Strong disorder for Stochastic Heat Flow and 2D Directed Polymers

TL;DR

This work analyzes the critical two-dimensional Stochastic Heat Flow (SHF) and its connection to 2D directed polymers, establishing quantitative extinction in strong disorder and large time regimes. A novel coarse-grained chaos proxy, together with size-biasing and scale-change arguments, enables sharp exponential-in-time and doubly-exponential-in-disorder bounds for the SHF’s one-time marginal, and translates these results into precise free-energy estimates for 2D directed polymers. The results extend to the supercritical stochastic heat equation under space-time discretization, revealing a superdiffusive scale at which nontrivial fluctuations emerge. The combination of coarse-graining, proxy construction, and finite-volume criteria provides a robust framework applicable to other singular SPDE and polymer-measure contexts. Overall, the paper delivers sharp extinction/mass-transition analyses and sharpened free-energy bounds in the critical and supercritical regimes in dimension two.

Abstract

The critical 2D Stochastic Heat Flow (SHF) is a universal measure-valued process providing a notion of solution to the ill-defined 2D stochastic heat equation. We investigate the SHF in the regime of large time and large disorder strength, proving a sharp form of local extinction: we identify the rate at which the distribution collapses to zero. We also identify the spatial scale governing the transition from vanishing to diverging mass, and from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, which yield precise free energy estimates. Our proof refines classical change of measure and coarse-graining techniques, introducing new ideas of independent interest. Our findings provide novel insight into the 2D stochastic heat equation regularized via space-time discretization: for any regime of supercritical disorder strength $β$, including the case where $β> 0$ is kept fixed, the solution exhibits fluctuations on a superdiffusive scale.

Figures (2)

• Figure 1: Illustration of an "interaction diagram". Pairwise interactions are grouped in stretches of space-time points belonging to the same set $C_{ij}$, i.e. with the same label $\mathsf{d}=ij$. A labeled diagram corresponds to a collections of stretches, where each stretch has a label $\mathsf{d}_p$, a size (cardinality) $k_p$ and ordered starting and ending points $(a_p,x_p)$, $(b_p,y_p)$. In the above diagram, there are $\ell=5$ stretches.
• Figure 2: Illustration of the renewal interpretation of the formula \ref{['def:Jell']} for $J_{\ell}$. The first points of $\tau,\tau'$ are chosen uniformly in $\llbracket 1,3\tilde{N}\rrbracket$ and they the two renewals have inter-arrival distribution $K(m)$ defined above. The stretches alternate between $\tau$ and $\tau'$ and have starting and ending point denoted by $a_p$ (or $\sigma_p$) and $b_p$, in reference to the interaction diagrams (see \ref{['fig:diagram']}). The number of alternating stretches is denoted $\mathcal{L}_{3\tilde{N}}(\tau,\tau')$: in the above picture we have $\mathcal{L}_{3\tilde{N}}(\tau,\tau') \geq \ell= 5$ and the two renewals $\tau\cap \tau'$ do not intersect before the beginning $\sigma_5$ of the $5^{\rm th}$ stretch (but they might intersect afterwards).

Theorems & Definitions (75)

• Theorem 1.1: Strong disorder and large time for the SHF
• Remark 1: Lower bounds and second moment
• Remark 2: Truncated vs. fractional moments
• Remark 3: Scaling covariance, strong disorder and large time
• Theorem 1.2: Transition for the SHF mass in large balls
• Conjecture 1
• Theorem 1.3: Supercritical rescaling of SHF
• Conjecture 2
• Remark 4
• Theorem 2.1: Strong disorder for 2D directed polymers