(\log\tfrac{1}{\epsilon})^{{h\choose 2}}$.

The Gist

The Big Picture: A Storm That Never Settles

Imagine you are standing in a field during a very strange, chaotic storm. This isn't a normal storm with rain; it's a "Stochastic Heat Flow." Think of it as a magical, invisible fluid spreading across a flat surface (a 2D plane).

In a normal world, if you pour water on the ground, it spreads out evenly. But in this "Critical" storm, the fluid behaves wildly. It doesn't spread smoothly; instead, it clumps together into incredibly dense, tiny spikes. Most of the time, the ground looks empty, but if you look very closely, you find these intense, concentrated bursts of "mass" (or fluid).

The authors of this paper are trying to answer a specific question: If we take a magnifying glass (a shrinking ball) and look at a tiny spot on this ground, how much "mass" do we find inside it?

The Problem: The "Empty" Illusion

The paper starts with a surprising fact: If you look at a tiny circle on this ground and ask, "How much mass is here compared to the size of the circle?" the answer is zero.

• The Analogy: Imagine a beach. If you scoop up a bucket of sand, you get a lot of sand. But if you have a magical beach where the sand is actually made of invisible dust that clumps into microscopic, super-dense pebbles, and you try to scoop up a bucket, you might find the bucket is mostly empty air.
• The Math: The authors prove that as your "bucket" (the ball of radius $\epsilon$) gets smaller and smaller, the amount of mass inside it shrinks to zero faster than the size of the bucket itself. To a casual observer, the field looks empty.

The Twist: The "Intermittency" Surprise

If the field looks empty, why study it? Because the authors are looking at the moments (a statistical way of measuring how wild the fluctuations are).

They ask: "If we look at the average of the mass raised to a high power (like squaring it or cubing it), what happens?"

• The Analogy: Imagine you are betting on a lottery. Most days, you win nothing. But once in a blue moon, you win a billion dollars.
  • If you just look at the "average" win, it might look small.
  • But if you look at the "average of the squares" (which punishes big wins heavily), that number explodes.
• The Discovery: The authors found that while the mass looks like it's disappearing, the statistical spikes are getting infinitely huge as you zoom in. The "clumps" are so dense that when you do the math on their intensity, the numbers go to infinity.

The Main Result: The Logarithmic Explosion

The core finding of the paper is about how fast these numbers grow.

They discovered that the growth isn't just random; it follows a very specific, predictable pattern related to logarithms.

• The Formula: The growth is roughly proportional to $(\log \frac{1}{\epsilon})^{h/2}$.
• The Metaphor: Imagine you are zooming in on a fractal image (like a Mandelbrot set). Every time you zoom in by a factor of 10, the complexity doesn't just double; it increases in a specific, layered way.
  • Here, as you shrink your ball radius ($\epsilon$) by a factor of 10, the "intensity" of the mass doesn't just grow linearly. It grows like a power of the logarithm.
  • Think of it like a Russian Nesting Doll. Inside every tiny doll, there is another doll, but the "weight" of the dolls inside grows faster than the dolls get smaller. The paper calculates exactly how heavy those inner dolls get.

How Did They Do It? (The "Collision" Game)

To solve this, the authors used a clever mathematical trick involving Brownian Motion (random walks).

• The Analogy: Imagine you release $h$ (say, 10) drunk ants on a table. They wander around randomly.
  • Usually, in 2D, two ants might bump into each other, but 10 ants rarely all bump into each other at the exact same spot at the exact same time.
  • However, in this "Critical" model, the ants are attracted to each other with a very specific, "critical" force. It's like they have a magnetic pull that is just strong enough to make them meet, but not so strong that they stick forever.
• The Calculation: The authors realized that the "mass" of the fluid is mathematically equivalent to the total time these ants spend colliding with each other.
  • They drew diagrams (like the ones in the paper) showing the paths of the ants and where they bumped.
  • They had to count every possible way the ants could collide.
  • The Challenge: Because the ants are attracted "critically," the math is extremely unstable. It's like trying to balance a pencil on its tip. A tiny change in the math makes the answer blow up.
  • The Solution: They used a technique called "diagrammatic expansion." They broke the problem down into simple collisions (two ants bumping) and complex collisions (three or more). They proved that even though the math is messy, the dominant factor is the simple pairwise collisions, and they managed to bound the "messy" parts to prove the final formula.

