The critical two-dimensional KPZ process

Overview

The two-dimensional Kardar–Parisi–Zhang equation describes a growing random surface, but its noise is too irregular for the equation to have a classical meaning. Berger, Caravenna, Clark, Dunlap, Gu, Liu, Nakajima, Sun, Tsai, Xie, and Zygouras construct a critical limiting height field by centering logarithms of directed-polymer partition functions. The same field is a renormalized logarithm of the critical stochastic heat flow. Its spatial averages are non-Gaussian, although a further limit at shrinking spatial scales is Gaussian (Berger et al., 2026).

The construction rests on Gaussian convexity, a decomposition at two intermediate times, and estimates for products of random kernels. Scott Armstrong’s Lean development supplies formal counterparts and four additional results; the final sections explain these results and the precise scope of the Palomar registration.

🏷️ The equation and its critical dimension

The Kardar–Parisi–Zhang equation, with the normalization used in the paper, is formally

$$\begin{array}{c}{\partial }_{t}h=\frac{1}{2}\Delta h+\frac{1}{2}|\nabla h{|}^2+\beta \xi ,\\ x\in {\mathbb{R}}^2.\end{array}$$

Here $h(t,x)$ is a height and $\xi$ is space-time white noise. White noise is a centered Gaussian random distribution: for smooth compactly supported space-time tests $f,g$,

$$\begin{array}{c}\mathbb{E}[\xi (f)\xi (g)]\\ ={\int }_0^{\infty }{\int }_{{\mathbb{R}}^2}f(t,x)g(t,x)dxdt.\end{array}$$

Thus $\xi (t,x)$ is symbolic notation, not a random function with values at individual points. The resulting height is also rough, and the square of its gradient cannot simply be formed pointwise.

For smooth functions, the substitution $u=e^h$ gives

$$\frac{\Delta e^h}{e^h}=\Delta h+|\nabla h{|}^2.$$

This is the Cole–Hopf transformation. It suggests constructing $h$ from the multiplicative stochastic heat equation

$${\partial }_{t}u=\frac{1}{2}\Delta u+\beta u\xi .$$

In a stochastic regularization, Itô’s formula also produces a deterministic correction. Both the regularization and the correction must be specified before the transformation becomes a mathematical definition.

Dimension two is critical under diffusive scaling. In $d$ spatial dimensions, the rescaled noise $\xi (at,\sqrt{a}x)$ has the law of $a^{-(d+2)/4}\xi (t,x)$. Rescaling the equation therefore multiplies its noise coefficient by $a^{(2-d)/4}$. This power is zero when $d=2$. The coupling changes on logarithmic scales, and the construction below tunes it inside a logarithmically narrow critical window.

🏷️ Gaussian directed polymers and the critical window

Let $S$ be the nearest-neighbor simple random walk on ${\mathbb{Z}}^2$. Independently, let

$$\omega (j,z),j\in \mathbb{N},z\in {\mathbb{Z}}^2,$$

be independent standard Gaussian variables. We write ${\mathrm{E}}_{a,w}$ for expectation over a walk started at $w$ at time $a$, and $\mathbb{E}$ for expectation over the environment $\omega$.

The point-to-point partition function is

$$\begin{array}{c}{V}_j=\beta \omega (j,S_j)-\frac{1}{2}{\beta }^2,\\ Z_{a,b}^{\beta }(w,z)\\ ={\mathrm{E}}_{a,w}\left[e^{{\sum }_{j=a+1}^b{V}_j}{1}_{\{S_b=z\}}\right].\end{array}$$

The terminal environment variable is included and the initial one is excluded. Summing over $z$ gives the point-to-plane partition function $Z_{a,b}^{\beta }(w)$, whose endpoint is unrestricted. The subtraction ${\beta }^2/2$ ensures

$$\begin{array}{c}\mathbb{E}{Z}_{a,b}^{\beta }(w,z)={\mathrm{P}}_{a,w}(S_b=z),\\ \mathbb{E}{Z}_{a,b}^{\beta }(w)=1.\end{array}$$

The relevant measure of recurrence is the expected overlap of two independent walks:

$$\begin{array}{rl}{R}_N& =\sum _{n=1}^N\mathrm{P}(S_n=S_n^{\prime })\\ & =\sum _{n=1}^N\mathrm{P}(S_{2n}=0)\\ & =\frac{1}{\pi }\log N+O(1).\end{array}$$

Fix $\vartheta \in \mathbb{R}$. For sufficiently large $N$, choose ${\beta }_N\ge 0$ by

$$\boxed{\begin{array}{rl}{\sigma }_N^2& :=e^{{\beta }_N^2}-1\\ & =\frac{1}{R_N}\left(1+\frac{\vartheta }{\log N}\right).\end{array}}$$

The quantity ${\sigma }_N^2$ is the variance of the mean-one random weight $e^{{\beta }_N\omega -{\beta }_N^2/2}$. In particular, ${\beta }_N^2\sim \pi /\log N$. The parameter $\vartheta$ specifies the next-order tuning; retaining only the leading equivalence would lose the distinction between different critical limits.

Diffusive normalization

Each coordinate of one walk step has variance $1/2$. Consequently, macroscopic space is represented by lattice sites near $\sqrt{N/2}x$. To handle parity, let $[r]=2\lceil r/2\rceil$ for time and let $[x]$ denote a fixed nearest point in

$${\mathbb{Z}}_{\mathrm{even}}^2=\{z\in {\mathbb{Z}}^2:z_1+z_2\in 2\mathbb{Z}\}.$$

Write $x_N=[\sqrt{N/2}x]$ and $y_N=[\sqrt{N/2}y]$. The rescaled kernel is

$$\begin{array}{c}{\mathcal{Z}}_{s,t}^{\vartheta ,(N)}(x,y)\\ =\frac{N}{4}{Z}_{[Ns],[Nt]}^{{\beta }_N}(x_N,y_N).\end{array}$$

The factor $N/4$ is the reciprocal of a rescaled even-lattice cell’s area. With this convention, the limiting mean kernel is

$$g_t(x)=\frac{1}{2\pi t}{e}^{-|x{|}^2/(2t)}.$$

For a positive initial profile $u_0$, define the forward field by

$$\begin{array}{c}{\mathcal{Z}}_t^{\vartheta ,(N)}(u_0,x)\\ ={\int }_{{\mathbb{R}}^2}{u}_0(y){\mathcal{Z}}_{0,t}^{\vartheta ,(N)}(y,x)dy.\end{array}$$

This is the discrete approximation to a stochastic heat evolution from $u_0$.

