New theoretical #physics preprint https://arxiv.org/abs/2412.08617
We looked at the asymptotic growth rate of the beta function in #quantumFieldTheory , and the relative importance of subdivergence-free #Feynmangraph s. These graphs correspond to integrals, and the size of the graph is measured by its loop number, which also indicates how hard it is to solve the integral. State of the art computations in realistic theories are anywhere between 1 and 6 loops. The asymptotics of the perturbation series is known from instanton calculations. We now showed (in a model theory), that the leading asymptotics describes the true growth rate only for more than 25 loops, way beyond anything that can realistically be computed.

This is good news: It tells us that asymptotic instanton calculations provide non-trivial additional information that can not be trivially inferred from low-order perturbation theory.
In the plot, the red dots are numerical data points for the subdivergence-free graphs in phi^4 theory up to 18 loops, the green lines are the leading instanton asymptotics.

My article together with Kimia Shaban has appeared in JHEP today. We have examined the #statistics of #Feynmangraph s in #QFT , and how they can be exploited to efficiently compute #amplitudes at high loop order.
The article is open access, and the dataset is freely available from my website if you want to explore statistics and correlations yourself. Predicting the values of these Feynman integrals could also be interesting as a test case for #machinelearning
https://link.springer.com/article/10.1007/JHEP11(2024)038

I recently discovered an excellent #math article by Krajewski and Martinetti that I had overlooked so far: https://arxiv.org/abs/0806.4309
Basically, renormalization of #Feynmangraph s in #qft is organized in terms of rooted trees, and so are solutions to differential equations, concatenation of differential operators, numerical integration schemes, and the Hopf algebra of power series. It is intuitively clear that all these things must be closely related, but I wasn't aware that there is this article where the relations are actually spelled out in detail. They also include a derivation of Wigner's semicircle law for Gaussian random matrices in the rooted-tree formalism, which is something I didn't think about at all. Learned another unexpected connection .

Here is a curious finding from our statistical analysis https://arxiv.org/abs/2403.16217 :
A #Feynmangraph is a graphical short hand notation for a complicated integral that computes the probability for scattering processes in #quantum field theory.
An electrical circuit can also be described as a graph. What happens if we interpret the Feynman graph as an #electrical network, where each edge is a 1 Ohm resistor? We can then compute the resistance between any pair of vertices and collect all these values in a "resistance matrix", as shown below. The average of all these resistances is called "Kirchhoff index". Now it turns out that this average resistance is correlated fairly strongly with the Feynman integral of that graph: A graph with large contribution to quantum scattering amplitudes on average also has a large electrical resistance. Isn't that a nice connection between two seemingly distinct branches of theoretical #physics ?

The #Feynmangraph s that contribute to scattering amplitudes in #quantum field theory come in all shapes and sizes. Can one guess how much their Feynman integral will contribute just from looking at them ? It turns out one can! Let's consider subdivergence-free graphs at 12 loops. The pictures show the two largest and the two smallest contributors. All graphs have the same number of edges and vertices. The graphs that contribute strongly look "larger", and the small contributors look "more dense", but drawings are of course arbitrary. A closer examination shows that "symmetry", as measured by graph automorphisms, is not clearly related to the value of the Feynman integral.