Critical 2D Stochastic Heat Flow

• Critical 2D Stochastic Heat Flow is a universal measure-valued flow defined at the critical scaling limit for directed polymers and the mollified stochastic heat equation.
• It exhibits precisely characterized moment identities, enhanced noise sensitivity, and double-exponential moment growth that set it apart from Gaussian fields.
• The framework integrates both discrete polymer constructions and continuum SPDE formulations, revealing unique spatial singularity, regularity, and martingale properties.

The critical $2d$ Stochastic Heat Flow (SHF) is a universal, measure-valued stochastic flow that arises in two spatial dimensions at the critical disorder scaling for directed polymers and for the stochastic heat equation with spatially mollified multiplicative noise. In the notation used in the recent literature, it is a process of random measures

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$

defined on $\mathbb{R}^2\times\mathbb{R}^2$, where $\theta\in\mathbb{R}$ indexes the critical window. It is not the solution to a naive white-noise-driven SPDE of the form $\partial_t u=\frac12\Delta u+u\xi$: enhanced noise sensitivity and black-noise results show that, in the scaling limit, the SHF is independent of the white noise arising from the original disorder (Caravenna et al., 14 Jul 2025, Gu et al., 19 Jun 2025).

1. Construction from critical directed polymers and mollified SHE

A standard discrete origin of the SHF is the $2d$ directed polymer in i.i.d. disorder. The environment is $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$ with

$$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$

and the underlying path is a symmetric simple random walk $S=(S_n)_{n\in\mathbb{N}}$ on $\mathbb{Z}^2$. The polymer measure is defined by

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$0

so the subtraction of $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$1 is the discrete renormalization. In dimension two, the expected replica overlap

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$2

produces the critical window

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$3

At this scaling, the point-to-point partition functions are degenerate, and the relevant observables are spatially averaged partition functions. For $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$4 and $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$5,

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$6

and the critical convergence theorem states that

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$7

jointly over times and test functions, in the sense of finite-dimensional distributions (Caravenna et al., 14 Jul 2025).

An equivalent continuum construction starts from the $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$8 stochastic heat equation with spatially mollified spacetime white noise. In that setting, the prelimit fundamental solution $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$9 solves

$\mathbb{R}^2\times\mathbb{R}^2$0

with a critically tuned coupling $\mathbb{R}^2\times\mathbb{R}^2$1. In the axiomatic formulation, $\mathbb{R}^2\times\mathbb{R}^2$2 converges in law to $\mathbb{R}^2\times\mathbb{R}^2$3 in the path space of continuous $\mathbb{R}^2\times\mathbb{R}^2$4-valued flows, providing existence from the continuum side as well (Tsai, 2024).

2. Flow structure, axioms, and moment characterization

The SHF is formulated as a stochastic flow of positive kernels. In the axiomatic approach, $\mathbb{R}^2\times\mathbb{R}^2$5 is an $\mathbb{R}^2\times\mathbb{R}^2$6-valued continuous process on $\mathbb{R}^2\times\mathbb{R}^2$7 satisfying four defining properties: continuity, an approximate Chapman–Kolmogorov property via mollified composition, independent increments on disjoint time intervals, and exact moment identities up to order four in terms of the $\mathbb{R}^2\times\mathbb{R}^2$8 delta–Bose gas semigroup $\mathbb{R}^2\times\mathbb{R}^2$9 (Tsai, 2024).

The first moment is purely diffusive: $\theta\in\mathbb{R}$0 with $\theta\in\mathbb{R}$1 the $\theta\in\mathbb{R}$2 heat kernel. Higher moments are encoded by the semigroups $\theta\in\mathbb{R}$3. For $\theta\in\mathbb{R}$4,

$\theta\in\mathbb{R}$5

This finite collection of moment identities, together with the flow axioms, determines the law uniquely (Tsai, 2024).

A complementary description emphasizes the decomposition

$\theta\in\mathbb{R}$6

where $\theta\in\mathbb{R}$7 is the interaction correction coming from the $\theta\in\mathbb{R}$8 delta-Bose semigroup. Its kernel involves a function $\theta\in\mathbb{R}$9 satisfying

$\partial_t u=\frac12\Delta u+u\xi$0

which captures the logarithmic attenuation characteristic of criticality in two dimensions (Gu et al., 19 Jun 2025).