🏷️ The stochastic heat flow and the logarithm problem

The critical two-dimensional stochastic heat flow, abbreviated SHF, is a family of random locally finite measures

$${\mathcal{Z}}_{s,t}^{\vartheta }(dx,dy),0\le s<t.$$

Its two coordinates describe the initial and terminal positions. Caravenna, Sun, and Zygouras constructed it as the joint scaling limit of the polymer kernels, tested against compactly supported continuous functions (Caravenna, Sun & Zygouras, 2023). Its mean is

$$\mathbb{E}{\mathcal{Z}}_{s,t}^{\vartheta }(dx,dy)=g_{t-s}(y-x)dxdy.$$

For suitable functions $f,g$, write

$$\begin{array}{c}{\mathcal{Z}}_{s,t}^{\vartheta }(f,g)\\ =\int f(x)g(y){\mathcal{Z}}_{s,t}^{\vartheta }(dx,dy).\end{array}$$

The heat evolution from $u_0$ is the measure

$${\mathcal{Z}}_t^{\vartheta }(u_0,dy)=\int u_0(x){\mathcal{Z}}_{0,t}^{\vartheta }(dx,dy).$$

For flat data $u_0=1$, this measure is almost surely singular with respect to Lebesgue measure (Caravenna, Sun & Zygouras, 2025). It has no density to which an ordinary pointwise logarithm could be applied.

Even convergence of strictly positive densities as measures would not justify passing to their logarithms.

Weak convergence does not control logarithms

On the unit square, divide each of $n$ equal vertical strips into two halves. Let a density $f_n$ equal $e^{-n}$ on the first half and $2-e^{-n}$ on the second. Then $f_n(x)dx$ converges weakly to Lebesgue measure, but

$$\begin{array}{rl}& {\int }_{[0,1{]}^2}\log f_n(x)dx\\ & =\frac{1}{2}(-n+\log (2-e^{-n}))\\ & ⟶-\infty .\end{array}$$

A logarithm detects small values that weak convergence of measures does not control.

The new construction must therefore control lower tails and determine which diverging deterministic quantity to subtract.

🏷️ Renormalized Cole–Hopf construction

Take a bounded continuous initial height $h_0:{\mathbb{R}}^2\to \mathbb{R}$, so that $u_0=e^{h_0}$ is bounded above and away from zero. Define

$$\begin{array}{rl}& {\mathcal{H}}_t^{\vartheta ,(N)}(h_0,x)\\ & =\log {\mathcal{Z}}_t^{\vartheta ,(N)}(e^{h_0},x)-c_t^{\vartheta ,(N)},\\ & c_t^{\vartheta ,(N)}=\mathbb{E}\log {\mathcal{Z}}_t^{\vartheta ,(N)}(1,0).\end{array}$$

Translation invariance makes the flat-data mean independent of the spatial point. The same constant is used for every $h_0$, preserving the effect of changing the initial height. The paper proves

$$c_t^{\vartheta ,(N)}=-\frac{1}{2}\log \log N+O_{\vartheta ,t}(1).$$

This asymptotic identifies the divergence. The definition uses the exact mean: replacing it by its leading term alone need not give the same limiting deterministic offset.

For the continuum construction, set $g_{\delta }^{(x)}(y)=g_{\delta }(y-x)$. For $0<\delta <t$, define

$$\begin{array}{rl}& {\mathcal{H}}_t^{\vartheta ,(\delta )}(h_0,x)\\ & =\log {\mathcal{Z}}_{0,t-\delta }^{\vartheta }(e^{h_0},g_{\delta }^{(x)})-c_t^{\vartheta ,(\delta )},\\ & c_t^{\vartheta ,(\delta )}=\mathbb{E}\log {\mathcal{Z}}_{0,t-\delta }^{\vartheta }(1,g_{\delta }^{(0)}).\end{array}$$

This evolves with the SHF until $t-\delta$ and then with the deterministic heat kernel for duration $\delta$. It removes the final short interval of disorder. The paper also proves that directly smoothing ${\mathcal{Z}}_{0,t}^{\vartheta }$ in its terminal spatial variable gives the same centered logarithmic limit.

Random distributions and local negative Sobolev spaces

A distribution $T$ acts continuously and linearly on tests $\phi \in C_c^{\infty }({\mathbb{R}}^2)$; its evaluation is denoted $T(\phi )$. This permits a height field to have well-defined averages without having pointwise values.

For $a>0$, the negative Sobolev norm has the Fourier representation

$$\begin{array}{c}\Vert f{\Vert }_{H^{-a}}^2\\ =\frac{1}{(2\pi {)}^2}{\int }_{{\mathbb{R}}^2}(1+|\xi {|}^2{)}^{-a}|\hat{f}(\xi ){|}^{2}d\xi .\end{array}$$

where $\hat{f}(\xi )=\int e^{-i\xi \cdot x}f(x)dx$. A distribution belongs to $H_{\mathrm{loc}}^{-a}$ when $\chi T\in H^{-a}$ for every smooth compactly supported cutoff $\chi$.

