The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos

The critical $2d$ Stochastic Heat Flow (SHF) is a stochastic process of random measures on ${\mathbb{R}}^2$, recently constructed in [CSZ23]. We show that this process falls outside the class of Gaussian Multiplicative Chaos (GMC), in the sense that it cannot be realised as the exponential of a (generalised) Gaussian field. We achieve this by deriving strict lower bounds on the moments of the SHF that are of independent interest.