<p>The asymptotics of the solution of the Gell-Mann–Low equation and their corrections are studied in terms of discrete-time Markov chains of random operators derived from systems of differential equations on specific operator algebras of stretched graphons. Thanks to the interconnection between the Gell-Mann–Low renormalization group equation, underlying momentum cutoff regularization, and the Connes–Kreimer renormalization group equation, underlying dimensional regularization, the beta function of a theory is discussed in the topological Hopf algebra of renormalization. The asymptotics of the beta function and their corrections are studied in terms of discrete-time Markov chains of random operators. This new mathematical setting is applied for (i) the study of the triviality of QED, and (ii) the computation of first-order corrections associated with the asymptotic of the sum of a perturbative series of the beta function of QCD when running coupling constants become strong enough to generate non-perturbative effects.</p>