Construction and convergence — Theorem 1.2

Fix $\vartheta \in \mathbb{R}$. There is an SHF-measurable family of random distributions ${\mathcal{H}}_t^{\vartheta }(h_0)$, for positive times and bounded continuous initial heights, such that for every $a>0$,

$$\begin{array}{c}{\mathcal{H}}_t^{\vartheta ,(\delta )}(h_0)\underset{\delta \downarrow 0}{\overset{\mathbb{P}}{\to }}{\mathcal{H}}_t^{\vartheta }(h_0),\\ {\mathcal{H}}_t^{\vartheta ,(N)}(h_0)\underset{N\to \infty }{\overset{\mathrm{law}}{\to }}{\mathcal{H}}_t^{\vartheta }(h_0),\\ \text{in }{H}_{\mathrm{loc}}^{-a}({\mathbb{R}}^2).\end{array}$$

Both convergences hold jointly for finitely many positive times and initial conditions. For each test $\phi$, the continuum convergence of pairings also holds in every $L^p(\mathbb{P})$, $1\le p<\infty$ (Berger et al., 2026).

There is no additional amplitude normalization in this theorem. A fixed macroscopic spatial average has a nontrivial limit after deterministic centering alone. The factor $\sqrt{\log \log (1/\epsilon )}$ appears later, when the averaging scale itself tends to zero.

Dependence on the initial height

Positivity of the heat kernels immediately gives a useful comparison. If $M=\Vert h_0-h_0^{\prime }{\Vert }_{\infty }$, then

$$e^{-M}{e}^{h_0^{\prime }}\le e^{h_0}\le e^M{e}^{h_0^{\prime }}.$$

Integrate against the positive kernel, take logarithms, and subtract the common flat-data centering. Passing to the limit yields, for fixed data and test $\psi$, almost surely,

$$\begin{array}{c}\left|{\mathcal{H}}_t^{\vartheta }(h_0,\psi )-{\mathcal{H}}_t^{\vartheta }(h_0^{\prime },\psi )\right|\\ \le \Vert \psi {\Vert }_1\Vert h_0-h_0^{\prime }{\Vert }_{\infty }.\end{array}$$

The paper highlights this comparison for bounded Lipschitz initial data. Adding a constant $b$ to the initial height likewise adds $b\int \psi$ to the tested field, because the partition function is multiplied by $e^b$.

🏷️ The convergence argument

The proof is easiest to describe at time $1$, with flat data and a fixed test $\phi$. It uses the time-reversed, point-to-plane formulation. Reversing the lattice model changes its endpoint noise convention; the resulting vanishing correction is accounted for when returning to the forward statement.

Write $X_N$ for the centered tested log-polymer field. Let $X_{N,\delta }$ be its version with disorder removed up to the even time $m=[\delta N]$, and let $X_{\delta }$ be the corresponding tested logarithm of the SHF.

Two estimates carry the construction:

$$\begin{array}{c}{X}_{N,\delta }\underset{N\to \infty }{\overset{\mathrm{law}}{\to }}{X}_{\delta },\\ \text{for each fixed }\delta >0.\end{array}$$

and

$$\begin{array}{rl}{ϵ}_{\delta }& :=\underset{N\to \infty }{\mathrm{limsup}}\Vert X_N-X_{N,\delta }{\Vert }_2\\ & \le C_{\phi }|\log \delta {|}^{-1/40}⟶0.\end{array}$$

The first estimate uses SHF convergence together with control of the logarithm near zero. The second is the main new approximation theorem.

flowchart TD
  A["Gaussian convexity<br/>and block factorization"]
  B["Moments, covariances,<br/>and endpoint estimates"]
  C["Two-time decomposition<br/>at m and n"]
  D["Uniform L²<br/>smoothing estimate"]
  E["Cauchy limit of<br/>centered log SHF"]
  F["Convergence in local<br/>negative Sobolev spaces"]
  A --> B --> C --> D --> E --> F

The fixed-cutoff convergence is joint in different cutoffs. Hence Portmanteau’s inequality, applied to the squared difference, and the triangle inequality give

$$\Vert X_{\delta }-X_{{\delta }^{\prime }}{\Vert }_2\le {ϵ}_{\delta }+{ϵ}_{{\delta }^{\prime }}.$$

Thus $X_{\delta }$ is Cauchy in $L^2$ and defines a limit $X$. This simultaneously constructs the continuum logarithm and identifies the discrete limit.

For completeness, let $d_{\mathrm{BL}}$ be the bounded-Lipschitz distance between probability laws. Since $d_{\mathrm{BL}}(U,V)\le \mathbb{E}|U-V|$ for variables on the same probability space,

$$\begin{array}{rl}& d_{\mathrm{BL}}(X_N,X)\\ & \le \Vert X_N-X_{N,\delta }{\Vert }_2\\ & +d_{\mathrm{BL}}(X_{N,\delta },X_{\delta })\\ & +\Vert X_{\delta }-X{\Vert }_2.\end{array}$$

First send $N\to \infty$ at fixed $\delta$, then send $\delta \downarrow 0$. The order of these limits is essential; the uniform smoothing estimate permits the second passage.

🏷️ Gaussian convexity and lower-tail control

The logarithm of a finite polymer partition function is a convex function of its Gaussian environment: it is a logarithm of a positive sum of exponentials of affine functions. It is also increasing in each environment coordinate.

Chen’s Gaussian convexity theorem supplies a stronger property than ordinary convexity of a log moment-generating function. If $F$ is a finite convex function of a finite-dimensional Gaussian vector and $\mathbb{E}{e}^{qF}<\infty$ for every $q>0$, then

$${\Phi }_F(q)=\{\begin{array}{ll}{q}^{-1}\log \mathbb{E}{e}^{qF},& q\ne 0,\\ \mathbb{E}F,& q=0\end{array}$$

is convex on the entire real line (Chen, 2026). Liu and Xie developed its moment-comparison consequences for polymers and the SHF (Liu & Xie, 2026).