Why Does This Matter?

This isn't just about math puzzles. This model describes real-world phenomena where things cluster unpredictably:

1. Physics: It relates to how particles interact in certain quantum systems.
2. Finance: It helps model how stock prices might have extreme, rare crashes (intermittency).
3. Nature: It describes how forests might grow in patches or how turbulence works in fluids.

Summary in One Sentence

The paper proves that while a chaotic, critical fluid looks empty when you zoom in, its hidden "spikes" of mass are actually infinitely dense, growing in a precise, logarithmic pattern that reveals a deep, fractal-like structure to the universe's randomness.

Technical Summary

1. Problem Statement

The paper investigates the intermittency properties of the Critical 2d Stochastic Heat Flow (SHF), denoted as $Z^\vartheta$. This is a measure-valued stochastic process on $\mathbb{R}^2$ that serves as a non-trivial solution to the ill-posed 2D Stochastic Heat Equation (SHE) with multiplicative space-time white noise:
$$\partial_t u = \frac{1}{2}\Delta u + \beta \xi u$$
The critical regime is defined by a specific scaling of the inverse temperature parameter $\beta$ (or $\beta_N$ in the discrete Directed Polymer in Random Environment model) as the regularization parameter vanishes.

Key Phenomenon:
It is known that the one-time marginals of the Critical 2d SHF are almost surely singular with respect to the Lebesgue measure. Specifically, for a Euclidean ball $B(x, \varepsilon)$ of radius $\varepsilon$, the mass assigned by the measure, $Z^\vartheta_t(B(x, \varepsilon))$, decays to zero faster than the volume of the ball ($\varepsilon^2$) as $\varepsilon \to 0$.
$$\lim_{\varepsilon \to 0} \frac{Z^\vartheta_t(B(x, \varepsilon))}{\text{Vol}(B(x, \varepsilon))} = 0 \quad \text{a.s.}$$

The Research Question:
While the mass vanishes almost surely, the paper asks about the behavior of the moments of this mass. Do the moments $E[(Z^\vartheta_t(B(x, \varepsilon)))^h]$ also vanish, or do they exhibit "intermittency" (blow up) as $\varepsilon \to 0$? If they blow up, what is the precise growth rate?

2. Methodology

The authors employ a combination of probabilistic tools, diagrammatic expansions, and asymptotic analysis of collision times.

A. Moment Representation via Collision Diagrams

The $h$-th moment of the SHF is expressed using the Laplace transform of the total collision time of a system of $h$ independent Brownian motions interacting via a critical delta potential. This is linked to the delta-Bose gas Hamiltonian.
The moment formula is expanded into a series of collision diagrams (replica overlap diagrams). Each diagram represents a specific sequence of pairwise collisions between the Brownian paths. The weight of a diagram involves:

• Heat kernels ($g_t$) for free propagation.
• A specific kernel $G_\vartheta(t)$ related to the critical delta interaction (derived from the Volterra function).
• Combinatorial factors counting the ways to assign pairs of Brownian motions to collision events.

B. Regularization and Approximation

To handle the singularity of the indicator function of a ball, the authors approximate the uniform density on the ball $U_{B(0, \varepsilon)}$ with a heat kernel approximation $g_{\varepsilon^2/2}$. They establish that the moments of the ball mass are comparable to the moments of this smoothed approximation.

C. Upper Bound Strategy

The upper bound proof involves a rigorous estimation of the infinite series of collision diagrams:

1. Integration of Replica Variables: The authors integrate out the spatial variables and the "collision duration" variables ($v_r$) using multipliers and the Laplace transform of $G_\vartheta$.
2. Recursive Bounds: They derive recursive bounds for the integrals over the time intervals between collisions ($u_r$). This involves a novel combinatorial analysis of the coefficients arising from the critical scaling.
3. Optimization: A key technical innovation is the introduction of a parameter $\lambda$ (multiplier) and optimizing the truncation of the series sum. They split the sum into a "small $m$" part and a "large $m$" part, using different bounds for each to control the growth.
4. Log-Log Corrections: The analysis reveals that the growth is dominated by terms of the form $(\log \frac{1}{\varepsilon})^{h/2}$, but with subtle sub-logarithmic corrections involving $\log \log \log \frac{1}{\varepsilon}$.