This framework clarifies why the name stochastic heat flow is used. The object is not merely a random field at fixed time: it is a measure-valued flow with Chapman–Kolmogorov composition, independent increments, and semigroup-controlled moments.

3. Noise sensitivity, black noise, and martingale structure

A major development is the identification of the SHF as asymptotically independent of the original disorder’s white noise. The key input is an extension of Benjamini–Kalai–Schramm noise sensitivity from Boolean inputs to functions of general independent random variables. For a function $\partial_t u=\frac12\Delta u+u\xi$1 of i.i.d. inputs, the relevant quantities are the probabilistic gradients

$\partial_t u=\frac12\Delta u+u\xi$2

the $\partial_t u=\frac12\Delta u+u\xi$3-influences

$\partial_t u=\frac12\Delta u+u\xi$4

and the total influence

$\partial_t u=\frac12\Delta u+u\xi$5

Applied to the polymer observables $\partial_t u=\frac12\Delta u+u\xi$6, one obtains

$\partial_t u=\frac12\Delta u+u\xi$7

and, for finite-valued disorder, enhanced noise sensitivity yields asymptotic independence for smooth functionals of finite collections of such observables. Combined with the diffusive rescaling of the disorder field

$\partial_t u=\frac12\Delta u+u\xi$8

this gives the joint convergence

$\partial_t u=\frac12\Delta u+u\xi$9

in the sense of finite-dimensional distributions (Caravenna et al., 14 Jul 2025).

At the level of the limiting object, this independence is strengthened by the black-noise theorem. The SHF defines a homogeneous continuous product of probability spaces in the sense of Tsirelson, and its first chaos

$2d$0

is trivial: $2d$1 Equivalently, the SHF is a black noise. As a corollary, if $2d$2 denotes the critically tuned mollified $2d$3 SHE and $2d$4 the mollified spacetime white noise, then

$2d$5

and $2d$6 is independent of $2d$7 in the limit (Gu et al., 19 Jun 2025).

A third viewpoint is the martingale one. For a fixed nonnegative compactly supported test function $2d$8, define the projected measure

$2d$9

For $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$0,

$$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$1

is a continuous martingale. Its quadratic variation is given by the renormalized diagonal limit

$$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$2

and the resulting orthogonal martingale measure is defined in the sense of Walsh (Nakashima, 26 Mar 2025). This formulation suggests an SPDE-level interpretation, but only after critical renormalization of the noise intensity.

4. Spatial singularity, regularity, and local mass

The one-time marginals of the critical SHF are singular with respect to Lebesgue measure. For fixed $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$3 and $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$4, almost surely

$$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$5

where $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$6 denotes the normalized indicator of the ball $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$7. At the same time, the field has negative Hölder regularity of every order: $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$8 This implies the absence of atoms. The same work also proves local convergence to zero in the long-time limit: for every bounded set $$\omega=(\omega(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^2}$$9,

$$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$0

(Caravenna et al., 8 Apr 2025).

The small-scale intermittency of these singular measures is quantified by moments of shrinking balls. If

$$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$1

denotes the normalized mass of a shrinking ball, then for every integer $$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$2,

$$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$3

Equivalently,

$$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$4

The exponent $$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$5 is the signature of logarithmic intermittency and reflects the pairwise interaction structure of the critical $$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$6 theory (Liu et al., 2024).

Local positivity and lower-tail control have also been established. For the mass of balls

$$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$7

one has a quantitative upper bound on the lower tail: for every $$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$8, there are constants $$E[\omega(n,x)]=0,\qquad E[\omega(n,x)^2]=1,\qquad \lambda(\beta):=\log E[e^{\beta\omega(n,x)}]<\infty,$$9 and $S=(S_n)_{n\in\mathbb{N}}$0 such that, for all sufficiently large $S=(S_n)_{n\in\mathbb{N}}$1,

$S=(S_n)_{n\in\mathbb{N}}$2

As consequences,

$S=(S_n)_{n\in\mathbb{N}}$3

and, for compactly supported nonnegative initial data $S=(S_n)_{n\in\mathbb{N}}$4,

$S=(S_n)_{n\in\mathbb{N}}$5

These results give partial answers to open questions on the local behavior of the SHF (Nakashima, 24 Jul 2025).