Normalize $F$ so that $\mathbb{E}{e}^F=1$. Then

$${\Phi }_F(1)=0,{\Phi }_F^{\prime }(0)=\frac{1}{2}\mathrm{Var}(F).$$

The supporting-line inequality at $q=0$, evaluated at $q=1$, gives

$$\mathrm{Var}(F)\le -2\mathbb{E}F.$$

The midpoint inequality at $0,1,2$ gives

$$-\mathbb{E}F\le \frac{1}{2}\log \mathbb{E}{e}^{2F}.$$

At $q=-p$, multiplication of the supporting-line inequality by the negative number $-p$ reverses its direction. For $p\ge 0$ this yields

$$\begin{array}{rl}\log \mathbb{E}{e}^{-pF}& \le -p\mathbb{E}F+\frac{p^2}{2}\mathrm{Var}(F)\\ & \le \frac{p(p+1)}{2}\log \mathbb{E}{e}^{2F}.\end{array}$$

Consequently, a mean-one partition function $Z=e^F$ satisfies

$$\boxed{\mathbb{E}{Z}^{-p}\le (\mathbb{E}{Z}^2{)}^{p(p+1)/2}.}$$

This converts second-moment information into control of arbitrarily small values of $Z$. Such control is exactly what is needed to pass to $\log Z$.

Covariance comparison and the Gaussianity defect

Define

$$\Delta (F)=-\mathbb{E}F-\frac{1}{2}\mathrm{Var}(F)\ge 0.$$

It vanishes for a Gaussian $F$ normalized by $\mathbb{E}{e}^F=1$. The paper proves

$$\Delta (F)\le \log \frac{\mathbb{E}{e}^{2F}}{\mathbb{E}{e}^{-F}}.$$

For two normalized convex functions $F_x,F_y$ of the same Gaussian vector,

$$\begin{array}{rl}\mathrm{Cov}(F_x,F_y)& \le \log \mathbb{E}{e}^{F_x+F_y}\\ & +\Delta (F_x)+\Delta (F_y).\end{array}$$

To see the mechanism, apply $\mathrm{Var}(H)\le -2\mathbb{E}H$ to

$$H=F_x+F_y-\log \mathbb{E}{e}^{F_x+F_y},$$

and expand its variance. For increasing functions of independent Gaussian coordinates, positive association also gives a nonnegative covariance. Bounds on ordinary polymer moments and on $\Delta$ therefore become bounds on the covariance of logarithms.

🏷️ Factorization at two intermediate times

Let $Z_N(z)=Z_{0,N}^{{\beta }_N}(z)$ and let $q_m^{(z)}$ be the law of $S_m$ started from $z$. Removing disorder from $(0,m]$ gives

$${\hat{Z}}_{m,N}(z)=Z_{m,N}(q_m^{(z)}),$$

where $Z_{a,b}(\nu )={\sum }_w\nu (w)Z_{a,b}(w)$ for a starting distribution $\nu$.

A decomposition directly at $m$ leaves two very different endpoint distributions. The proof allows them to evolve through a common environment until a later time

$$m=[\delta N],n=[\sqrt{\delta }N].$$

As $N\to \infty$ and then $\delta \downarrow 0$, the ratios $n/m$ and $N/n$ both diverge. Define the endpoint probabilities

$$\begin{array}{rl}{\mu }_n^{(z)}(w)& =\frac{Z_{0,n}(z,w)}{Z_n(z)},\\ {\hat{\mu }}_{m,n}^{(z)}(w)& =\frac{Z_{m,n}(q_m^{(z)},w)}{{\hat{Z}}_{m,n}(z)}.\end{array}$$

The Markov property gives the exact identities

$$\begin{array}{rl}{Z}_N(z)& =Z_n(z)Z_{n,N}({\mu }_n^{(z)}),\\ {\hat{Z}}_{m,N}(z)& ={\hat{Z}}_{m,n}(z)Z_{n,N}({\hat{\mu }}_{m,n}^{(z)}).\end{array}$$

Taking their logarithmic ratio separates two errors:

$$\begin{array}{c}\log \frac{Z_N(z)}{{\hat{Z}}_{m,N}(z)}\\ =L_{m,n}(z)+G_{m,n,N}(z).\end{array}$$

where

$$\begin{array}{rl}{L}_{m,n}(z)& =\log \frac{Z_n(z)}{{\hat{Z}}_{m,n}(z)},\\ G_{m,n,N}(z)& =\log \frac{Z_{n,N}({\mu }_n^{(z)})}{Z_{n,N}({\hat{\mu }}_{m,n}^{(z)})}.\end{array}$$

The local term depends only on disorder up to $n$. Its correlations occur on a spatial scale comparable to $\sqrt{n}$, much shorter than the macroscopic averaging scale $\sqrt{N}$. Its individual logarithms become small after centering and averaging, despite their large pointwise fluctuations.

The global term compares the future masses produced by two profiles that have already evolved through the same noise on $(m,n]$. Their remaining discrepancy is small in the estimates appropriate to the future evolution. Negative moments allow this comparison of masses to survive taking logarithms.

Let ${\mathcal{L}}_{N,\delta }(\phi )$ and ${\mathcal{G}}_{N,\delta }(\phi )$ denote these terms after centering and spatial averaging with the same cell weights as $X_N$. The paper obtains

$$\begin{array}{c}\underset{N\to \infty }{\mathrm{limsup}}\mathbb{E}|{\mathcal{L}}_{N,\delta }(\phi ){|}^2\le \frac{C_{\phi }}{|\log \delta {|}^{1/4}},\\ \underset{N\to \infty }{\mathrm{limsup}}\mathbb{E}|{\mathcal{G}}_{N,\delta }(\phi ){|}^2\le \frac{C_{\phi }}{|\log \delta {|}^{1/20}}.\end{array}$$

Taking square roots and applying the triangle inequality gives the exponent $1/40$ in the smoothing estimate. This is an explicit sufficient rate for convergence, not a claimed optimal rate.