D. Lower Bound Strategy

The lower bound relies on the Gaussian Correlation Inequality (GCI) and a specific property of the Critical 2d SHF established in previous work ([CSZ23b]):
$$E[(Z^\vartheta_t(g_{\varepsilon^2/2}))^h] \geq (1+\eta) (E[(Z^\vartheta_t(g_{\varepsilon^2/2}))^2])^{h/2}$$
This inequality suggests that the $h$-th moment is essentially determined by the square of the second moment raised to the power $h/2$. Since the second moment is known to grow as $\log \frac{1}{\varepsilon}$, the $h$-th moment grows as $(\log \frac{1}{\varepsilon})^{h/2}$. The authors carefully control the error terms when replacing the heat kernel approximation with the indicator function of the ball.

3. Key Contributions and Results

Main Theorem (Theorem 1.2)

For any integer $h \geq 2$, time $t > 0$, and parameter $\vartheta \in \mathbb{R}$, there exists a constant $C$ such that as $\varepsilon \to 0$:
$$C \left( \log \frac{1}{\varepsilon} \right)^{h/2} \leq E\left[ \left( Z^\vartheta_t(U_{B(0, \varepsilon)}) \right)^h \right] \leq \left( \log \frac{1}{\varepsilon} \right)^{h/2 + o(1)}$$
More precisely, the upper bound includes a correction term:
$$E\left[ \left( Z^\vartheta_t(U_{B(0, \varepsilon)}) \right)^h \right] \leq \left( \log \frac{1}{\varepsilon} \right)^{\frac{h}{2} + \frac{C'}{\log \log \log \frac{1}{\varepsilon}}}$$

Technical Insights

• Growth Rate: The moments grow as a power of the logarithm of the inverse radius, specifically $(\log \frac{1}{\varepsilon})^{h/2}$. This contrasts with the sub-critical case where moments might remain bounded or grow differently.
• Critical Delta Interaction: The result confirms that even at the critical scaling (where the noise strength is tuned to the phase transition point), the system exhibits strong intermittency.
• Correlation Structure: The growth rate $(\log \frac{1}{\varepsilon})^{h/2}$ suggests that pairwise collisions are "almost independent" (as the exponent is exactly $h/2$), though the presence of sub-logarithmic corrections hints at a more intricate correlation structure than in the sub-critical regime.

4. Significance and Implications

• Multifractality: The results provide evidence that the Critical 2d SHF exhibits multifractality at a logarithmic scale. The non-linear growth of moments with respect to $h$ (specifically the exponent $\xi(h) \approx h/2$ in the context of small ball asymptotics) is a hallmark of multifractal measures.
• Phase Transitions: The paper bridges the gap between the sub-critical regime (where moments are bounded or grow slowly) and the super-critical regime. It characterizes the precise behavior at the critical point, which is often the most difficult to analyze due to the singularity of the noise.
• Methodological Advancement: The paper introduces refined techniques for estimating high-order collision diagrams in the critical regime, specifically handling the singularities that arise when the regularization is removed. The use of Laplace multipliers and specific combinatorial optimizations for the critical delta interaction is a significant methodological contribution.
• Future Directions: The authors conjecture that the asymptotic behavior extends to fractional moments $h \in [0, 1]$, which would fully characterize the multifractal spectrum. They also pose questions about the transition between the logarithmic growth (for small balls) and the exponential growth ($e^{ch}$) observed for fixed-size balls in recent work by Ganguly and Nam.

In summary, this paper provides a rigorous quantitative description of how the mass of the Critical 2d Stochastic Heat Flow concentrates on small scales, revealing a delicate logarithmic intermittency structure that is distinct from both the Gaussian and the sub-critical polymer regimes.