5. Non-Gaussianity, moment growth, and strong disorder

The SHF is not a Gaussian multiplicative chaos (GMC). If one matches its first and second moments with a GMC $S=(S_n)_{n\in\mathbb{N}}$6, then for natural observables such as heat kernels or ball indicators, the third moment of the SHF is strictly larger: $S=(S_n)_{n\in\mathbb{N}}$7 More generally, for every integer $S=(S_n)_{n\in\mathbb{N}}$8 there exists $S=(S_n)_{n\in\mathbb{N}}$9 such that

$\mathbb{Z}^2$0

whereas the matched GMC satisfies asymptotic factorization of the form

$\mathbb{Z}^2$1

This strict moment mismatch rules out any representation of the SHF as the exponential of a generalized Gaussian field (Caravenna et al., 2022).

The intermittency is even more pronounced in high moments at fixed scales. For a smooth nonnegative test function $\mathbb{Z}^2$2 with $\mathbb{Z}^2$3 and

$\mathbb{Z}^2$4

there exists an absolute constant $\mathbb{Z}^2$5 such that, for all sufficiently large integers $\mathbb{Z}^2$6,

$\mathbb{Z}^2$7

This matches the long-standing prediction that the critical $\mathbb{Z}^2$8 theory has double-exponential moment growth with exponent linear in $\mathbb{Z}^2$9. The same work gives two-sided upper-tail bounds: for all sufficiently large $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$00,

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$01

The proof introduces a new connection between SHF moments and the Gaussian Free Field on Feynman diagrams (Ganguly et al., 29 Jul 2025).

A distinct regime appears when the disorder parameter is sent to the strong-disorder or super-critical side. In that regime the SHF vanishes locally with an optimal doubly-exponential decay rate in the disorder intensity. More precisely, for fixed $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$02 and compactly supported $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$03,

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$04

and the corresponding truncated means decay on the scale $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$05. The polymer analogue yields sharp free-energy bounds, identifying the exact exponential scale $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$06 in the small-$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$07 asymptotics (Berger et al., 4 Aug 2025).

6. Continuum polymer measures and current directions

The SHF supports a continuum polymer interpretation. Using a Chapman–Kolmogorov relation and a conditional expectation formula for the flow, one can construct path-space measures $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$08 on $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$09 such that their finite-dimensional distributions are exactly the multi-interval compositions generated by the SHF. The resulting family satisfies a pathwise conditional expectation identity of the form

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$10

and its second moments are governed by an intersection-time functional $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$11 on path pairs. Across disorder strengths, the second moments satisfy the Radon–Nikodym relation

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$12

which is consistent with a conditional Gaussian multiplicative chaos interrelationship at the level of second moments (Clark et al., 2024).

Several open directions are already sharply formulated in the literature. One is to extend enhanced noise sensitivity beyond finite-valued environments; this is explicitly identified as a natural open question in the general theory (Caravenna et al., 14 Jul 2025). Another is the renormalized $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$13 KPZ problem suggested by the small-ball analysis: the conjectured limit of centered and rescaled fields of the form

$$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$14

remains open (Nakashima, 24 Jul 2025). A further problem is to sharpen the known upper bounds on high moments to match the lower bound $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$15 with a corresponding upper bound of the same form (Ganguly et al., 29 Jul 2025). On the structural side, the martingale problem associated with the SHF is not well posed as currently formulated, and identifying a complete characterization that yields uniqueness in law remains open (Nakashima, 26 Mar 2025).

Taken together, these results place the critical $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$16 SHF in a narrow and distinctive class of scaling limits: universal but non-Gaussian, measure-valued but atomless, singular with respect to Lebesgue measure yet in $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$17 for every $$\mathcal{Z}^{\theta}=\big(\mathcal{Z}_{s,t}^{\theta}(dx,dy)\big)_{0\le s<t<\infty},$$18, and independent of the white noise from which it originates in the scaling limit.