Geometric blocks and cancellation

The profile estimates adapt Gu and Tsai’s geometric factorization (Gu & Tsai, 2026). Fix a sufficiently large ratio $\rho >1$ and divide $(0,N]$ at times

$$\begin{array}{c}K=\lfloor {\log }_{\rho }N\rfloor +1,\\ N_0=0,N_K=N,\\ N_i=[N{\rho }^{i-K}],1\le i<K.\end{array}$$

Write $q_k=q_k^{(0)}$ for the walk’s distribution at time $k$ when started at the origin. The random kernels ${\mathsf{Z}}_i=Z_{N_{i-1},N_i}$ use disjoint pieces of the environment. Replace each kernel by one that preserves its total mass and resets its endpoint to the deterministic heat profile:

$$\begin{array}{rl}{\mathsf{Z}}_i^{\mathrm{rew}}(x,y)& ={\mathsf{Z}}_i(x,1)q_{N_i}(y),\\ D_i(x,y)& ={\mathsf{Z}}_i(x,y)-{\mathsf{Z}}_i^{\mathrm{rew}}(x,y).\end{array}$$

The discrepancy has the exact cancellation

$$\sum _y{D}_i(x,y)=0.$$

Expanding products of the kernels into reset terms and discrepancy terms isolates strings of consecutive discrepancies. Their estimates decay geometrically in string length. This makes the product of independent block masses a controlled approximation to the full partition function.

There is also a quantitative exceptional-event estimate. Put $B_i={\mathsf{Z}}_i(q_{N_{i-1}},1)$ and let ${\mathcal{R}}_{r,s}$ be the ratio of the full mass on blocks $r,\dots ,s$, started from $q_{N_{r-1}}$, to ${\prod }_{i=r}^s{B}_i$. On

$${\Omega }_{r,s}=\bigcap _{i=r}^s\{|B_i-1|\le 1/2\},$$

the estimates, for $r\le s\le K-1$, are

$$\begin{array}{rl}\mathbb{P}({\Omega }_{r,s}^c)& \le \frac{C}{K-s},\\ \mathbb{E}[1_{{\Omega }_{r,s}}|{\mathcal{R}}_{r,s}-1{|}^2]& \le \frac{C}{K-s}.\end{array}$$

To handle reciprocal masses, the proof also changes measure by a density proportional to ${\prod }_{i=r}^s{B}_i^{-1}$. Under that probability law, both bounds have order $(K-s{)}^{-1/2}$. This weighted estimate is a key addition: ordinary second-moment factorization alone does not control rare small denominators.

🏷️ From tested limits to spatial distributions

Convergence for each test does not by itself establish convergence in $H_{\mathrm{loc}}^{-a}$. The paper obtains tightness from uniform spatial covariance bounds of the form

$$\begin{array}{c}0\le \mathrm{Cov}(H_{\delta }(x),H_{{\delta }^{\prime }}(y))\\ \le C[1+\log (1+{\log }^{+}|x-y{|}^{-1})].\end{array}$$

for the flat-data regularizations at fixed time $t>0$, with $0<\delta ,{\delta }^{\prime }<t/2$ and $x\ne y$. Here ${\log }^{+}r=\max (\log r,0)$.

The Fourier multiplier in the Sobolev norm has the integral representation

$$\begin{array}{c}(1+|\xi {|}^2{)}^{-a}\\ =\frac{1}{\Gamma (a)}{\int }_0^{\infty }{u}^{a-1}{e}^{-u}{e}^{-u|\xi {|}^2}du.\end{array}$$

Fourier inversion converts the expected squared localized norm into a heat-kernel integral of the covariance. For $0<a<1$, the resulting Bessel kernel behaves near zero like $|x-y{|}^{2a-2}$. The radial singularity is integrable because

$$\begin{array}{c}{\int }_0^{1/2}{r}^{2a-1}\left(1+\log \log \frac{e}{r}\right)dr\\ <\infty ,a>0.\end{array}$$

This gives uniform expected $H^{-a}$ bounds for every positive $a$; larger orders also follow by weakening the norm. To obtain compactness in $H_{\mathrm{loc}}^{-a}$, use a bound at the stronger order $-a/2$, localize, and apply the compact Sobolev embedding. The already constructed test-function limits identify every subsequential limit. The initial-data comparison transfers the argument from flat heights to bounded initial heights.

🏷️ Local fluctuations and the log-log covariance

For the rest of the spatial results, fix $t>0$, $\vartheta \in \mathbb{R}$, and flat initial data, and abbreviate ${\mathcal{H}}_t^{\vartheta }(0,\cdot )$ to ${\mathcal{H}}_t^{\vartheta }$.

For a test $\phi$ and scale $\epsilon >0$, define

$${\phi }_{\epsilon }^{(x)}(y)={\epsilon }^{-2}\phi \left(\frac{y-x}{\epsilon }\right).$$

Its integral equals $\int \phi$, independently of scale.

Local structure — Theorem 1.4

For every fixed $x$ and nonnegative $\phi \in C_c^{\infty }({\mathbb{R}}^2)$,

$$\begin{array}{c}\frac{{\mathcal{H}}_t^{\vartheta }({\phi }_{\epsilon }^{(x)})}{\sqrt{\log \log (1/\epsilon )}}\\ \underset{\epsilon \downarrow 0}{\overset{\mathrm{law}}{\to }}\mathcal{N}\left(0,{\left(\int \phi \right)}^2\right).\end{array}$$

There is a nonnegative radial covariance kernel $C_t^{\vartheta }\in L_{\mathrm{loc}}^2({\mathbb{R}}^2)$, continuous away from zero. Writing $H={\mathcal{H}}_t^{\vartheta }$ and $C=C_t^{\vartheta }$, for real smooth compactly supported $f,g$ we have

$$\begin{array}{c}\mathrm{Cov}(H(f),H(g))\\ =\iint f(x)g(y)C(x-y)dxdy,\\ \underset{|z|\downarrow 0}{\lim }\frac{C(z)}{\log \log (1/|z|)}=1.\end{array}$$

