Recent algorithmic improvements have made it possible to evaluate subdivergence-free (=prim-
itive=skeleton) Feynman integrals in $\phi^4$ theory numerically up to 18 loops. By now, all such
integrals up to 13 loops and several hundred thousand at higher loop order have been computed.
This data enables a statistical analysis of the typical behaviour of Feynman integrals at large
loop order. We find that the average value grows exponentially, but the observed growth rate is
accurately described by its leading asymptotics only upwards of 25 loops. This is in contrast with
the 𝑁-dependence of the 𝑂 (𝑁)-symmetric 𝜙4 theory, which is close to its large-order asymptotics
already around 10 loops.
Secondly, the distribution of integrals has a largely continuous inner part but a few extreme
outliers. This makes uniform random sampling inefficient. We find that the value of the integral
is correlated with many features of the graph, which can be used for importance sampling. With
a naive test implementation we obtained an approximately 1000-fold speedup compared with
uniform sampling. This suggests that in future work, Feynman amplitudes at large loop order
might be computed numerically with statistical methods, rather than through enumerating and
evaluating every individual integral.
The talk is based on the articles [1], [2], [3].