Feynman integrals at large loop order and the $\log$-$\mathrm{\Gamma }$ distribution

We find empirically that the value of Feynman integrals follows a $\log$-$\mathrm{\Gamma }$ distribution at large loop order. This result opens up a new avenue towards the large-order behavior in perturbative quantum field theory. Our study of the primitive contribution to the scalar ${\phi }^4$ beta function in four dimensions up to 17 loops provides accompanying evidence. Guided by instanton considerations, we discuss the extrapolation of this contribution to all loop orders.