Matching temporal and spatial scales

The local central limit theorem uses the heat scale $\delta ={\epsilon }^2$. In the reversed time-one notation, let

$$F_{\delta }(x)=\log {\mathcal{Z}}_{\delta ,1}^{\vartheta }(g_{\delta }^{(x)},1).$$

Gu–Tsai log-normal fluctuations, with the centering adjusted to the expectation, give

$$\frac{F_{{\epsilon }^2}(x)-\mathbb{E}{F}_{{\epsilon }^2}(x)}{\sqrt{\log \log (1/\epsilon )}}⟹\mathcal{N}(0,1).$$

For a nonnegative test of integral one and barycenter zero, the KPZ paper proves the scale-matched comparison

$$\begin{array}{c}{\Vert \mathcal{H}({\phi }_{\epsilon })-(F_{{\epsilon }^2}(0)-\mathbb{E}{F}_{{\epsilon }^2}(0))\Vert }_{L^1}\\ \le C_{\phi }.\end{array}$$

The bounded error disappears after division by $\sqrt{\log \log (1/\epsilon )}$. Translation and normalization recover the general statement. This is stronger information than the fixed-test smoothing bound, whose constant can depend on the test being rescaled.

For a probability-density test, the variance also satisfies

$$\mathrm{Var}({\mathcal{H}}_t^{\vartheta }({\phi }_{\epsilon }^{(x)}))\sim \log \log (1/\epsilon ).$$

The covariance singularity is much weaker than the logarithmic singularity of a planar Gaussian free field. Nevertheless, the height is too singular to be a locally finite signed measure.

Proof mechanism for the absence of a signed-measure representative

Take a nonnegative smooth test supported in the unit ball with integral one. The local central limit theorem implies, for each fixed $k$, that

$$\mathbb{P}(|\mathcal{H}({\phi }_{\epsilon }^{(x)})|\le k)⟶0.$$

Spatial stationarity lets us choose a single sequence ${\epsilon }_k\downarrow 0$ for which these probabilities are at most $2^{-k}$ for every $x$. Fubini and summability imply that, almost surely, $|\mathcal{H}({\phi }_{{\epsilon }_k}^{(x)})|>k$ eventually for Lebesgue-almost every $x$.

If the field agreed with a locally finite signed measure $\mu$ on an open ball, then on a smaller ball

$$\begin{array}{c}|\mathcal{H}({\phi }_{\epsilon }^{(x)})|\\ \le \Vert \phi {\Vert }_{\infty }{\epsilon }^{-2}|\mu |(B(x,\epsilon )).\end{array}$$

The Lebesgue differentiation theorem makes the right-hand side bounded as $\epsilon \downarrow 0$ for almost every $x$. This contradicts the preceding divergence. A countable exhaustion handles the possible location of the ball. In particular, the field has no locally integrable representative.

🏷️ Non-Gaussianity through rigidity

The local central limit theorem concerns a limit of rescaled observables. At each fixed nonzero averaging scale, a stronger structural theorem holds.

Non-Gaussianity — Theorem 1.6

For every nonzero nonnegative $\psi \in C_c^{\infty }({\mathbb{R}}^2)$, the random variable ${\mathcal{H}}_t^{\vartheta }(\psi )$ has a non-Gaussian law. Constant Gaussian laws are excluded as well (Berger et al., 2026).

The proof uses a class of laws that retains convex Gaussian geometry after passage to a limit. A real random variable has Ehrhard type if it has the same law as

$$h(G),G\sim \mathcal{N}(0,1),$$

for a finite nondecreasing convex function $h:\mathbb{R}\to \mathbb{R}$. Constant functions are allowed. Convex functions of finite-dimensional Gaussian vectors have Ehrhard-type laws, and this class is closed under weak convergence. In particular, nonnegative tested averages of the limiting logarithmic field have this property.

The decisive rigidity statement is:

Gaussian rigidity under bounded perturbations

Suppose $X,Y$ are centered, $X$ is Gaussian, and $|Y-X|\le K$ almost surely for a deterministic finite $K$. If both $Y$ and $X+Y$ have Ehrhard-type laws, then $Y=X$ almost surely.

The condition on $X+Y$ retains information about the joint law. Separate knowledge of the two marginal distributions would not suffice.

To apply rigidity, normalize $\int \psi =1$ and work in the reversed formulation. For spatial smoothing radius $r$, define the continuum polymer endpoint probability

$${\mu }_{t,r}^x(dy)=\frac{{\mathcal{Z}}_{0,t}^{\vartheta }(g_r^{(x)},dy)}{{\mathcal{Z}}_{0,t}^{\vartheta }(g_r^{(x)},1)}.$$

Changing the terminal weight from $1$ to a bounded Lipschitz function $f$ satisfying $0<c\le f\le C$ changes the logarithm by

$$\begin{array}{c}\log \frac{{\mathcal{Z}}_{0,t}^{\vartheta }(g_r^{(x)},f)}{{\mathcal{Z}}_{0,t}^{\vartheta }(g_r^{(x)},1)}=\log {\mu }_{t,r}^x(f).\end{array}$$

This difference lies in $[\log c,\log C]$. Center each tested observable by its own expectation for this argument. Their difference is then bounded by $\log (C/c)$, and their joint limits have the Ehrhard properties required by the rigidity lemma.

If the flat observable were Gaussian, rigidity would force every such centered perturbation to vanish in the limit. Choosing $f=1+\eta v$, then controlling the first-order expansion in $\eta$, implies that the averaged endpoint measures

$${\mu }_{t,r}^{\psi }(dy)=\int \psi (x){\mu }_{t,r}^x(dy)dx$$

have deterministic subsequential limits. A separate estimate ensures these limits are nonzero; this cannot be inferred from vague convergence of probability measures alone, because mass could escape to infinity.

