An upper bound of the lower tail of the mass of balls under the critical $2d$ stochastic heat flow

TL;DR

This paper establishes an upper bound on the lower tail of the mass of balls under the critical 2D stochastic heat flow, proving its strict positivity and integrability of its logarithm.

Contribution

It provides the first upper bound on the lower tail of the mass of balls in the critical 2D stochastic heat flow, addressing open questions about its local behavior.

Findings

Proved an upper bound on the lower tail of the mass of balls.

Established the strict positivity of the mass.

Demonstrated the integrability of the logarithm of the mass.

Abstract

We study the critical two-dimensional stochastic heat flow $\mathscr{Z}_t^{\vartheta}$, recently constructed as the scaling limit of directed polymers in a random environment and as the weak limit of the solution to a mollified stochastic heat equation. Focusing on the mass of balls $\mathscr{Z}_t^{\vartheta}(B_r(0),B_r(a))$ ($a\in \mathbb{R}^2$, $r>0$), we establish an upper bound on its lower tail. As a consequence, we prove the integrability of the logarithm of $\mathscr{Z}_t^{\vartheta}(B_r(0),B_r(a))$ and its strict positivity. These results provide partial answers to open questions concerning the local behavior of $\mathscr{Z}_t^\vartheta$.