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

We study the critical two-dimensional stochastic heat flow ${\mathcal{Z}}_t^{\vartheta }$, 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 ${\mathcal{Z}}_t^{\vartheta }(B_r(0),B_r(a))$ ($a\in {\mathbb{R}}^2$, $r>0$), we establish an upper bound on its lower tail. As a consequence, we prove the integrability of the logarithm of ${\mathcal{Z}}_t^{\vartheta }(B_r(0),B_r(a))$ and its strict positivity. These results provide partial answers to open questions concerning the local behavior of ${\mathcal{Z}}_t^{\vartheta }$.