The contradiction comes from the singularity of the SHF. The paper proves that such a nonzero deterministic limit ${\mu }_t$ would overlap, with positive probability, with the later heat-flow measure

$$M(dy)={\mathcal{Z}}_{2t/3,t}^{\vartheta }(1,dy).$$

Yet $M$ is almost surely mutually singular with every deterministic finite positive measure. Indeed, write a deterministic $\mu$ as its absolutely continuous and singular parts. Almost-sure singularity of $M$ separates it from the first part. The second part is supported on a deterministic Lebesgue-null set $A$, and $\mathbb{E}M(A)=|A|=0$ forces $M(A)=0$ almost surely. These two separations imply $M\perp \mu$.

The substantial comparison lemma leading to the overlap uses the flow over successive time intervals and integrates out an intervening interval of disorder. It is the probabilistic ingredient that turns scalar Gaussian rigidity into a contradiction with spatial singularity.

🏷️ Spatial symmetry and scaling covariance

For flat data, the process is invariant in law under translations and rotations. Its scaling relation also changes the critical parameter. For $a>0$, put

$$(S_a\phi )(x)=a^{-1}\phi (x/\sqrt{a}).$$

Then, jointly for finitely many times and tests,

$$({\mathcal{H}}_{a{t}_i}^{\vartheta }(S_a{\phi }_i){)}_i\stackrel{\mathrm{law}}{=}({\mathcal{H}}_{t_i}^{\vartheta +\log a}({\phi }_i){)}_i.$$

The factor $a^{-1}$ preserves the integral of a test in two dimensions. The shift $\vartheta \mapsto \vartheta +\log a$ is the remaining effect of the logarithmic critical tuning. Thus the scaling law relates different members of the critical family.

🏷️ The Lean formalization and Palomar record

Scott Armstrong’s Lean repository at the registered revision contains formal statements for the construction, its principal properties, selected supporting results, and four extensions (Armstrong, 2026). Its statement catalogue has 26 entries: 22 covering the paper and inputs, and four for Armstrong’s companion note.

The supplied Palomar entry, version 1, registers the theorem Critical2DKPZAudit.criticalKPZ_main at commit f1c98dacb2a83737096bd976019dfa2c5986ee5a. It uses Lean 4.35.0-rc2. The recorded verification report reports acceptance by Lean’s default kernel and NanoDa, together with an accepted con-ron replay. The permitted axioms are the standard propext, Classical.choice, and Quot.sound.

The formal statement’s mathematical content

The standalone Challenge file imports only Mathlib and defines the walk, Gaussian environment, critical tuning, partition functions, random distributions, and local Sobolev energies. Its theorem takes an SHF satisfying the predicate IsCriticalSHF and constructs the KPZ field with the stated convergence properties.

Mathematical object or assertion	Formal counterpart
Gaussian product environment and critical lattice model	disorderLaw, criticalBeta, pointToPoint
Joint polymer limit defining the SHF	IsCriticalSHF
Existence of a flow satisfying that predicate	exists_isCriticalSHF, separately proved in the library
Centered logarithmic approximations	kpzRegularized, kpzLattice
Finite localized negative Sobolev energy	MemLocalNegSobolev
SHF-measurable KPZ construction and convergence	criticalKPZ_main

The SHF hypothesis is a specified joint scaling-limit property, rather than an assumption that the desired KPZ conclusions already hold. The separate existence theorem establishes that this class of flows is nonempty. That existence declaration is outside the theorem selection in this particular Palomar entry.

The Challenge’s final proof is a placeholder used to freeze the statement. The Solution file supplies the proof from the development; the verification applies to that solution.

Scope of the registered comparison

Version 1 selects one theorem and has an empty definition_names list. Its editorial review explicitly notes that the README’s claim about separately comparing all 40 definitions is broader than this configuration. The repository contains Lean-checked definition-equality lemmas, but these are not separately selected for comparator checking or independent replay in this entry. The 26-result catalogue and the four additional results therefore describe the wider development, not 26 independently registered conclusions of this one record.

Conventions and correspondence

The errata and deviations document makes several mathematically useful distinctions explicit.

The exported lattice statements fix $\vartheta$; they do not include the paper’s extension to all deterministic sequences ${\vartheta }_N\to \vartheta$. The local Sobolev carrier is implemented as the full distribution space with a topology generated by extended Sobolev energies. Reading its convergence as convergence in $H_{\mathrm{loc}}^{-a}$ uses the finite-energy support results and the identification of the restricted topology.

The terminal-noise convention also produces an endpoint correction under lattice time reversal. The formalization bounds the discrepancy between the two flat logarithmic means by ${\beta }_N^2/2$, which vanishes and preserves the $-\frac{1}{2}\log \log N+O(1)$ asymptotic. This illustrates why the endpoint convention matters even when it does not alter the limiting theorem.

Kernel checking establishes the formal implications. Relating the precise SHF predicate to the earlier literature, and reading the chosen topology and null-set conventions against the manuscript, remain mathematical correspondence questions. The repository documents these choices and states that the paper’s authors have not reviewed the formalization.

🏷️ Four additional results in the formal development

Armstrong’s companion note, Four results on the critical 2D KPZ process, goes beyond the original paper (Armstrong, 2026a). The following statements fix $\vartheta$ and a positive physical time $t$; their scope should be distinguished from the original construction and from the single theorem selected in the Palomar entry. The note gives complete arguments for the first two results and combines proofs with sketches for the last two, referring to the Lean development for the full formal proofs.

Strict positive skewness

For nonzero nonnegative smooth compactly supported $\psi$, the flat-data variable $X={\mathcal{H}}_t^{\vartheta }(\psi )$ has a finite third absolute moment and

$$\mathbb{E}[(X-\mathbb{E}X{)}^3]>0.$$

This strengthens non-Gaussianity by identifying a strictly positive centered moment. The distinction matters because a non-Gaussian distribution can have zero third centered moment.

Twice differentiable response to initial data

For bounded continuous $h,v$ and a real smooth compactly supported test $\psi$, the response

$$\begin{array}{rl}R(q)& ={\mathcal{H}}_t^{\vartheta }(h+qv,\psi )\\ & -{\mathcal{H}}_t^{\vartheta }(h,\psi ).\end{array}$$

has a $C^2(\mathbb{R})$ version, unique up to almost-sure equality. With $A=\Vert \psi {\Vert }_1$ and $M=\Vert v{\Vert }_{\infty }$,

$$\begin{array}{c}|R(q)|\le AM|q|,\\ |R^{\prime }(q)|\le AM,\\ |R^{\prime \prime }(q)|\le A{M}^2.\end{array}$$

It is convex when $\psi \ge 0$. For this theorem, the approximating responses retain the disorder over the full interval $[0,t]$ and smooth the terminal position. At a positive smoothing scale, the first derivative of the logarithm is the polymer expectation of $v$, and its second derivative is the variance of $v$. The theorem justifies the limiting response and the convergence of the first two derivatives in every finite $L^p$, uniformly on compact $q$ intervals.

Brownian motion across spatial scales

Fix $x$, a nonnegative smooth compactly supported $\phi$ with $\int \phi =1$, and $S>0$. Set

$$\begin{array}{c}{\epsilon }_L(s)=\exp (-e^{Ls}),\\ X_L(s)=\frac{1}{\sqrt{L}}{\mathcal{H}}_t^{\vartheta }({\phi }_{{\epsilon }_L(s)}^{(x)}).\end{array}$$

As real $L\to \infty$, the continuous paths $X_L$ converge in law in $C([0,S])$, with the supremum norm, to standard Brownian motion. The clock is natural because

$$\log \log \frac{1}{{\epsilon }_L(s)}=Ls.$$

The Brownian parameter is the logarithm of a logarithmic spatial scale. Physical time $t$ remains fixed.

A Gaussian free field tangent on zero-integral tests

For finitely many real smooth compactly supported $f_i$ with $\int f_i=0$,

$$\begin{array}{c}{\left(\sqrt{\log (1/\epsilon )}{\mathcal{H}}_t^{\vartheta }(f_{i,\epsilon }^{(x)})\right)}_i\\ ⟹\mathcal{N}(0,Q).\end{array}$$

where $Q_{ij}$ equals

$$-\iint f_i(u)f_j(v)\log |u-v|dudv.$$

This is the whole-plane Gaussian free field covariance on zero-integral tests, with the normalization displayed above. Degenerate Gaussian vectors are allowed. The assertion is finite-dimensional convergence of these pairings, rather than convergence of an entire rescaled distribution in a function-space topology.

The different normalizations reflect cancellation. A nonzero-integral test sees the common fluctuation of size $\sqrt{\log \log (1/\epsilon )}$. A zero-integral test removes that contribution and detects a much smaller fluctuation, which must be multiplied by $\sqrt{\log (1/\epsilon )}$. Proving this finer limit requires more than the leading covariance equivalence $C(z)\sim \log \log (1/|z|)$.

🏷️ Scope and further questions

The construction is a precise notion of critical KPZ based on Gaussian polymers and the SHF. Its convergence is joint at finitely many positive physical times. It does not establish convergence in a path space over physical time, almost-sure temporal continuity, or a uniqueness theorem for every possible interpretation of the formal SPDE.

The paper also discusses the KPZ equation driven by spatially mollified white noise. It expects the corresponding centered logarithms to converge to the same process, but omits the required adaptations. This differs from spatially smoothing the already constructed SHF: the latter convergence is proved in Corollary 6.2. Extension to more general disorder laws, walks, and unbounded initial data requires additional results.

Appendix C records how the authors developed the analytic proof through interactions with generative AI. It attributes contributions to the tightness, convergence, and non-Gaussianity arguments, and describes an abandoned approach whose contraction argument failed an audit. The authors supplied mathematical inputs, directed revisions, and take responsibility for the manuscript. Armstrong’s subsequent Lean development is a separate contribution with its own statements, proofs, and correspondence documentation.

Links

• on Slepian’s lemma and Gaussian comparison — Gaussian comparison supplies useful background for covariance arguments; the central input here is instead convexity of normalized logarithmic moment-generating functions.
• on Liouville quantum gravity and KPZ — Gaussian free fields and renormalized random measures provide a comparison with the SHF and its GFF tangent. The KPZ relation in that title is the Knizhnik–Polyakov–Zamolodchikov relation, a different use of the acronym from the growth equation here.
• on limit laws for two-dimensional torus cover times — another setting where two-dimensional recurrence and singular random measures determine limiting observables, with a different underlying stochastic mechanism.

📚 References

🐻  Armstrong, S. 2026a. Four results on the critical 2D KPZ process.

🐻  Armstrong, S. 2026b. The Critical 2D KPZ Process: a Lean formalization.

🐻  Berger, Q., Caravenna, F., Clark, J., Dunlap, A., Gu, Y., Liu, Z., Nakajima, S., Sun, R., Tsai, L.-C., Xie, Z. & Zygouras, N. 2026. The Critical 2D KPZ Process.

🐻  Caravenna, F., Sun, R. & Zygouras, N. 2023. The critical 2d stochastic heat flow. Inventiones Mathematicae 233(1), 325–460.

🐻  Caravenna, F., Sun, R. & Zygouras, N. 2025. Singularity and regularity of the critical 2D Stochastic Heat Flow.

🐻  Chen, W.-K. 2026. A Gaussian convexity for logarithmic moment generating functions with applications in spin glasses. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 62(1), 195–206.

🐻  Gu, Y. & Tsai, L.-C. 2026. Log Log Fluctuations of the Stochastic Heat Flow.

🐻  Liu, Z. & Xie, Z. 2026. Moments Comparison Inequalities for Critical 2d Stochastic Heat Flow, Polymers and GMC.